Springer Handbook of Experimental Solid Mechanics

Springer Handbook of Experimental Solid Mechanics

Springer Handbooks provide a concise compilation of approved key information on methods of research, general principles, and functional relationships in physical sciences and engineering. The world’s leading experts in the fields of physics and engineering will be assigned by one or several renowned editors to write the chapters comprising each volume. The content is selected by these experts from Springer sources (books, journals, online content) and other systematic and approved recent publications of physical and technical information. The volumes are designed to be useful as readable desk reference books to give a fast and comprehensive overview and easy retrieval of essential reliable key information, including tables, graphs, and bibliographies. References to extensive sources are provided.

Springer

Handbook of Experimental Solid Mechanics Sharpe (Ed.) With DVD-ROM, 874 Figures, 58 in four color and 50 Tables

123

Editor: Professor William N. Sharpe, Jr. Department of Mechanical Engineering Room 126, Latrobe Hall The Johns Hopkins University 3400 North Charles Street Baltimore, MD 21218-2681, USA [email protected]

Library of Congress Control Number:

ISBN: 978-0-387-26883-5

2008920731

e-ISBN: 978-0-387-30877-7

c 2008, Springer Science+Business Media, LLC New York All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC New York, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. The use of designations, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product liability: The publisher cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Production and typesetting: le-tex publishing services oHG, Leipzig Senior Manager Springer Handbook: Dr. W. Skolaut, Heidelberg Typography and layout: schreiberVIS, Seeheim Illustrations: Hippmann GbR, Schwarzenbruck Cover design: eStudio Calamar Steinen, Barcelona Cover production: WMXDesign GmbH, Heidelberg Printing and binding: Stürtz GmbH, Würzburg Printed on acid free paper SPIN 11510079

9085/3820/YL

543210

V

Preface to Handbook on Experimental Stress Analysis

Experimental science does not receive Truth from superior Sciences: she is the Mistress and the other sciences are her servants ROGER BACON: Opus Tertium. Stress analysis has been regarded for some time as a distinct professional branch of engineering, the object of which is the determination and improvement of the mechanical strength of structures and machines. Experimental stress analysis strives to achieve these aims by experimental means. In doing so it does not remain, however, a mere counterpart of theoretical methods of stress analysis but encompasses those, utilizing all the conclusions reached by theoretical considerations, and goes far beyond them in maintaining direct contact with the true physical characteristics of the problems under considerations. Many factors make the experimental approach indispensable, and often the only means of access, in the investigation of problems of mechanical strength. At our present state of knowledge it is remarkable how quickly we can reach the limit of applicability of mathematical methods of stress analysis, and there is a multitude of comparatively simple, and in practice frequently occurring, stress problems for which no theoretical solutions have yet been obtained. In addition to this, theoretical considerations are usually based on simplifying assumptions which imply certain detachment from reality, and it can be decided only by experimentation whether such idealization has not resulted in an undue distortion of the essential features of the problem. No such doubt needs to enter experimental stress analysis, especially if it is done under actual service conditions, where all the factors due to the properties of the employed materials, the methods of manufacture, and the conditions of operation are fully represented. The advantage of the experimental approach becomes especially obvious if we consider that it is possible to determine experimentally the stress distribution in a machine part in actual operation without knowing the nature of the forces acting on the part under these circumstances, which proposition is clearly inaccessible to any theoretical method of analysis. To these major advantages we may add one more, from the point of view of the average practicing engineer, whose mathematical preparation is not likely to enable him to

deal theoretically with some of the complex strength problems which he, nevertheless, is expected to settle satisfactorily. To these men experimental methods constitute a recourse that is more readily accessible and that, with proper care and perseverance, is most likely to furnish the needed information. Several principal methods and literally hundreds of individual tools and artifices constitute the “arsenal” of the experimental stress analyst. It is interesting to observe, however, that each of these devices, no matter how peculiar it sometimes appears to be, has its characteristic feature and, with it, some unique advantage that may render this tool most suitable for the investigation of a particular problem. The stress analyst cannot afford, therefore, to ignore any of these possibilities. This circumstance, together with the ever-increasing demand on mechanical strength, will always tend to keep experimental stress analysis a distinct entity in the field of technical sciences. There has been a long-felt need of a comprehensive reference book of this nature, but, at the same time, it was recognized that no one person could possibly write with authority on all the major experimental procedures that are being used at present in the investigation of mechanical strength. It was proposed therefore that the problem could be solved only by a concerted effort which might be initiated most suitably under the aegis of the Society for Experimental Stress Analysis, and the writer was appointed as editor with complete freedom to proceed with the organization of this undertaking. Invitations were sent to thirty eminent engineers and scientists who were best known for their outstanding contributions in one or more of the specific branches of experimental stress analysis. It was most impressive to witness the readiness and understanding with which these men, many of them not even associated with the Society, responded to the request and joined the editor in contributing their work, without remuneration, to the furtherance of the aims of the Society, which thus becomes the sole recipient of all royalties from this publication. This being the first comprehensive publication in its field, it may be of general interest to say a few words about the method used in the planning and coordina-

VI

tion of the material. In inviting the contributors, I first briefly out-lined the subject to be covered requesting, in return, from each author a more detailed outline of what he would propose on his respective subject. These authors’ outlines were subsequently collected in a booklet, a copy of which was sent to each participant, thus informing him in advance of projected contents of all the other parts of the book. This scheme proved of considerable help in assuring adequate coverage of all matters of interest, without undue overlaps, repetition, or need of frequent cross references. In the final plan, as seen in the table of contents, the main body of the book was divided into 18 chapters, each dealing with either a principal method, from mechanical gages to x-ray analysis, or a major topic of interest, such as residual stresses, interpretation of service fractures, or analogies. In addition to these, an appendix was devoted to the discussion of three theoretical subjects which are of fundamental importance in the planning and interpretation of experimental stress work. In the final outcome, not only the book as a whole but also most of the individual chap-

ters turned out to be pioneering ventures in their own rights, often constituting the first systematic exposition of their respective subject matter. Another innovation was undertaken in the treatment of bibliographical references, where an effort was made to review briefly the contents of each entry, since it was found that the mere titles of technical articles seldom convey a satisfactory picture of their respective contents. Despite all precautions the book is bound to have errors and shortcomings, and it is the sincere hope of the editor that users of the book will not hesitate to inform him of possibilities of improvement which may be incorporated in a later edition. In the course of this work the editor was greatly aided by advice from numerous friends and colleagues, among whom he wishes to acknowledge in particular the invaluable help received from B. F. Langer, R. D. Mindlin, W. M. Murray, R. E. Peterson, and G. Pickett. Evanston, Illinois April 1950

M. Hetenyi

VII

Preface to the Handbook on Experimental Mechanics, First Edition

The Handbook on Experimental Stress Analysis, which was published under the aegis of the Society for Experimental Stress Analysis in 1950, has been the comprehensive and authoritative reference in our field for more than thirty years. Under the able editorship of the late M. Herenyi, 31 authors contributed without compensation 18 chapters and 3 appendices to this handbook. It received international acclaim and brought considerable income to the Society for Experimental Mechanics. Since 1950, new experimental techniques, such as holography, laser speckle interferometry, geometric moire, moire interferometry, optical heterodyning, and modal analysis, have emerged as practical tools in the broader field of experimental mechanics. The emergence of new materials and new disciplines, such as composite materials and fracture mechanics, resulted in the evolution of traditional experimental techniques to new fields such as orthotropic photoelasticity and experimental fracture mechanics. These new developments, together with the explosive uses of on/off-line computers for rapid data processing and the combined use of experimental and numerical techniques, have expanded the capabilities of experimental mechanics far beyond those of the 1950s. Sensing the need to update the handbook, H. F. Brinson initiated the lengthy process of revising the

handbook during his 1978-79 presidency of the Society. Since M. Hetenyi could not undertake the contemplated revision at that time, the decision was made to publish a new handbook under a new editor. Opinions ranging from topical coverage to potential contributors were solicited from various SEM members, and after a short respite I was chosen as editor by the ad hoc Handbook Committee chaired by J. B. Ligon. Despite the enormous responsibility, our task was made easier by inheriting the legacy of the Herenyi Handbook and the numerous suggestions that were collected by H. Brinson. The new handbook, appropriately entitled Handbook on Experimental Mechanics, is dedicated to Dr. Hetenyi. Twenty-five authors have contributed 21 chapters that include, among others, the new disciplines and developments that are mentioned above. The handbook emphasizes the principles of the experimental techniques and de-emphasizes the procedures that evolve with time. I am grateful to the contributors, who devoted many late afterhours in order to meet the manuscript deadlines and to J. B. Ligon who readily provided welcomed assistance during the trying times associated with this editorship.

Albert S. Kobayashi 1987

VIII

Preface to the Handbook on Experimental Mechanics, Second Edition

Since the publication of the first edition, considerable progress has been made in automated image processing, greatly reducing the heretofore laborious task of evaluating photoelastic and moire fringe patterns. It is therefore appropriate to add Chapter 21: “Digital Image Processing” before the final chapter, “Statistical Analysis of Experimental Data.” Apart from the new chapter, this second edition is essentially same as the first edition with minor corrections and updating. Exceptions to this are the addition of a section on optical fiber sensors in Chapter 2: “Strain Gages,” and extensive additions to

Chapter 14, which is retitled “Thermal Stress Analysis,” and to Chapter 16: “Experimental Modal Analysis.” To reiterate, the purpose of this handbook is to document the principles involved in experimental mechanics rather than the procedures and hardware, which evolve over time. To that extent, we, the twenty-seven authors, judging from the many appreciative comments which were received upon the publication of the first edition, have succeeded. Albert S. Kobayashi April 1993

IX

Preface

This handbook is a revision and expansion of the Handbook on Experimental Mechanics published by the Society for Experimental Mechanics in 1987 with a second edition in 1993 – both edited by Albert Kobayashi. All three of these trace a direct lineage to the seminal Handbook of Experimental Stress Analysis conceived and edited by Miklós Hetényi in 1950 and they encapsulate the history of the field. In 1950, the capability of measuring strains on models and structures was just becoming widely available. Engineers were still making their own wire resistance foil gages, and photoelasticity measurements required film processing. Conversion of these measurements to stresses relied on slide rules and graph paper. Now, foil resistance gages are combined with automatic data acquisition, and photoelasticity is just as automated. Input from both experimental methods is combined with finite element analysis to present stress variations in color on a computer screen. The focus then was on large structures such as airframes; in fact, the efforts of the Society for Experimental Stress Analysis (founded in 1938) were crucial to the rapid development of aircraft in the 1940s. While measurements on large structures continue to be important, researchers today also measure the mechanical properties of specimens smaller than a human hair. The field is completely different now. Experimental techniques and applications have expanded (or contracted if you prefer) from stress analysis of large structures to include the electromechanical analysis of micron-sized sensors and actuators. Those changes – occurring gradually over the early years but now more rapidly – led to a change in the society name to the Society for Experimental Mechanics. Those changes also have led to the expansion of the current volume with the deletion of some topics and the addition of others

in order to address these emerging topics in the “micro world”. This volume presents experimental solid mechanics as it is practiced in the early part of the 21st century. It is a field that is important as a technology and rich in research opportunities. A striking feature of this handbook is that 20 of the 36 chapters are on topics that have arisen or matured in the 15 years since the last edition; and, in most cases, these have been written by relatively young researchers and practitioners. Consider microelectromechanical systems (MEMS), for example. That technology, originated by electrical engineers only 25 years ago, now permeates our lives. It was soon learned that designers and manufacturers needed better understanding of the mechanical properties of the new materials involved, and experimental mechanists became involved only 15 years ago. That is just one example; several of the chapters speak to it as well as similar completely new topics. The reader will find in this volume not only information on the traditional areas of experimental solid mechanics, but on new and emerging topics as well. This revision was initiated by the Executive Board of the Society and managed by the very capable staff at Springer, in particular Elaine Tham, Werner Skolaut, and Lauren Danahy. Sound advice was provided over the course of the effort by Jim Dally and Tom Proulx. However, the real work was done by the authors. Each chapter was written by authors, who are not only experts, but who volunteered to contribute to this Handbook. Although they are thoroughly familiar with the technical details, it still required a major effort on their part to prepare a chapter. On behalf of the Society and Springer, I acknowledge and thank them. Baltimore June 2008

William N. Sharpe, Jr.

XI

About the Editor

William N. Sharpe, Jr. holds the Alonzo G. Decker Chair in Mechanical Engineering at the Johns Hopkins University where he served as department chairman for 11-1/2 years. His early research focused on laser-based strain measurement over very short gage lengths to study plasticity and fracture behavior. The past fifteen years have been devoted to development and use of experimental methods to measure the mechanical properties of materials used in microelectromechanical systems. He is a Fellow and Past President of the Society for Experimental Mechanics from which he received the Murray Medal. The American Society of Mechanical Engineers awarded him the Nadai Award as well as Fellow grade. He has received an Alexander von Humboldt Award along with the Roe Award from the American Society of Engineering Education.

XIII

List of Authors Jonathan D. Almer Argonne National Laboratory 9700 South Cass Avenue Argonne, IL 60439, USA e-mail: [email protected] Archie A.T. Andonian The Goodyear Tire & Rubber Co. D/410F 142 Goodyear Boulevard Akron, OH 44305, USA e-mail: [email protected] Satya N. Atluri University of California Department of Mechanical & Aerospace Engineering, Center for Aeorspace Research & Education 5251 California Avenue, Suite 140 Irvine, CA 92612, USA David F. Bahr Washington State University Mechanical and Materials Engineering Pullman, WA 99164, USA e-mail: [email protected] Chris S. Baldwin Aither Engineering, Inc. 4865 Walden Lane Lanham, MD 20706, USA e-mail: [email protected] Stephen M. Belkoff Johns Hopkins University International Center for Orthopaedic Advancement, Department of Orthopaedic Surgery, Bayview Medical Center 5210 Eastern Avenue Baltimore, MD 21224, USA e-mail: [email protected] Hugh Bruck University of Maryland Department of Mechanical Engineering College Park, MD 20742, USA e-mail: [email protected]

Ioannis Chasiotis University of Illinois at Urbana-Champaign Aerospace Engineering Talbot Lab, 104 South Wright Street Urbana, IL 61801, USA e-mail: [email protected] Gary Cloud Michigan State University Mechanical Engineering Department East Lansing, MI 48824, USA e-mail: [email protected] Wendy C. Crone University of Wisconsin Department of Engineering Physics 1500 Engineering Drive Madison, WI 53706, USA e-mail: [email protected] James W. Dally University of Maryland 5713 Glen Cove Drive Knoxville, TN 37919, USA e-mail: [email protected] James F. Doyle Purdue University School of Aeronautics & Astronautics West Lafayette, IN 47907, USA e-mail: [email protected] Igor Emri University of Ljubljana Center for Experimental Mechanics Cesta na brdo 85 Lubljana, SI-1125, Slovenia e-mail: [email protected] Yimin Gan Universität GH Kassel Fachbereich 15 – Maschinenbau Mönchebergstr. 7 34109 Kassel, Germany e-mail: [email protected]

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List of Authors

Ashok Kumar Ghosh New Mexico Tech Mechanical Engineering and Civil Engineering Socorro, NM 87801, USA e-mail: [email protected]

Peter G. Ifju University of Florida Mechanical and Aerospace Engineering Gainesville, FL 32611, USA e-mail: [email protected]

Richard J. Greene The University of Sheffield Department of Mechanical Engineering Mappin Street Sheffield, S1 3JD, UK e-mail: [email protected]

Wolfgang G. Knauss California Institute of Technology-GALCIT 105-50 1201 East California Boulevard Pasadena, CA 91125, USA e-mail: [email protected]

Bongtae Han University of Maryland Mechanical Engineering Department College Park, MD 20742, USA e-mail: [email protected]

Albert S. Kobayashi University of Washington Department of Mechanical Engineering Seattle, Washington 98195-2600, USA e-mail: [email protected]

M. Amanul Haque Pennsylvania State University Department of Mechanical Engineering 317A Leonhard Building University Park, PA 16802, USA e-mail: [email protected]

Sridhar Krishnaswamy Northwestern University Center for Quality Engineering & Failure Prevention Evanston, IL 60208-3020, USA e-mail: [email protected]

Craig S. Hartley El Arroyo Enterprises LLC 231 Arroyo Sienna Drive Sedona, AZ 86332, USA e-mail: [email protected] Roger C. Haut Michigan State University College of Osteopathic Medicine, Orthopaedic Biomechanics Laboratories A407 East Fee Hall East Lansing, MI 48824, USA e-mail: [email protected] Jay D. Humphrey Texas A&M University Department of Biomedical Engineering 335L Zachry Engineering Center, 3120 TAMU College Station, TX 77843-3120, USA e-mail: [email protected]

Yuri F. Kudryavtsev Integrity Testing Laboratory Inc. PreStress Engineering Division 80 Esna Park Drive Markham, Ontario L3R 2R7, Canada e-mail: [email protected] Pradeep Lall Auburn University Department of Mechanical Engineering Center for Advanced Vehicle Electronics 201 Ross Hall Auburn, AL 36849-5341, USA e-mail: [email protected] Kenneth M. Liechti University of Texas Aerospace Engineering and Engineering Mechanics Austin, TX 78712, USA e-mail: [email protected]

List of Authors

Hongbing Lu Oklahoma State University School of Mechanical and Aerospace Engineering 218 Engineering North Stillwater, OK 74078, USA e-mail: [email protected] Ian McEnteggart Instron Coronation Road, High Wycombe Buckinghamshire, HP12 3SY, UK Dylan J. Morris National Institute of Standards and Technology Materials Science and Engineering Laboratory Gaithersburg, MD 20877, USA e-mail: [email protected] Sia Nemat-Nasser University of California Department of Mechanical and Aerospace Engineering 9500 Gilman Drive La Jolla, CA 92093-0416, USA e-mail: [email protected] Wolfgang Osten Universität Stuttgart Institut für Technische Optik Pfaffenwaldring 9 70569 Stuttgart, Germany e-mail: [email protected] Eann A. Patterson Michigan State University Department of Mechanical Engineering 2555 Engineering Building East Lansing, MI 48824-1226, USA e-mail: [email protected] Daniel Post Virginia Polytechnic Institute and State University (Virginia Tech) Department of Engineering Science and Mechanics Blacksburg, VA 24061, USA e-mail: [email protected]

Ryszard J. Pryputniewicz Worcester Polytechnic Institute NEST – NanoEngineering, Science, and Technology CHSLT – Center for Holographic Studies and Laser Micro-Mechatronics Worcester, MA 01609, USA e-mail: [email protected] Kaliat T. Ramesh Johns Hopkins University Department of Mechanical Engineering 3400 North Charles Street Baltimore, MD 21218, USA e-mail: [email protected] Krishnamurthi Ramesh Indian Institute of Technology Madras Department of Applied Mechanics Chennai, 600 036, India e-mail: [email protected] Krishnaswamy Ravi-Chandar University of Texas at Austin 1 University Station, C0600 Austin, TX 78712-0235, USA e-mail: [email protected] Guruswami Ravichandran California Institute of Technology Graduate Aeronautical Laboratories Pasadena, CA 91125, USA e-mail: [email protected] Robert E. Rowlands University of Wisconsin Department of Mechanical Engineering 1415 Engineering Drive Madison, WI 53706, USA e-mail: [email protected] Taher Saif University of Illinois at Urbana-Champaign Micro and Nanotechnology Laboratory, 2101D Mechanical Engineering Laboratory 1206 West Green Street Urbana, IL 61801, USA e-mail: [email protected] Wolfgang Steinchen (deceased)

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Jeffrey C. Suhling Auburn University Department of Mechanical Engineering 201 Ross Hall Auburn, AL 36849-5341, USA e-mail: [email protected] Michael A. Sutton University of South Carolina Center for Mechanics, Materials and NDE Department of Mechanical Engineering 300 South Main Street Columbia, SC 29208, USA e-mail: [email protected]

Robert B. Watson Vishay Micro-Measurements Sensors Engineering Department Raleigh, NC, USA e-mail: [email protected] Robert A. Winholtz University of Missouri Department of Mechanical and Aerospace Engineering E 3410 Lafferre Hall Columbia, MO 65211, USA e-mail: [email protected]

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Contents

List of Abbreviations ................................................................................. XXVII

Part A Solid Mechanics Topics 1 Analytical Mechanics of Solids Albert S. Kobayashi, Satya N. Atluri .......................................................... 1.1 Elementary Theories of Material Responses ..................................... 1.2 Boundary Value Problems in Elasticity ............................................ 1.3 Summary ...................................................................................... References ..............................................................................................

3 4 11 14 14

2 Materials Science for the Experimental Mechanist Craig S. Hartley ........................................................................................ 2.1 Structure of Materials .................................................................... 2.2 Properties of Materials................................................................... References ..............................................................................................

17 17 33 47

3 Mechanics of Polymers: Viscoelasticity Wolfgang G. Knauss, Igor Emri, Hongbing Lu ............................................ 3.1 Historical Background.................................................................... 3.2 Linear Viscoelasticity ..................................................................... 3.3 Measurements and Methods .......................................................... 3.4 Nonlinearly Viscoelastic Material Characterization ........................... 3.5 Closing Remarks ............................................................................ 3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known ... References ..............................................................................................

49 49 51 69 84 89 90 92

4 Composite Materials Peter G. Ifju ............................................................................................. 4.1 Strain Gage Applications ................................................................ 4.2 Material Property Testing ............................................................... 4.3 Micromechanics ............................................................................ 4.4 Interlaminar Testing ...................................................................... 4.5 Textile Composite Materials............................................................ 4.6 Residual Stresses in Composites ..................................................... 4.7 Future Challenges.......................................................................... References ..............................................................................................

97 98 102 107 111 114 117 121 121

5 Fracture Mechanics Krishnaswamy Ravi-Chandar ................................................................... 5.1 Fracture Mechanics Based on Energy Balance ................................. 5.2 Linearly Elastic Fracture Mechanics .................................................

125 126 128

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Contents

5.3 Elastic–Plastic Fracture Mechanics .................................................. 5.4 Dynamic Fracture Mechanics .......................................................... 5.5 Subcritical Crack Growth ................................................................ 5.6 Experimental Methods................................................................... References ..............................................................................................

132 137 140 140 156

6 Active Materials Guruswami Ravichandran ........................................................................ 6.1 Background .................................................................................. 6.2 Piezoelectrics ................................................................................ 6.3 Ferroelectrics ................................................................................ 6.4 Ferromagnets ................................................................................ References ..............................................................................................

159 159 161 162 166 167

7 Biological Soft Tissues Jay D. Humphrey ..................................................................................... 7.1 Constitutive Formulations – Overview ............................................ 7.2 Traditional Constitutive Relations ................................................... 7.3 Growth and Remodeling – A New Frontier ...................................... 7.4 Closure ......................................................................................... 7.5 Further Reading ............................................................................ References ..............................................................................................

169 171 172 178 182 182 183

8 Electrochemomechanics of Ionic Polymer–Metal Composites Sia Nemat-Nasser .................................................................................... 8.1 Microstructure and Actuation ......................................................... 8.2 Stiffness Versus Solvation............................................................... 8.3 Voltage-Induced Cation Distribution .............................................. 8.4 Nanomechanics of Actuation ......................................................... 8.5 Experimental Verification .............................................................. 8.6 Potential Applications ................................................................... References ..............................................................................................

187 188 191 193 195 197 199 199

9 A Brief Introduction to MEMS and NEMS Wendy C. Crone ........................................................................................ 9.1 Background .................................................................................. 9.2 MEMS/NEMS Fabrication ................................................................. 9.3 Common MEMS/NEMS Materials and Their Properties ....................... 9.4 Bulk Micromachining versus Surface Micromachining ...................... 9.5 Wafer Bonding .............................................................................. 9.6 Soft Fabrication Techniques ........................................................... 9.7 Experimental Mechanics Applied to MEMS/NEMS.............................. 9.8 The Influence of Scale.................................................................... 9.9 Mechanics Issues in MEMS/NEMS ..................................................... 9.10 Conclusion .................................................................................... References ..............................................................................................

203 203 206 206 213 214 215 217 217 221 224 225

Contents

10 Hybrid Methods James F. Doyle ......................................................................................... 10.1 Basic Theory of Inverse Methods .................................................... 10.2 Parameter Identification Problems ................................................. 10.3 Force Identification Problems ........................................................ 10.4 Some Nonlinear Force Identification Problems ................................ 10.5 Discussion of Parameterizing the Unknowns ................................... References ..............................................................................................

229 231 235 240 246 255 257

11 Statistical Analysis of Experimental Data James W. Dally ........................................................................................ 11.1 Characterizing Statistical Distributions ............................................ 11.2 Statistical Distribution Functions .................................................... 11.3 Confidence Intervals for Predictions ............................................... 11.4 Comparison of Means .................................................................... 11.5 Statistical Safety Factor .................................................................. 11.6 Statistical Conditioning of Data ...................................................... 11.7 Regression Analysis ....................................................................... 11.8 Chi-Square Testing ........................................................................ 11.9 Error Propagation .......................................................................... References ..............................................................................................

259 260 263 267 270 271 272 272 277 278 279

Part B Contact Methods 12 Bonded Electrical Resistance Strain Gages Robert B. Watson ..................................................................................... 12.1 Standardized Strain-Gage Test Methods ......................................... 12.2 Strain and Its Measurement ........................................................... 12.3 Strain-Gage Circuits....................................................................... 12.4 The Bonded Foil Strain Gage .......................................................... 12.5 Semiconductor Strain Gages ........................................................... References ..............................................................................................

283 284 284 285 291 325 332

13 Extensometers Ian McEnteggart ...................................................................................... 13.1 General Characteristics of Extensometers ........................................ 13.2 Transducer Types and Signal Conditioning ...................................... 13.3 Ambient-Temperature Contacting Extensometers............................ 13.4 High-Temperature Contacting Extensometers ................................. 13.5 Noncontact Extensometers............................................................. 13.6 Contacting versus Noncontacting Extensometers ............................. 13.7 Conclusions ................................................................................... References ..............................................................................................

335 336 337 338 341 343 345 346 346

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Contents

14 Optical Fiber Strain Gages Chris S. Baldwin ...................................................................................... 14.1 Optical Fiber Basics........................................................................ 14.2 General Fiber Optic Sensing Systems ............................................... 14.3 Interferometry .............................................................................. 14.4 Scattering ..................................................................................... 14.5 Fiber Bragg Grating Sensors ........................................................... 14.6 Applications of Fiber Optic Sensors ................................................. 14.7 Summary ...................................................................................... References ..............................................................................................

347 348 351 354 359 361 367 368 369

15 Residual Stress Yuri F. Kudryavtsev .................................................................................. 15.1 Importance of Residual Stress ........................................................ 15.2 Residual Stress Measurement ......................................................... 15.3 Residual Stress in Fatigue Analysis ................................................. 15.4 Residual Stress Modification .......................................................... 15.5 Summary ...................................................................................... References ..............................................................................................

371 371 373 381 383 386 386

16 Nanoindentation: Localized Probes of Mechanical Behavior

of Materials David F. Bahr, Dylan J. Morris .................................................................. 16.1 Hardness Testing: Macroscopic Beginnings...................................... 16.2 Extraction of Basic Materials Properties from Instrumented Indentation ..................................................... 16.3 Plastic Deformation at Indentations ............................................... 16.4 Measurement of Fracture Using Indentation ................................... 16.5 Probing Small Volumes to Determine Fundamental Deformation Mechanisms ..................... 16.6 Summary ...................................................................................... References ..............................................................................................

402 404 404

17 Atomic Force Microscopy in Solid Mechanics Ioannis Chasiotis ..................................................................................... 17.1 Tip–Sample Force Interactions in Scanning Force Microscopy ........... 17.2 Instrumentation for Atomic Force Microscopy.................................. 17.3 Imaging Modes by an Atomic Force Microscope ............................... 17.4 Quantitative Measurements in Solid Mechanics with an AFM ........... 17.5 Closing Remarks ............................................................................ 17.6 Bibliography ................................................................................. References ..............................................................................................

409 411 412 423 432 438 439 440

389 389 392 396 399

Contents

Part C Noncontact Methods 18 Basics of Optics Gary Cloud ............................................................................................... 18.1 Nature and Description of Light ..................................................... 18.2 Interference of Light Waves ........................................................... 18.3 Path Length and the Generic Interferometer................................... 18.4 Oblique Interference and Fringe Patterns ....................................... 18.5 Classical Interferometry ................................................................. 18.6 Colored Interferometry Fringes ....................................................... 18.7 Optical Doppler Interferometry ....................................................... 18.8 The Diffraction Problem and Examples ........................................... 18.9 Complex Amplitude ....................................................................... 18.10 Fraunhofer Solution of the Diffraction Problem ............................... 18.11 Diffraction at a Clear Aperture ........................................................ 18.12 Fourier Optical Processing .............................................................. 18.13 Further Reading ............................................................................ References .............................................................................................. 19 Digital Image Processing for Optical Metrology Wolfgang Osten ....................................................................................... 19.1 Basics of Digital Image Processing .................................................. 19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology ...................................................................... 19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology ...................................................................... References ..............................................................................................

447 448 449 451 453 455 461 464 468 470 472 474 476 479 479

481 483 485 545 557

20 Digital Image Correlation for Shape

and Deformation Measurements Michael A. Sutton .................................................................................... 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10

Background .................................................................................. Essential Concepts in Digital Image Correlation ............................... Pinhole Projection Imaging Model.................................................. Image Digitization ......................................................................... Intensity Interpolation .................................................................. Subset-Based Image Displacements ............................................... Pattern Development and Application ............................................ Two-Dimensional Image Correlation (2-D DIC)................................. Three-Dimensional Digital Image Correlation.................................. Two-Dimensional Application: Heterogeneous Material Property Measurements............................. 20.11 Three-Dimensional Application: Tension Torsion Loading of Flawed Specimen.................................. 20.12 Three-Dimensional Measurements – Impact Tension Torsion Loading of Single-Edge-Cracked Specimen ..................................... 20.13 Closing Remarks ............................................................................

565 566 568 569 573 573 575 577 579 581 585 588 593 597

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Contents

20.14 Further Reading ............................................................................ References ..............................................................................................

597 599

21 Geometric Moiré Bongtae Han, Daniel Post ........................................................................ 21.1 Basic Features of Moiré .................................................................. 21.2 In-Plane Displacements ................................................................ 21.3 Out-Of-Plane Displacements: Shadow Moiré .................................. 21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT) ............... 21.5 Increased Sensitivity...................................................................... References ..............................................................................................

601 601 607 611 617 623 626

22 Moiré Interferometry Daniel Post, Bongtae Han ........................................................................ 22.1 Current Practice ............................................................................. 22.2 Important Concepts ....................................................................... 22.3 Challenges .................................................................................... 22.4 Characterization of Moiré Interferometry ........................................ 22.5 Moiré Interferometry in the Microelectronics Industry ..................... References ..............................................................................................

627 630 634 644 645 646 652

23 Speckle Methods Yimin Gan, Wolfgang Steinchen (deceased)............................................... 23.1 Laser Speckle ................................................................................ 23.2 Speckle Metrology ......................................................................... 23.3 Applications .................................................................................. 23.4 Bibliography ................................................................................. References ..............................................................................................

655 655 658 668 672 672

24 Holography Ryszard J. Pryputniewicz .......................................................................... 24.1 Historical Development.................................................................. 24.2 Fundamentals of Holography ......................................................... 24.3 Techniques of Hologram Interferometry.......................................... 24.4 Representative Applications of Holography ..................................... 24.5 Conclusions and Future Work ......................................................... References ..............................................................................................

675 676 677 679 685 696 697

25 Photoelasticity Krishnamurthi Ramesh ............................................................................ 25.1 Preliminaries ................................................................................ 25.2 Transmission Photoelasticity .......................................................... 25.3 Variants of Photoelasticity ............................................................. 25.4 Digital Photoelasticity.................................................................... 25.5 Fusion of Digital Photoelasticity Rapid Prototyping and Finite Element Analysis ........................................................... 25.6 Interpretation of Photoelasticity Results .........................................

701 704 705 710 719 732 734

Contents

25.7 Stress Separation Techniques ......................................................... 25.8 Closure ......................................................................................... 25.9 Further Reading ............................................................................ 25.A Appendix ...................................................................................... References ..............................................................................................

735 737 737 738 740

26 Thermoelastic Stress Analysis Richard J. Greene, Eann A. Patterson, Robert E. Rowlands ......................... 26.1 History and Theoretical Foundations .............................................. 26.2 Equipment .................................................................................... 26.3 Test Materials and Methods ........................................................... 26.4 Calibration .................................................................................... 26.5 Experimental Considerations.......................................................... 26.6 Applications .................................................................................. 26.7 Summary ...................................................................................... 26.A Analytical Foundation of Thermoelastic Stress Analysis .................... 26.B List of Symbols .............................................................................. References ..............................................................................................

743 744 745 747 749 749 753 759 760 762 763

27 Photoacoustic Characterization of Materials Sridhar Krishnaswamy ............................................................................. 27.1 Elastic Wave Propagation in Solids ................................................. 27.2 Photoacoustic Generation .............................................................. 27.3 Optical Detection of Ultrasound...................................................... 27.4 Applications of Photoacoustics ....................................................... 27.5 Closing Remarks ............................................................................ References ..............................................................................................

769 770 777 783 789 798 798

28 X-Ray Stress Analysis Jonathan D. Almer, Robert A. Winholtz ..................................................... 28.1 Relevant Properties of X-Rays ........................................................ 28.2 Methodology ................................................................................. 28.3 Micromechanics of Multiphase Materials ........................................ 28.4 Instrumentation ............................................................................ 28.5 Experimental Uncertainties ............................................................ 28.6 Case Studies .................................................................................. 28.7 Summary ...................................................................................... 28.8 Further Reading ............................................................................ References ..............................................................................................

801 802 804 807 809 810 813 817 818 818

Part D Applications 29 Optical Methods Archie A.T. Andonian ............................................................................... 29.1 Photoelasticity .............................................................................. 29.2 Electronic Speckle Pattern Interferometry .......................................

823 824 828

XXIII

XXIV

Contents

29.3 Shearography and Digital Shearography ......................................... 29.4 Point Laser Triangulation ............................................................... 29.5 Digital Image Correlation ............................................................... 29.6 Laser Doppler Vibrometry ............................................................... 29.7 Closing Remarks ............................................................................ 29.8 Further Reading ............................................................................ References ..............................................................................................

830 831 832 834 835 836 836

30 Mechanical Testing at the Micro/Nanoscale M. Amanul Haque, Taher Saif ................................................................... 30.1 Evolution of Micro/Nanomechanical Testing .................................... 30.2 Novel Materials and Challenges...................................................... 30.3 Micro/Nanomechanical Testing Techniques ..................................... 30.4 Biomaterial Testing Techniques ...................................................... 30.5 Discussions and Future Directions .................................................. 30.6 Further Reading ............................................................................ References ..............................................................................................

839 840 841 842 856 859 862 862

31 Experimental Methods in Biological Tissue Testing Stephen M. Belkoff, Roger C. Haut ............................................................ 31.1 General Precautions ...................................................................... 31.2 Connective Tissue Overview ............................................................ 31.3 Experimental Methods on Ligaments and Tendons.......................... 31.4 Experimental Methods in the Mechanical Testing of Articular Cartilage ...................................................................... 31.5 Bone ............................................................................................ 31.6 Skin Testing .................................................................................. References ..............................................................................................

871 871 872 873 876 878 883 884

32 Implantable Biomedical Devices

and Biologically Inspired Materials Hugh Bruck ............................................................................................. 32.1 Overview....................................................................................... 32.2 Implantable Biomedical Devices..................................................... 32.3 Biologically Inspired Materials and Systems .................................... 32.4 Conclusions ................................................................................... 32.5 Further Reading ............................................................................ References ..............................................................................................

891 892 899 909 923 924 924

33 High Rates and Impact Experiments Kaliat T. Ramesh...................................................................................... 33.1 High Strain Rate Experiments ......................................................... 33.2 Wave Propagation Experiments ...................................................... 33.3 Taylor Impact Experiments ............................................................. 33.4 Dynamic Failure Experiments ......................................................... 33.5 Further Reading ............................................................................ References ..............................................................................................

929 930 945 949 949 953 954

Contents

34 Delamination Mechanics Kenneth M. Liechti ................................................................................... 34.1 Theoretical Background ................................................................. 34.2 Delamination Phenomena ............................................................. 34.3 Conclusions ................................................................................... References ..............................................................................................

961 962 968 980 980

35 Structural Testing Applications Ashok Kumar Ghosh ................................................................................. 985 35.1 Past, Present, and Future of Structural Testing ................................ 987 35.2 Management Approach to Structural Testing ................................... 990 35.3 Case Studies .................................................................................. 997 35.4 Future Trends ................................................................................ 1012 References .............................................................................................. 1013 36 Electronic Packaging Applications Jeffrey C. Suhling, Pradeep Lall ................................................................. 36.1 Electronic Packaging...................................................................... 36.2 Experimental Mechanics in the Field of Electronic Packaging ........... 36.3 Detection of Delaminations ........................................................... 36.4 Stress Measurements in Silicon Chips and Wafers ............................ 36.5 Solder Joint Deformations and Strains ............................................ 36.6 Warpage and Flatness Measurements for Substrates, Components, and MEMS .......................................... 36.7 Transient Behavior of Electronics During Shock/Drop ....................... 36.8 Mechanical Characterization of Packaging Materials ........................ References ..............................................................................................

Acknowledgements ................................................................................... About the Authors ..................................................................................... Detailed Contents...................................................................................... Subject Index.............................................................................................

1015 1017 1019 1022 1024 1031 1036 1039 1041 1042 1045 1049 1059 1083

XXV

XXVII

List of Abbreviations

A AASHTO ABS ACES AC AFM AFNOR AISC ALT AREA ASTM

Association of State Highway Transportation Officials American Bureau of Shipping analytical, computational, and experimental solution alternating current atomic force microscopy Association Française de National American Institute of Steel Construction accelerated lift testing American Railway Engineering Association American Society for Testing and Materials

B bcc BGA BHN BIL BSI BS BrUTS

body-centered cubic ball grid array Brinell hardness number boundary, initial, and loading British Standards Institute beam-splitter bromoundecyltrichlorosilane

C CAD CAE CAM CAR CBGA CCD CCS CDC CD CGS CHS CMC CMM CMOS CMP CMTD CNC CNT COD COI CP

computer-aided design computer-aided engineering computer-aided manufacturing collision avoidance radar ceramic ball grid array charge-coupled device camera coordinate system Centers for Disease Control compact disc coherent gradient sensing coefficient of hygroscopic swelling ceramic matrix composite coordinate measuring machine complementary metal–oxide-semiconductor chemical–mechanical polishing composite materials technical division computer numerical control carbon nanotube crack opening displacement crack opening interferometry conductive polymer

CSK CSM CTE CTOD CVD CVFE CW CZM

cytoskeleton continuous stiffness module coefficient of thermal expansion crack-tip opening displacement chemical vapor deposition cohesive-volumetric finite elements continuous wave cohesive zone model

D DBS DCB DCPD DC DEC DGL DIC DIN DMA DMD DMT DNP DPH DPN DPV DRIE DR DSC DSPI DSPSI DSP DTA DTS DTS

directional beam-splitter double-cantilever beam dicyclopentadiene direct current diffraction elastic constant Devonshire–Ginzburg–Landau digital image correlation Deutsches Institut für Normung dynamical mechanical analyzers digital micromirror device Derjaguin–Muller–Toporov distance from the neutral point diamond pyramid hardness dip-pen nanolithography Doppler picture velocimetry deep reactive-ion etching deviation ratio digital speckle correlation digital speckle-pattern interferometry digital speckle-pattern shearing interferometry digital speckle photography differential thermal analyzer distributed temperature sensing dodecyltrichlorosilane

E EAP ECM ECO EDM EDP EFM EFPI EMI ENF EPFM EPL

electroactive polymer extracellular matrix poly(ethylene carbon monoxide) copolymer electric-discharge machining ethylenediamine pyrochatechol electrostatic force microscopy extrinsic Fabry–Pérot interferometer electromagnetic interference end-notched flexure elastic–plastic fracture mechanics electron-beam projection lithography

XXVIII

List of Abbreviations

ESIS ESPI ESSP ES ET EW EXAFS

European Structural Integrity Society electronic speckle pattern interferometry electrostatically stricted polymer expert system emerging technologies equivalent weight extended x-ray absorption fine structure

IEC IFM IMU IPG IPMC IP IR-GFP ISDG ISO

focal adhesion complex fine-pitch ball grid array fiber Bragg grating flip-chip ball grid array face-centered cubic flip-chip plastic ball grid array Food and Drug Administration finite element modeling finite element fast Fourier transform full field of view functionally graded material focused ion beam force modulation microscopy Fillers–Moonan–Tschoegl frequency modulation friction stir welds Fourier-transform method field of view

ISTS IT

F FAC FBGA FBG FC-BGA fcc FC-PBGA FDA FEM FE FFT FFV FGM FIB FMM FMT FM FSW FTE FoV

G GF GMR GPS GRP GUM G&R

growth factor giant magnetoresistance global positioning sensing glass-reinforced plastic Guide for the Expression of Uncertainty in Measurement growth and remodeling

H HAZ hcp HDI HEMA HFP HNDE HRR HSRPS HTS

heat-affected zone hexagonal close-packed high-density interconnect 2-hydroxyethyl methacrylate half-fringe photoelasticity holographic nondestructive evaluation Hutchinson–Rice–Rosengreen high strain rate pressure shear high-temperature storage

I IBC IC

International Building Code integrated circuits

ion exchange capacity interfacial force microscope inertial measurement unit ionic polymer gel ionomeric polymer–metal composite image processing infrared grey-field polariscope interferometric strain/displacement gage International Organization for Standardization impulsive stimulated thermal scattering interferometer

J JIS

Japanese Industrial Standards

L LAN LASIK LCE LDV LED LEFM LFM LIGA LMO LORD LPCVD LPG LVDT LVDT

local area network laser-based corneal reshaping liquid crystal elastomer laser Doppler vibrometry light-emitting diode linear elastic fracture mechanics lateral-force AFM lithography galvanoforming molding long-working-distance microscope laser occlusive radius detector low-pressure CVD long-period grating linear variable differential transformer linear variable displacement transducer

M MBS MEMS MFM micro-CAT MITI MLF MMC MOEMS MOSFET MO M-TIR μPIV MPB MPCS MPODM MW(C)NT

model-based simulation micro-electromechanical system magnetic force microscopy micro computerized axial tomography Ministry of International Trade and Industry metal leadframe package metal matrix composites microoptoelectromechanical systems metal-oxide semiconductor field-effect transistor microscope objective modified total internal reflection microparticle image velocimetry morphotropic phase boundary most probable characteristics strength multipoint overdeterministic method multiwalled (carbon) nanotubes

List of Abbreviations

N NA NC-AFM NCOD NC NDE NDT/NDE NEMS NEPA NHDP NIST NMR NSF NSOM NVI

numerical aperture noncontact AFM normal crack opening displacement nanocrystalline nondestructive evaluation nondestructive testing/evaluation nanoelectromechanical system National Environmental Policy Act The National Highways Development Project National Institute of Standard and Technology nuclear magnetic resonance National Science Foundation near-field scanning optical microscopy normal velocity interferometer

OFDR OIML OIM OMC ONDT OPD OPS OSHA

optical/digital fringe multiplication object coordinate system orientation distribution function optoelectronic holography optoelectronic laser interferometric microscope optical frequency-domain reflectometry Organisation Internationale de Metrologie Legale orientation imaging microscopy organic matrix composites optical nondestructive testing optical path difference operations per second Occupational Safety and Health Administration

P PBG PBS PCB PCF PDMS PECVD PE PID PIN PIV PLA PLD PLZT PL PM fiber PMC

photonic-bandgap phosphate-buffered saline printed circuit board photonic-crystal fiber polydimethylsiloxane plasma-enhanced CVD pulse-echo proportional-integral-derivative p-type intrinsic n-type particle image velocimetry polylactic acid path length difference Pb(La,Zr,Ti)O3 path length polarization maintaining fiber polymer matrix composites

phase-measurement interferometry polymethyl methacrylate Pb(Mgx Nb1−x )O3 polarization maintaining power meter plastic quad flat package photorefractive crystal point spread function phase shifting technique plated-through holes polyvinyl chloride polyvinylidene fluoride physical vapor deposition phase velocity scanning Pb(Zr,Ti)O3

Q QC QFP QLV

O O/DFM OCS ODF OEH OELIM

PMI PMMA PMN PM PM PQFP PRC PSF PST PTH PVC PVDF PVD PVS PZT

quality-control quad flat pack quasilinear viscoelastic theory

R RCC RCRA RF RIBS RMS RPT RSG RSM RS RVE R&D

reinforced cement concrete Resource Conservation and Recovery Act radiofrequency replamineform inspired bone structures root-mean-square regularized phase tracking resistance strain gage residual stress management residual stress representative volume element research and development

S S-T-C S/N SAC SAM SAM SAR SAW SCF SCS SC SDF SD SEM SEM SHPB SIF SI SLC

self-temperature compensation signal to noise Sn–Ag–Cu scanning acoustic microscopy self-assembled monolayer synthetic-aperture radar surface acoustic wave stress concentration factor sensor coordinate system speckle correlation structure data files standard deviation scanning electron microscopy Society for Experimental Mechanics split-Hopkinson pressure bar stress intensity factor speckle interferometry surface laminar circuit

XXIX

XXX

List of Abbreviations

SLL SMA SNL SNR SOI SOTA SPATE SPB SPM SPSI SPS SP SQUID SRM SSM STC STM SThM ST SW(C)NT

surface laminar layer shape-memory alloy Sandia National Laboratories signal-to-noise ratio silicon on insulator state-of-the-art stress pattern analysis by thermal emissions space–bandwidth product scanning probe microscope speckle pattern shearing interferometry spatial phase shifting speckle photography superconducting quantum interface device sensitivity response method sacrificial surface micromachining self-temperature compensated scanning tunneling microscope scanning thermal microscopy structural test single-walled (carbon) nanotubes

TIG TIM TIR TMAH TM TPS TSA TS TWSME Terfenol-D TrFE

thermal barrier coatings temperature coefficient of expansion temperature coefficient of resistance transverse displacement interferometer time division multiplexing thermoelectric cooler transmission electron microscopy thermal evaluation for residual stress analysis tetrafluoroethylene

VLE

T TBC TCE TCR TDI TDM TEC TEM TERSA TFE

tungsten inert gas thermal interface materials total internal reflection tetramethylammonium hydroxide thermomechanical temporal phase shifting thermoelastic stress analysis through scan two-way shape memory effect Tb0.3 Dy0.7 Fe2 trifluoroethylene

U UCC UDL UHV ULE UP

ultrasonic computerized complex uniformly distributed load ultrahigh vacuum ultralow expansion ultrasonic peening

V VCCT VDA VISAR

virtual crack closure technique video dimension analysis velocity interferometer system for any reflector vector loop equations

W WCS WDM WFT WLF WM WTC

world coordinate system wavelength-domain multiplexing windowed Fourier transform Williams–Landel–Ferry wavelength meter World Trade Center

1

Part A

Solid Mec Part A Solid Mechanics Topics

1 Analytical Mechanics of Solids Albert S. Kobayashi, Seattle, USA Satya N. Atluri, Irvine, USA 2 Materials Science for the Experimental Mechanist Craig S. Hartley, Sedona, USA 3 Mechanics of Polymers: Viscoelasticity Wolfgang G. Knauss, Pasadena, USA Igor Emri, Lubljana, Slovenia Hongbing Lu, Stillwater, USA 4 Composite Materials Peter G. Ifju, Gainesville, USA 5 Fracture Mechanics Krishnaswamy Ravi-Chandar, Austin, USA 6 Active Materials Guruswami Ravichandran, Pasadena, USA

7 Biological Soft Tissues Jay D. Humphrey, College Station, USA 8 Electrochemomechanics of Ionic Polymer–Metal Composites Sia Nemat-Nasser, La Jolla, USA 9 A Brief Introduction to MEMS and NEMS Wendy C. Crone, Madison, USA 10 Hybrid Methods James F. Doyle, West Lafayette, USA 11 Statistical Analysis of Experimental Data James W. Dally, Knoxville, USA

3

Albert S. Kobayashi, Satya N. Atluri

In this chapter we consider certain useful fundamental topics from the vast panorama of the analytical mechanics of solids, which, by itself, has been the subject of several handbooks. The specific topics that are briefly summarized include: elementary theories of material response such as elasticity, dynamic elasticity, viscoelasticity, plasticity, viscoplasticity, and creep; and some useful analytical results for boundary value problems in elasticity.

1.1

Elementary Theories of Material Responses ........................... 1.1.1 Elasticity ..................................... 1.1.2 Viscoelasticity ..............................

1.1.3 Plasticity ..................................... 1.1.4 Viscoplasticity and Creep ...............

7 9

Boundary Value Problems in Elasticity .... 1.2.1 Basic Field Equations .................... 1.2.2 Plane Theory of Elasticity............... 1.2.3 Basic Field Equations for the State of Plane Strain ............................. 1.2.4 Basic Field Equations for the State of Plane Stress ............................. 1.2.5 Infinite Plate with a Circular Hole... 1.2.6 Point Load on a Semi-Infinite Plate

11 11 12

Summary .............................................

14

References ..................................................

14

1.2

4 4 6

1.3

12 12 13 13

Herein, we employ Cartesian coordinates exclusively. We use a fixed Cartesian system with base vectors ei (i = 1, 2, 3). The coordinates of a material particle before and after deformation are xi and yi , respectively. The deformation gradient, denoted as Fij , is defined to be ∂yi ≡ yi, j . (1.1) ∂x j

A wide variety of other strain measures may be derived [1.1–4]. Let ( da) be a differential area in the deformed body, and let n i be direction cosines of a unit outward normal to ( da). If the differential force acting on this area is d f i , the true stress or Cauchy stress τij is defined from the relation

The displacement components will be denoted by u i (= yi − xi ), such that

Thus τij is the stress per unit area in the deformed body. The nominal stress (or the transpose of the so-called first Piola–Kirchhoff stress) tij and the second Piola– Kirchhoff stress Sij are defined through the relations

Fij = δij + u i, j ,

(1.2)

where δij is a Kronecker delta. The Green–Lagrange strain tensor εij is given by 1 1 εij = (Fki Fk j − δij ) ≡ (u i, j + u j,i + u k,i u k, j ) . 2 2 (1.3)

When displacements and their gradients are infinitesimal, (1.3) may be approximated as 1 (1.4) εij = (u i, j + u j,i ) ≡ u (i, j) . 2

d f i = ( da)n j τij .

d f i = ( da)N j t ji = ( dA)N j S jk yi,k ,

(1.5)

(1.6) (1.7)

where ( dA)N j is the image in the undeformed configuration, of the oriented vector area ( da)n j in the deformed configuration. Note that both t ji and S ji are stresses per unit area in the undeformed configuration, and t ji is unsymmetric, while S ji is, by definition, symmetric [1.3,4]. It should also be noted that a wide variety of other stress measures may be derived [1.3, 4].

Part A 1

Analytical Me 1. Analytical Mechanics of Solids

4

Part A

Solid Mechanics Topics

Part A 1.1

From the geometric theory of deformation [1.5], it follows that   ∂xi (1.8) , ( da)n j = (J)( dA)Nk ∂y j where ρ0 dv (1.9) = . J= d∀ ρ In the above dv is a differential volume in the deformed body, and d∀ is its image in the undeformed body. From (1.5) through (1.9), it follows that ∂x j ∂xi ∂xi τm j and Sij = J τmn . (1.10) tij = J ∂ym ∂ym ∂yn

Another useful stress tensor is the so-called Kirchhoff stress tensor, denoted by σij and defined as σij = Jτij .

(1.11)

When displacements and their gradients are infinitesimal, J ≈ 1, ∂xi /∂yk = δik and so on, and thus the distinction between all the stress measures largely disappears. Hence, in an infinitesimal deformation theory, one may speak of the stress tensor σij . For more on finite deformation mechanism of solids see [1.6, 7].

1.1 Elementary Theories of Material Responses The mathematical characterization of the behavior of solids is one of the most complex aspects of solid mechanics. Most of the time, the general behavior of a material defies our mathematical ability to characterize it. The theories discussed below must be viewed simply as idealizations of regimes of material response under specific types of loading and/or environmental conditions.

invariants of εij . These invariants may be defined as

1.1.1 Elasticity

where eijk is equal to 1 if (ijk) take on values 1, 2, 3 in a cyclic order, and equal to −1 if in anticyclic order, and is zero if two of the indices take on identical values. Sometimes, invariants J1 , J2 , and J3 , defined as

In this idealization, the underlying assumption is that stress is a single-valued function of strain and is independent of the history of straining. Also, for such materials, one may define a potential for stress in terms of strain, in the form of a strain-energy density function, denoted here by W. It is customary [1.4] to measure W per unit of the undeformed volume. In the general case of finite deformations, different stress measures are related to the derivative of W with respect to specific strain measures, labeled as conjugate strain measures, i. e., strains conjugate to the appropriate form of stress. Thus it may be shown [1.3] that tij =

∂W , ∂F ji

Sij =

∂W . ∂εij

(1.12)

Note that, for finite deformations, the Cauchy stress τij does not have a simple conjugate strain measure. When W does not depend on the location of the material particle (in the undeformed conjugation), the material is said to be homogeneous. A material is said to be isotropic if W depends on εij only through the basic

I1 = 3 + 2εkk , I2 = 3 + 4εkk + 2(εkk εmm − εkm εkm ) , and I3 = det|δmn + 2εmn | ≡ 1 + 2εkk + 2(εkk εmm − εkm εkm ) ,

J1 = (I1 − 3), J2 = (I2 − 2I1 + 3) , J3 = (I3 − I2 + I1 − 1)

(1.13)

(1.14)

are also used. When the material is isotropic, the Kirchhoff stress tensor may be shown to be the derivative of W with respect to a certain logarithmic strain measure [1.4]. Also, by decomposing the deformation gradient Fij into pure stretch and rigid rotation [1.4, 8], one may derive certain other useful stress measures, such as the Biot–Lure stress, Jaumann stress, and so on [1.4]. An isotropic nonlinearly elastic material may be characterized, in its behavior at finite deformations, by W=

∞ 

Crst (I1 − 3)r (I2 − 3)s (I3 − 1)t ,

r,s,t=0

C000 = 0 .

(1.15)

The ratio of volume change due to deformation, dv/ d∀, is given, for finite deformations, by I3 . Thus, for

Analytical Mechanics of Solids

¯ = W(εij ) + p(I3 − 1) , W

(1.16)

that is, tij =

∂W ∂I3 +p , ∂F ji ∂F ji

Sij =

∂W ∂I3 +p . (1.17) ∂εij ∂εij

For isotropic, incompressible, elastic materials, W(εij ) = W(I1 , I2 ) .

(1.18)

Thus (1.17) and (1.18) yield, for instance, ∂W ∂W δij + 4[δij (1 + εmm ) − δim δ jn εmn ] ∂I1 ∂I2 + p[δij (1 + 2εmm ) − 2δim δ jn εmn + 2eimn e jrs εmr εns ] . (1.19)

Sij = 2

A well-known representation of (1.18) is due to Mooney [1.8], where W(I1 , I2 ) = C1 (I1 − 3) + C2 (I2 − 3) .

(1.20)

So far, we have discussed isotropic materials. In general, for a homogeneous solid, one may write W = E ij εij + 12 E ijmn εij εmn + 13 E ijmnrs εij εmn εrs + · · · .

(1.21)

We use, for convenience of presentation, Sij and εij as conjugate measures of stress and strain. Since Sij and εij are both symmetric, one must have E ij = E ji , E ijmn = E jinm = E ijnm = E mnij , E ijmnrs = E jimnrs = E ijnmrs = E ijmnsr = · · · = Ersijmn = · · · . (1.22) Thus Sij = E ij + E ijmn εmn + E ijmnrs εmn εrs + · · · . (1.23) Henceforth, we will consider the case when deformations are infinitesimal. Thus εij ≈ (1/2)(u i, j + u j,i ). Further, the differences in the definitions of various stress measures disappear, and one may speak of the stress σij . Thus (1.23) may be rewritten as σij = E ij + E ijmn εmn + E ijmnrs εmn εrs + · · · . (1.24)

A material is said to be linearly elastic if a linear approximation of (1.24) is valid for the magnitude of strains under consideration. For such a material, σij = E ij + E ijmn εmn .

(1.25)

The stress at zero strain (i. e., E ij ) most commonly is due to temperature variation from a reference state. The simplest assumption in thermal problems is to set E ij = −βij ΔT where ΔT [= (T − T0 )] is the temperature increment from the reference value T0 . For an anisotropic linearly elastic solid, in view of the symmetries in (1.23), one has 21 independent elastic constants E ijkl and six constants βij . In the case of isotropic linearly elastic materials, an examination of (1.13) through (1.15) reveals that the number of independent elastic constants E ijkl is reduced to two, and the number of independent βs to one. Thus, for an isotropic elastic material, σij = λεkk δij + 2μεij − βΔT δij ,

(1.26a)

where λ and μ are Lamé parameters, which are related to the Young’s modulus E and the Poisson’s ratio ν through E Eν , μ= . λ= (1 + ν)(1 − 2ν) 2(1 + ν) The bulk modulus K is defined as 3λ + 2μ . K= 3 The inverse of (1.26a) is ν 1+ν (1.26b) σij + αΔT δij , εij = − σmn δij + E E where β and α are related through Eα β= (1.26c) 1 − 2ν and α is the linear coefficient of thermal expansion. The state of plane strain is characterized by the conditions that u r = u r (X s ), r, s = 1, 2, and u 3 = 0. Thus ε3i = 0, i = 1, 2, 3. In plane strain,   1 − ν2 ν σ22 + α(1 + ν)ΔT , σ11 − ε11 = E 1−ν (1.27a)

  1 − ν2 ν ε22 = σ22 − σ11 + α(1 + ν)ΔT , E 1−ν

(1.27b)

1+ν σ12 , E σ33 = ν(σ11 + σ) − αEΔT . ε12 =

(1.27c) (1.27d)

5

Part A 1.1

incompressible materials, I3 = 1. For incompressible materials, stress is determined from strain only to within a scalar quantity (function of material coordinates) called the hydrostatic pressure. For such materials, one ¯ in may define a modified strain-energy function, say W, which the incompressibility condition, I3 = 1, is introduced as a constraint through the Lagrange multiplier p. Thus

1.1 Elementary Theories of Material Responses

6

Part A

Solid Mechanics Topics

Part A 1.1

The state of plane stress is characterized by the conditions that σ3k = 0, k = 1, 2, 3. Here one has 1 (1.28a) ε11 = (σ11 − νσ22 ) + αΔT , E 1 ε22 = (σ22 − νσ11 ) + αΔT , (1.28b) E 1+ν ε12 = (1.28c) σ12 , E ν ε33 = − (σ11 + σ22 ) + αΔT . (1.28d) E Note that in (1.27c) and (1.28c), ε12 is the tensor component of strain. Sometimes it is customary to use the engineering strain component γ12 = 2ε12 . Note also that, in the case of a linearly elastic material, the strainenergy density W is given by W= =

σij (t) = εkl (0+ )E ijkl +

t E ijkl (t − τ) 0

∂εkl dτ ∂τ (1.33a)

= E kl (0+ )εijkl +

t εijkl (t − τ) 0

∂E kl dτ . ∂τ (1.33b)

1 2 σij εij 1 2 (σ11 ε11 + σ22 ε22 + σ33 ε33

In the above, it has been assumed that σkl = εkl = 0 for t < 0 and that εij (t) and E ij (t) are piecewise continuous. E ijkl (t) is called the relaxation tensor for an anisotropic material. Conversely, one may write

+ 2ε12 σ12 + 2ε13 σ13 + 2ε23 σ23 ) ≡ 12 (σ11 ε11 + σ22 ε22 + σ33 ε33 + γ12 σ12 + γ23 σ23 + γ13 σ13 ) .

(1.29)

From (1.26b) it is seen that, for linearly elastic isotropic materials, 1 − 2ν σmm σmm + 3αΔT ≡ + 3αΔT . (1.30) εkk = E 3k When the bulk modulus k → ∞ (or ν → 12 ), it is seen that εkk → 3αΔT and is independent of the mean stress. Note also from (1.26c) that β → ∞ as ν → 12 . For such materials, the mean stress is indeterminate from deformation alone. In this case, the relation (1.26a) is replaced by σij = −ρδij + 2μεij

(1.31a)

with the constraint εkk = 3αΔT ,

(1.31b)

εij

where ρ is the hydrostatic pressure and is the deviator of the strain. Note that the strain-energy density of a linearly elastic incompressible material is W = μεij εij − p(εkk − 3αΔT ) ,

materials are those for which the current deformation is a function of the entire history of loading, and conversely, the current stress is a function of the entire history of straining. Linearly viscoelastic materials are those for which the hereditary relations are expressed in terms of linear superposition integrals, which, for infinitesimal strains, take the forms

(1.32)

wherein p acts as a Lagrange multiplier to enforce (1.31b).

1.1.2 Viscoelasticity A linearly elastic solid, by definition, is one that has the memory of only its unstrained state. Viscoelastic

+

t

εij (t) = σkl (0 )Cijkl +

Cijkl (t − τ) 0

∂σkl dτ , ∂τ (1.34)

where Cijkl (t) is called the creep compliance tensor. For isotropic linearly viscoelastic materials, E ijkl = μ(t)(δik δ jl + δlk δ jk ) + λ(t)δij δkl ,

(1.35)

where μ(t) is the shear relaxation modulus and B(t) ≡ [3α(t) + 2μ(t)]/3 is the bulk relaxation modulus. It is often assumed that B(t) is a constant; so that the material is assumed to have purely elastic volumetric change. In viscoelasticity, a Poisson function corresponding to the strain ratio in elasticity does not exists. However, for every deformation history there is computable Poisson contraction or expansion behavior. For instance, in a uniaxial tension test, let the stress be σ11 , the longitudinal strain ε11 , and the lateral strain ε22 . For creep at constant stress, the ratio of lateral contraction, denoted by νc (t), is νc (t) = −ε22 (t)/ε11 (t). On the other hand, under relaxation at constant strain, ε11 , the lateral contraction ratio is ν R (t) = −ε22 (t)/ε11 . It is often convenient, though not physically correct, to assume that Poisson’s ratio is a constant, which renders B(t) proportional to μ(t). A constant bulk modulus provides a much better and simple approximation for the material behavior than a constant Poisson’s ratio when properties over the whole time range are needed.

Analytical Mechanics of Solids

σ ij ( p) = pE ijkl ( p)εkl ( p)

(1.36a)

εij ( p) = pC ijkl ( p)σ kl ( p) ,

(1.36b)

and

deformation is assumed to be insensitive to hydrostatic pressure, the yield function is assumed, in general, to depend on the stress deviator, σij = σij − 13 σmm δij . The commonly used yield functions are von Mises

where (·) is the Laplace transform of (·) and p is the Laplace variable. From (1.36a) and (1.36b), it follows that p E ijkl C klmn = δim δn j . 2

M 

μm exp(−μm t) ,

m=1 M 

B(t) = B0 +

Bm exp(−βm t) .

m=1

1.1.3 Plasticity Most structural metals behave elastically for only very small values of strain, after which the materials yield. During yielding, the apparent instantaneous tangent modulus of the material is reduced from those in the prior elastic state. Removal of load causes the material to unload elastically with the initial elastic modulus. Such materials are usually labeled as elastic–plastic. Observed phenomena in the behavior of such materials include the so-called Bauschinger effect (a specimen initially loaded in tension often yields at a much reduced stress when reloaded in compression), cyclic hardening, and so on [1.9, 10]. (When a specimen a specimen is subjected to cyclic straining of amplitude −ε to +ε, the stress for the same value of tensile strain ε, prior to unloading, increases monotonically with the number of cycles and eventually saturates.) Various levels of sophistication of elastic-plastic constitutive theories are necessary to incorporate some or all of these observed phenomena. Here we give a rather cursory review of this still burgeoning literature. In most theories of metal plasticity, it is assumed that plastic deformations are entirely distortional in nature, and that volumetric strain is purely elastic. The elastic limit of the material is assumed to be specified by a yield function, which is a function of stress (or of strain, but most commonly of stress). Since plastic

J2 = 12 σij σij ,

(1.38)

Tresca    f (σij ) = (σ1 − σ2 )2 − 4k2 (σ2 − σ3 )2 − 4k2   (1.39) × (σ1 − σ3 )2 − 4k2 = 0 .

(1.37)

It is also customary to represent the relaxation moduli, μ(t) and B(t), in series form, as μ(t) = μ0 +

f (σij ) = J2 − k2 ,

In (1.38) and (1.39), k may be a function of the plastic strain. Both (1.38) and (1.39) represent a surface, which is defined as the yield surface, in the stress space. The two equations also imply the equality of the tensile and compressive yield stresses at all times – so-called isotropic hardening. The yield surface expands while its center remains fixed in the stress space.√ The relation of k to test data follows: in (1.38), k = σ¯ / 3, where σ¯ is the yield stress in uniaxial tension, which may be a function of plas√ tic strain for strain-hardening materials, or k = τ¯ / 2, where τ¯ is the yield stress in pure shear; or in (1.39), k = σ¯ /2 or τ¯ . Experimental data appear to favor the use of the Mises condition [1.11, 12]. To account for the Bauschinger effect, one may use the representation of the yield surface   f (σij − αij ) = 0 = 12 σij − αij σij − αij − 13 σ¯ 2 =0,

(1.40)

αij

where represents the center of the yield surface in the deviatoric stress space. The evolution equations suggested for αij by Prager [1.13] and Ziegler [1.14], respectively, are that the incremental dαij is proporp tional to the incremental plastic strain dεij or dαij = c dεij

(1.41)

dαij = dμ(σij − αij ) .

(1.42)

p

and

(·)

In the above, denotes the deviatoric part of the second-order tensor (·) where an additive decomposition of differential strain into elastic and plastic parts p (i. e., dεij = dεije + dεij ) was used. Elastic processes (with no increase in plastic strain) and plastic processes (with increase in plastic strain) are defined [1.12] as

7

Part A 1.1

The Laplace transforms of (1.33) and (1.34) may be written as

1.1 Elementary Theories of Material Responses

8

Part A

Solid Mechanics Topics

Part A 1.1

elastic process: f < 0 or

f = 0 and

∂f dσij ≤ 0 , ∂σij

(1.43)

plastic process: ∂f dσij > 0 . (1.44) ∂σij The flow rule for strain-hardening materials, arising out of consideration of stress working in a cyclic process and stability of the process, often referred to as Drucker’s postulates [1.15], is given by f = 0 and

p dεij

∂f = dλ . ∂σij

(1.45)

The scalar dλ is determined from the fact that d f = 0 during a plastic process, the so-called consistency condition. Using the isotropic-hardening (J2 flow) theory for which f is given in (1.38), this consistency condition leads to 9 σ  dσmn p (1.46) dεij = σij mn  2 , 4 H σ¯ where H  is the slope of the curve of stress versus plastic strain in uniaxial tension (or, more correctly, the slope of the curve of true stress versus logarithmic strain in pure tension). On the other hand, for Prager’s linearly kinematic hardening rules, given in (1.40) and (1.41), the consistency condition leads to  3   p  σmn − αmn dσmn σij − αij . (1.47) dεij = 2cσ¯ 2 For pressure-insensitive plasticity, the stress–strain laws may be written as dσmn = (3λ + 2μ) dεmn ,  p dσij = 2μ dεij − dεij .

(1.48a) (1.48b) p

Choosing a parameter ζ such that ζ = 1 when dεij = 0 p and α = 0 when dεij = 0, we have    dσ 9  σmn mn   dσij = 2μ dεij − σij α (1.49) 4 H  σ¯ 2 for isotropic hardening, and

3  dσij = 2μ dεij − (σ  − αmn ) 2cσ¯ 2 mn  × dσmn (σij − αij )α

(1.50)

for Prager’s linearly kinematic hardening. Taking the tensor product of both sides of (1.49) with σij (and

 dσ  noting that σmn dσmn = σmn mn by definition), we have   3α   σ dσ (1.51a) dσij σij = 2μ dεij σij − 2H  mn mn or 2μH  dσij σij =  dε σ  , when α = 1 . (1.51b) H + 3μ ij ij Use of (1.51b) in (1.49) results in

 9α    dσij = 2μ dεij − 2μ σ σ dε . 4(H  + 3μ)σ¯ 2 ij mn mn

(1.52)

Combining (1.48a) and (1.52), one may write the isotropic-hardening elastic-plastic constitutive law in differential form as  dσij = 2μδim δ jn + λδij δmn  9αμ   − 2μ σ σ (1.53) dεmn , (2H  + 6μ)σ¯ 2 ij mn  dε ≡ σ  dε wherein σmn mn has been noted. Simimn mn larly, by taking the tensor product of (1.50) with (σij − αij ) and repeating steps analogous to those in (1.51) through (1.53), one may write the kinematic-hardening elastoplastic constitutive law as

3μ dσij = 2δim δ jn + λδij δmn − 2μ (c + 2μ)σ¯ 2    × (σij − αij )(σmn − αmn ) dεmn . (1.54) Note that all the developments above are restricted to the infinitesimal strain and small-deformation case. Discussion of finite-deformation plasticity is beyond the scope of this summary. (Even in small deformation plasticity, if the current tangent modulus of the stress–strain relation are of the same order of magnitude as the current stress, one must use an objective stress rate, instead of the material rate dσij , in (1.54).) Here the objectivity of the stress–strain relation plays an important role. We refer the reader to [1.4, 16, 17]. We now briefly examine the elastic-plastic stress– strain relations in the isotropic-hardening case, for plane strain and plane stress, leaving it to the reader to derive similar relations for kinematic hardening. In the planestrain case, dε3n = 0, n = 1, 2, 3. Using this in (1.53), we have

dσαβ = 2μδαθ δβν + λδαβ δθν  9αμ   − 2μ σ σ (1.55) dεθν (2H  + 6μ)σ¯ 2 αβ θυ

Analytical Mechanics of Solids

dσ33 = λ dεθθ − 2μ

9αμ σ  σ  dεϑν , (2H  + 6μ)σ¯ 2 33 θυ

α, β, θ, ν = 1, 2 . (1.56) Note that, in the plane-strain case, σ33 , as integrated from (1.56), enters the yield condition. In the planestress case the stress–strain relation is somewhat tedious to derive. Noting that in the plane-stress case, dεαβ = dεeαβ + p dεαβ one may, using the elastic strain–stress relations as given in (1.33), write p

dεαβ = dεeαβ + dεαβ

1 ν p = dσαβ − dσθθ δαβ + dεαβ (1.57) 2μ 1+ν

1 ν dσαβ − = dσθθ δαβ 2μ 1+ν 9   , (1.58) + σαβ (σθυ dσθυ ) 4H  σ¯ 2 wherein (1.16) has been used. Equation (1.58) may be inverted to obtain dσαβ in terms of dεθν . This 3 × 3 matrix inversion may be carried out, leading to the result [1.18]   2  Q ) + 2P dε11 dσ11 = (σ22 E   + (−σ11 σ22 + 2ν P) dε22   σ11 + νσ22  − 2 dε12 , σ12 1+ν Q   σ22 + 2ν P) dε11 dσ22 = (−σ11 E   2  + (σ11 ) + 2P dε22  σ  + σ11 2 − 22 dε12 , σ12 1+ν     σ11 + νσ22 Q dσ12 = − σ12 dε11 E 1+ν  σ  + νσ11 − 22 σ12 dε22

1+ν  R 2H  + + (1 − ν)σ¯ dε12 , 2(1 + ν) 9E where σ2 2H  2 P= σ¯ + 12 , Q = R + 2(1 − ν2 )P , 9H 1+ν (1.59)

and     2 + 2νσ11 σ22 + σ22 2. R = σ11

(1.60)

As noted earlier, the classical plasticity theory has several limitations. Intense research is underway to im-

prove constitutive modeling in cyclic plasticity, and so on, some notable avenues of current research being multisurface plasticity model, endochronic theories, and related internal variable theories (see, e.g., [1.19–21]).

1.1.4 Viscoplasticity and Creep A viscoplastic solid is similar to a viscous fluid, except that the former can resist shear stress even in a rest configuration; but when the stresses reach critical values as specified by a yield function, the material flows. Consider, for instance, the loading case of simple shear with the only applied stress being σ12 . Restricting ourselves to infinitesimal deformations and strains, let the shearstrain rate be ε˙ 12 = ( dε12 / dt). Then ε˙ 12 = 0 until the magnitude of σ12 reaches a value k, called the yield stress. When |σ12 | > k, ε˙ 12 , by definition for a simple viscoplastic material, is proportional to |σ12 | − k and has the same sign as σ12 . Thus, defining a function F 1 for this one-dimensional problem as |σ12 | (1.61) −1 , k the viscoplastic property may be characterized by the equation F1 =

2η˙ε12 = k(F 1 )σ12 , where

(1.62)

F1

must have the property ⎧ ⎨0 if F 1 < 0 , F 1 = ⎩ F 1 if F 1 ≥ 0 ,

(1.63)

and where η is the coefficient of viscosity. The above relation for simple shear is due to Bing2 for simple shear, ham [1.22]. Recognizing that J = σ12 a generalization of the above for three-dimensional case was given by Hohenemser and Prager [1.23] as νp

η˙εij = 2k

F 1

∂F , ∂σij

(1.64a)

where

   1/2 1/2 σij σij /2 J2 F= −1 = −1 , k k

(1.64b)

and the specific function F is defined similar to F 1 of (1.63). For an elasto-viscoplastic solid undergoing infinitesimal straining, one may use the additive decomposition p

ε˙ ij = ε˙ ije + ε˙ ij

(1.65)

9

Part A 1.1

and

1.1 Elementary Theories of Material Responses

10

Part A

Solid Mechanics Topics

Part A 1.1

and the stress–strain rate relation  ν p σ˙ ij = E ijkl ε˙ kl − ε˙ kl ,

(1.66)

where E ijkl are the instantaneous elastic moduli. Note that the viscoplastic strains in (1.64a) are purely deviatoric, since ∂F/∂σij = σij /2 is deviatoric. Thus, for an isotropic solid, (1.66) may be written as σ˙ mn = (3λ + 2μ) ε˙ mn and



σ˙ ij = 2μ ε˙ ij − ε˙ ij .

(1.68)

(1.69a)

(1.69b)

The expression (1.69a) is often referred to as time hardening and (1.69b) as strain hardening. In as much as (1.68), (1.69a), and (1.69b), are valid for constant stress, (1.69a) and (1.69b), when integrated for variable stress histories, do not necessarily give the same results. Usually, strain hardening leads to better agreement with experimental findings for variable stresses. In the study of creep at a given temperature and for long times, called steady-state creep, the creep strain rate in uniaxial loading is usually expressed as ε˙ c = f (σ, T ) ,

(1.70)

where T is the temperature. Assuming that the effects of σ and T are separable, the relation ε˙ c = f 1 (σ) f 2 (T ) = Aσ n f 2 (T ) = Bσ n



3   σ σ 2 ij ij



 and ε˙ ceq =

2 ε˙ ij ε˙ ij 3

1/2 (1.73)

(1.67b)

or, equivalently, ε˙ c = g(σ, εc ) .

(1.72)

where the subscript “eq” denotes an “equivalent” quantity, defined, analogous to the case of plasticity, as σ˙ eq =

where σ is the uniaxial stress and t is the time. The creep rate may be written as ε˙ c = f (σ, t) ,

n , ε˙ ceq = Bσeq

(1.67a)

On the other hand, for metals operating at elevated temperatures, the strain in uniaxial tension is known to be a function of time, for a constant stress of magnitude even below the conventional elastic limit. Most often, based on extensive experimental data [1.24], the creep strain under constant stress, in uniaxial tests, is expressed as ε˙ c = Aσ n t m ,

involve no volume change. Thus, in the multiaxial case, ε˙ ijc is a deviatoric tensor. The relation (1.71) may be generalized to the multiaxial case as

(1.71a) (1.71b)

is usually employed, with B denoting a function of the temperature. The steady-state creep strains are associated largely with plastic deformations and are usually observed to

such that σeq ε˙ ceq = σij ε˙ ijc . Thus (1.72) implies that ε˙ ijc =

3 B(σeq )n−1 σij . 2

(1.74)

For the elastic-creeping solid, one may again write ε˙ ij = ε˙ ije + ε˙ ijc

(1.75)

and once again, the stress–strain rate relation may be written as  (1.76) σ˙ ij = E ijkl ε˙ kl − ε˙ ckl . In the above, the applied stress level has been assumed to be such that the material remains within the elastic limit. If the applied loads are of such a magnitude as to cause the material to exceed its yield limit, one must account for plastic or viscoplastic strains. An interesting unified viscoplastic/plastic/creep constitutive law has been proposed by Perzyna [1.25]. Under multiaxial conditions, the relation for inelastic strain rate suggested in [1.26] is ε˙ ija = Aψ( f )

∂q , ∂σij

(1.77)

where A is the fluidity parameter, the superscript “a” denotes an elastic strain rate, and f is a loading function, expressed, analogous to the plasticity case, as f (σij , k) = ϕ(σij ) − k = 0 ,

(1.78)

q is a viscoplastic potential defined as q = q(σij ) , and ψ( f ) is a specific function such that ⎧ ⎨0 if f < 0 ψ( f ) = ⎩ψ( f ) if f ≥ 0 .

(1.79)

(1.80)

Analytical Mechanics of Solids

ψ( f ) = f n and

 f =

3   σ σ 2 ij ij

(1.81a)

1/2 − σ¯ = σeq − σ¯ .

(1.81b)

By letting σ¯ = 0 and q = f , one may easily verify that ε˙ ija of (1.77) tends to the creep strain rate ε˙ ijc of (1.74). Letting σ¯ be a specified constant value and q = f , we obtain, using (1.81b) in (1.77), that ε˙ ija =

σij 3 A(σeq − σ¯ )n 2 σeq

for f > 0 (i. e., σeq > σ¯ ) .

(1.82)

The equivalent inelastic strain may be written as   2 a a 1/2 = A(σeq − σ¯ )n (1.83a) ε˙ ij ε˙ ij ε˙ aeq = 3 or   1 a 1/2 . (1.83b) ε˙ eq σeq − σ¯ = A Thus, if a stationary solution of the present inelastic model (i. e., when ε˙ aeq → 0) is obtained, it

is seen that σeq → σ. ¯ Thus a classical inviscid plasticity solution is obtained. This fact has been utilized in obtaining classical rate-independent plasticity solutions from the general model of (1.77), by Zienkiewicz and Corneau [1.26]. An alternative way of obtaining an inviscid plastic solution from Perzyna’s model is to let A → ∞. This concept has been implemented numerically by Argyris and Kleiber [1.27]. Also, as seen from (1.83b), σeq or equivalently the size of the yield surface is governed by isotropic work-hardening effects as characterized by the dependence on viscoplastic work, and the strain-rate effect as q characterized by the term (˙εeq )1/n . Thus rate-sensitive plastic problems may also be treated by Perzyna’s model [1.25]. Thus, by appropriate modifications, the general relation (1.77) may be used to model creep, rate-sensitive plasticity, and rate-insensitive plasticity. By a linear combination of strain rates resulting from these individual types of behavior, combined creep, plasticity, and viscoplasticity may be modeled. However, such a model is more or less formalistic and does not lead to any physical insights into the problem of interactive effects between creep, plasticity, and viscoplasticity. Modeling of such interactions is the subject of a large number of current research studies.

1.2 Boundary Value Problems in Elasticity 1.2.1 Basic Field Equations When deformations are finite and the material is nonlinear, the field equations governing the motion of a solid may become quite complicated. When the constitutive equation is of a differential form, such as in plasticity, viscoplasticity, and so on, it is often convenient to express the field equations in rate form as well. On the other hand, when stress is a single-valued function of strain as in nonlinear elasticity, the field equations may be written in a total form. In general, when numerical procedures are employed to solve boundary/initial value problems for arbitrary-shaped bodies, it is often convenient to write the field equations in rate form, for arbitrary deformations and general constitutive laws. A wide variety of equivalent but alternative forms of these equations is possible, since one may use a wide variety of stress and strain measures, a wide variety of rates of stress and strain, and a variety of coordinate

systems, such as those in the initial undeformed configuration (total Lagrangian), the currently deformed configuration (updated Lagrangian), or any other known intermediate configuration. For a detailed discussion see [1.3, 4]. Each of the alternative forms may offer advantages in specific applications. It is beyond the scope of this chapter to discuss the foregoing alternative forms. Here we state, for a finitely deformed nonlinearly elastic solid, the relevant field equations governing stress, strain, and deformation when the solid undergoes dynamic motion. For this purpose, let x j denote the Cartesian coordinates of a material particle in the undeformed solid. Let u k (xi ) be the arbitrary displacement of a material particle from the undeformed to the deformed configuration. Let Sij be the second Piola–Kirchhoff stress in the finitely deformed solid. Note that S jj is measured per unit area in the undeformed configuration. Let the Green–Lagrange strain tensor, which is work-conjugate to Sij [1.3],

11

Part A 1.2

If q ≡ f , one has the so-called associative law, and if q = f , one has a nonassociative law. Perzyna [1.25] suggests a fairly general form for ψ as

1.2 Boundary Value Problems in Elasticity

12

Part A

Solid Mechanics Topics

Part A 1.2

be εij . Let ρ0 be the mass density in the undeformed solid; b j be body forces per unit mass; ti be tractions measured per unit area in the undeformed solid, prescribed at surface St , of the undeformed solid; and let u i be prescribed displacements at Su . The field equations are [1.4]: linear momentum balance Sik (δ jk + u j,k ) + ρ0 b j = ρ0 u¨ j ,

(1.84)

angular momentum balance Sij = S ji ,

(1.85)

strain displacement relation εij = 12 (u i, j + u j,i + u k,i u k, j ) , constitutive law

(1.86)

∂W , ∂εij

(1.87)

at Si ,

(1.88)

Sij = traction boundary condition n j Sik (δ jk + u j,k ) = t j

displacement boundary condition u j = u¯ i at Su . (1.89) In the above, (·)k denotes ∂(·)/∂xk ; (¨·) denotes ∂ 2 (·)/∂t 2 ; n i are components of a unit normal to St ; and W is the strain-energy density, measured per unit volume in the undeformed body. When the deformations and strains are infinitesimal, the differences in the alternate stress and strain measures disappear. Further, considering only an isothermal linearly elastic solid, the equations above simplify as σij,i + ρ0 b j = ρ0 u¨ j , σij = σ ji , 1 εij = (u i, j + u j,i ) , 2 σij = E ijkl εkl , n i σij = t¯i at St , u i = u¯ i

at Su ,

(1.90) (1.91) (1.92) (1.93) (1.94) (1.95)

and the initial condition, u i (xk , 0) = u ∗ (xk ), u˙ i (xk , 0) = u˙ i∗ (xk , 0) at t = 0 . (1.96)

problems, unfortunately, are few and are limited to semifinite or finite domains; homogeneous and isotropic materials; and often relatively simple boundary conditions. In practice, however, it is common to encounter problems with finite but complex shape in which the material is neither homogeneous nor homogeneous and the boundary conditions are complex. The rapid development and easy accessibility of large-scale numerical codes in recent years are now providing engineers with numerical tools to analyze these practical problems, which are mostly three dimensional in nature and that commonly occur in engineering. Often, however, some three-dimensional problems can be reduced, as a first approximation, to two-dimensional problems for which analytical solutions exist. The utility of such twodimensional solutions lies not in their elegant analysis but in their use for physically understanding certain classes of problems and, more recently, as benchmarks for validating numerical modeling and computational procedures. In the following, we will reformulate the basic equations in two-dimensional Cartesian coordinates and provide analytical example solutions to two simple problems in terms of a polar coordinate system.

1.2.3 Basic Field Equations for the State of Plane Strain A plane state of strain is defined as the situation with zero displacement, say u 3 = 0. Equation (1.90) through (1.96) recast with this definition are σ11,1 + σ21,2 + ρ0 b1 = ρ0 u¨ 1 σ12,1 + σ22,2 + ρ0 b2 = ρ0 u¨ 2 , σ1,2 = σ2,1 , ε11 = u 1,1 ,

ε22 = u 2,2 ,

 1 − ν2

(1.97) (1.98)

ε12 =

1 2 (u 1,2 + u 2,1 ) ,



ν σ22 , E 1−ν   1 − ν2 ν ε22 = σ22 − σ11 , E 1−ν 1+ν ε12 = σ12 , E σ33 = ν(σ11 + σ22 ) . ε11 =

σ11 −

(1.99) (1.100a) (1.100b) (1.100c) (1.100d)

1.2.2 Plane Theory of Elasticity The basic field equations and boundary conditions for a three-dimensional boundary/initial value problem in linear elasticity are given in (1.90) through (1.96). Analytical (exact) solutions to idealized three-dimensional

1.2.4 Basic Field Equations for the State of Plane Stress The plane state of stress is defined with zero surface traction, say σ33 = σ31 = σ32 = 0, parallel to the

Analytical Mechanics of Solids

1 (1.101a) (σ11 − νσ22 ) , E 1 ε22 = (σ22 − νσ11 ) , (1.101b) E 1+ν ε12 = (1.101c) σ12 , E ν ε33 = − (σ11 + σ22 ) . (1.102) E In the following, we cite three classical analytical solutions to the linear boundary value problem in Eqs. (1.97) through (1.102) for specific cases that are often of interest. ε11 =

1.2.5 Infinite Plate with a Circular Hole Consider a plane problem of an infinite linearly elastic isotropic body containing a hole of radius a. Let the body be subjected to uniaxial tension, say o∞ 11 , along the x1 axis. Let the Cartesian coordinate system be located at the center of the hole and let r and θ be the corresponding polar coordinates with θ being the angle measured from the x1 axis. The state of stress near the hole is given by [1.28]  ∞ σ11 a2 σrr = 1− 2 2 r   ∞ σ a4 a2 + 11 1 − 4 2 + 3 4 cos 2θ , (1.103a) 2 r r     σ∞ σ∞ a2 a4 σθθ = 11 1 + 2 − 11 1 + 3 4 cos 2θ , 2 2 r r σrθ = −

∞ σ11

2

1+2

a2 r2

−3

 a4 r4

(1.103b)

sin 2θ .

(1.103c)

The solution for a biaxial stress state may be obtained by superposition. For a compendium of solutions of holes in isotropic and anisotropic bodies, and for shapes of holes other than circular, such as elliptical holes, see Savin [1.28].

A basic problem for a heterogeneous medium such as a composite, is that of an inclusion (or inclusions). Consider then a rigid inclusion of radius a and assume a perfect bonding between the medium and the inclusion. The solution for stresses near the inclusion due to a far-field uniaxial tension, say o∞ 11 , are   σ∞ a2 σrr = 11 1 − ν 2 2 r   ∞ σ a4 a2 + 11 1 − 2β 2 − 3δ 4 cos 2θ , (1.104a) 2 r r   ∞ ∞ σ11 σ11 a2 a4 σθθ = 1+ν 2 − 1 − 3δ 4 cos 2θ , 2 2 r r σrθ = −

∞ σ11

a2

 a4

(1.104b)

(1.104c) 1 + β 2 + 3δ 4 sin 2θ , 2 r r 2(λ + μ) μ λ+μ β=− ν=− δ= , λ + 3μ λ+μ λ + 3μ (1.104d)

where λ and μ are Lamé constants.

1.2.6 Point Load on a Semi-Infinite Plate Consider a concentrated vertical force P acting on a horizontal straight edge of a semi-infinitely large plate. The origin of a Cartesian coordinate is at the location of load application with x1 in the direction of the force. Consider again a polar coordinate of r and θ with θ being the angle measured from the x1 axis, positive in the counterclockwise direction. The state of stress is a very simple one given by 2P cos θ π r σθθ = σrθ = 0 . σrr = −

(1.105a) (1.105b)

This state of stress satisfies the natural boundary conditions as all three stress components vanish on the straight boundary, i. e., θ = π/2, except at the origin, where σrr → ∞ as r → 0. A contour integration of the vertical component of σrr over a semicircular arc from the origin yields the statically equivalent applied force P and thus all boundary conditions are satisfied. By using stress equations of transformations, which can be found in textbooks on solid mechanics, the stresses in polar coordinates can be converted into Cartesian coordinates as 2P (1.106a) cos4 θ , σ11 = σrr cos2 θ = − πa

13

Part A 1.2

x1 –x2 plane. This state shares the same stress equations of equilibrium and strain-displacement relations with the state of plane strain, i. e. (1.97) through (1.99). Equations (1.97) and (1.98) are necessary and sufficient to solve a two-dimensional elastostatic boundary value problem when only tractions are prescribed on the boundary. Thus the stresses of the plane-stress and plane-strain solutions coincide while the strains differ. The stress–strain relations for the state of plane stress are

1.2 Boundary Value Problems in Elasticity

14

Part A

Solid Mechanics Topics

Part A 1

2P (1.106b) sin2 θ cos2 θ , πa 2P σ12 = σrr sin θ cos θ = − sin θ cos3 θ . (1.106c) πa σ22 = σrr sin2 θ = −

Since strain is what is actually being measured, the corresponding strains can be computed by (1.100) for the plane-strain state or by (1.101) for the plane-stress state.

1.3 Summary It is obviously impossible in this extremely brief review even to mention all of the important subjects and recent developments in the theories of elasticity, plasticity, viscoelasticity, and viscoplasticity. For further details,

readers are referred to the many excellent books and survey articles (see for example [1.29] and [1.30]), many of which are referenced in the succeeding chapters, in each of the disciplines.

References 1.1

1.2

1.3

1.4

1.5 1.6 1.7 1.8 1.9

1.10 1.11 1.12 1.13

1.14

C. Truesdell (Ed.): Mechanics of Solids. In: Encyclopedia of Physics, Vol. VIa/2 (Springer, Berlin, Heidelberg 1972) C. Truesdell, W. Noll: The Nonlinear Field Theories of Mechanics. In: Encyclopedia of Physics, Vol. III/3, ed. by S. Flügge (Springer, Berlin, Heidelberg 1965) S.N. Atluri: Alternate stress and conjugate strain measures, and mixed foundations involving rigid rotations for computational analysis of finitely deformed solids, with application to plates and shells. I- Theory, Comput. Struct. 18(1), 93–116 (1986) S.N. Atluri: On some new general and complementary energy theorems for rate problems in finite strain, classical elastoplasticty, J. Struct. Mech. 8(1), 61–92 (1980) A.C. Eringen: Nonlinear Theory of Continuous Media (McGraw-Hill, New York 1962) R.W. Ogden: Nonlinear Elastic Deformation (Dover, New York 2001) Y.C. Fung, P. Tong: Classical and Computational Solid Mechanics (World Scientific, Singapore 2001) M. Mooney: A theory of large elastic deformation, J. Appl. Phys. 11, 582–592 (1940) A. Abel, R.H. Ham: The cyclic strain behavior of crystal aluminum-4% copper, the Bauschinger effect, Acta Metallur. 14, 1489–1494 (1966) A. Abel, H. Muir: The Bauschinger effect and stacking fault energy, Phil. Mag. 27, 585–594 (1972) G.I. Taylor, H. Quinney: The plastic deformation of metals, Phil. Trans. A 230, 323–362 (1931) R. Hill: The Mathematical Theory of Plasticity (Oxford Univ. Press, New York 1950) W. Prager: A new method of analyzing stress and strains in work-hardening plastic solids, J. Appl. Mech. 23, 493–496 (1956) H. Ziegler: A modification of Prager’s hardening rule, Q. Appl. Math. 17, 55–65 (1959)

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22 1.23

1.24 1.25

D.C. Drucker: A more fundamental approach to plane stress-strain relations, Proc. 1st U.S. Nat. Congr. Appl. Mech. (1951) pp. 487–491 S. Nemat-Nasser: Continuum bases for consistent numerical foundations of finite strains in elastic and inelastic structures. In: Finite Element Analysis of Transient Nonlinear Structural Behavior, AMD, Vol. 14, ed. by T. Belytschko, J.R. Osias, P.V. Marcal (ASME, New York 1975) pp. 85–98 S.N. Atluri: On constitutive relations in finite strain hypoelasticity and elastoplasticity with isotropic or kinematic hardening, Comput. Meth. Appl. Mech. Eng. 43, 137–171 (1984) Y. Yamada, N. Yoshimura, T. Sakurai: Plastic stressstrain matrix and its application to the solution of elastic-plastic problems by the finite element method, Int. J. Mech. Sci. 10, 343–354 (1968) K. Valanis: Fundamental consequences of a new intrinsic tune measure plasticity as a limit of the endochronic theory, Arch. Mech. 32(2), 171–191 (1980) Z. Mroz: An attempt to describe the behavior of metals under cyclic loads using a more general workhardening model, Acta Mech. 7, 199–212 (1969) O. Watanabe, S.N. Atluri: Constitutive modeling of cyclic plasticity and creep using an internal time concept, Int. J. Plast. 2(2), 107–134 (1986) E.C. Bingham: Fluidity and Plasticity (McGraw-Hill, New York 1922) K. Hohenemser, W. Prager: Über die Ansätze der Mechanik isotroper Kontinua, Z. Angew. Math. Mech. 12, 216–226 (1932) I. Finnie, W.R. Heller: Creep of Engineering Materials (McGraw-Hill, New York 1959) P. Perzyna: The constitutive equations for rate sensitive plastic materials, Quart. Appl. Mech. XX(4), 321–332 (1963)

Analytical Mechanics of Solids

1.27

O.C. Zienkiewicz, C. Corneau: Visco-plasticity, plasticity and creep in elastic solids – a unified numerical solution approach, Int. J. Numer. Meth. Eng. 8, 821–845 (1974) J.H. Argyris, M. Keibler: Incremental formulation in nonlinear mechanics and large strain elastoplasticity – natural approach – Part I, Comput. Methods Appl. Mech. Eng. 11, 215–247 (1977)

1.28 1.29 1.30

G.N. Savin: Stress Concentration Around Holes (Pergamon, Elmsford, 1961) A.S. Argon: Constitutive Equations in Plasticity (MIT Press, Cambridge, 1975) A.L. Anand: Constitutive equations for rate independent, isotropicelastic-plastic solid exhibitive pressure sensitive yielding and plastic dilatancy, J. Appl. Mech. 47, 439–441 (1980)

15

Part A 1

1.26

References

17

Materials Scie

2. Materials Science for the Experimental Mechanist

This chapter presents selected principles of materials science and engineering relevant to the interpretation of structure–property relationships. Following a brief introduction, the first section describes the atomic basis for the description of structure at various size levels. Types of atomic bonds form a basis for a classification scheme of materials as well as for the distinction between amorphous and crystalline materials. Crystal structures of elements and compounds are described. The second section presents the thermodynamic and kinetic basis for the formation of microstructures and describes the use of phase diagrams for determining the nature and quantity of equilibrium phases present in materials. Principal methods for the observation and determination of structure are described. The structural foundations for phenomenological descriptions of equilibrium, dissipative, and transport properties are described. The chapter includes examples of the relationships among physical phenomena responsible for various mechanical properties and the values of

2.1

Structure of Materials ........................... 2.1.1 Atomic Bonding ........................... 2.1.2 Classification of Materials .............. 2.1.3 Atomic Order ............................... 2.1.4 Equilibrium and Kinetics ............... 2.1.5 Observation and Characterization of Structure .................................

17 18 21 22 28

Properties of Materials .......................... 2.2.1 The Continuum Approximation ...... 2.2.2 Equilibrium Properties .................. 2.2.3 Dissipative Properties ................... 2.2.4 Transport Properties of Materials .... 2.2.5 Measurement Principles for Material Properties ..................

33 34 35 38 43

References ..................................................

47

2.2

31

46

these properties. In conclusion the chapter presents several useful principles for experimental mechanists to consider when measuring and applying values of material properties.

2.1 Structure of Materials Engineering components consist of materials having properties that enable the items to perform the functions for which they are designed. Measurements of the behavior of engineering components under various conditions of service are major objectives of experimental mechanics. Validation and verification of analytical models used in design require such measurements. All models employ mathematical relationships that require knowledge of the behavior of materials under a variety of conditions. Assumptions such as isotropy, homogeneity, and uniformity of materials affect both analytical calculations and the interpretation of experimental results. Regardless of the scale or purpose of the measurements, properties of materials that comprise the

components affect both the choice of experimental techniques and the interpretation of results. In measuring static behavior, it is important to know whether relevant properties of the constituent materials are independent of time. Similarly, measurements of dynamic behavior require information on the dynamic and dissipative properties of the materials. At best, the fundamental nature of materials, which is the ultimate determinant of their behavior, forms the basis of these models. The extent to which such assumptions represent the actual physical situation limits the accuracy and significance of results. The primary axiom of materials science and engineering states that the properties and performance of

Part A 2

Craig S. Hartley

18

Part A

Solid Mechanics Topics

Part A 2.1

a material depend on its structure at one or more levels, which in turn is determined by the composition and the processing, or thermomechanical history of the material. The meaning of structure as employed in materials science and engineering depends on the scale of reference. Atomic structure refers to the number and arrangement of the electrons, protons, and neutrons that compose each type of atom in a material. Nanostructure refers to the arrangement of atoms over distances of the order of 10−9 m. Analysis of the scattering of electrons, neutrons, or x-rays is the principal tool for measurements of structure at this scale. Microstructure refers to the spatial arrangement of groups of similarly oriented atoms as viewed by an optical or electron microscope at resolutions in the range 10−6 –10−3 m. Macrostructure refers to arrangements of groups of microstructural features in the range of 10−3 m or greater, which can be viewed by the unaided eye or under low-power optical magnification. Structure-insensitive properties, such as density and melting point, depend principally on composition, or the relative number and types of atoms present in a material. Structure-sensitive properties, such as yield strength, depend on both composition and structure, principally at the microscale. This survey will acquaint the experimental mechanist with some important concepts of materials science and engineering in order to provide a basis for informed selections and interpretations of experiments. The chapter consists of a description of the principal factors that determine the structure of materials, including techniques for quantitative measurements of structure, followed by a phenomenological description of representative material properties with selected examples of physically based models of the properties. A brief statement of some principles of measurement that acknowledge the influence of material structure on properties concludes the chapter. Additional information on many of the topics covered in the first two sections can be found in several standard introductory texts on materials science and engineering for engineers [2.1–4]. Since this introduction can only briefly survey the complex field of structure–property relationships, each section includes additional representative references on specific topics.

2.1.1 Atomic Bonding The Periodic Table The realization that all matter is composed of a finite number of elements, each consisting of atoms with a characteristic arrangement of elementary particles, be-

came widespread among scientists in the 19th and 20th century. Atomic theory of matter led to the discovery of primitive units of matter known as electrons, protons, and neutrons and laws that govern their behavior. Although discoveries through research in high-energy physics constantly reveal more detail about the structure of the atom, the planetary model proposed in 1915 by Niels Bohr, with some modifications due to later discoveries of quantum mechanics, suffices to explain most of the important aspects of engineering materials. In this model, atoms consist of a nucleus, containing protons, which have a positive electrical charge, and an approximately equal number of electrically neutral neutrons, each of which has nearly the same mass as a proton. Surrounding this nucleus is an assembly of electrons, which are highly mobile regions of concentrated negative charge each having substantially smaller mass than a proton or neutron. The number of electrons is equal to the number of protons in the nucleus, so each atom is electrically neutral. Elements differ from one another primarily through the atomic number, or number of protons in the nucleus. However, many elements form isotopes, which are atoms having identical atomic numbers but different numbers of neutrons. If the number of neutrons differs excessively from the number of protons, the isotope is unstable and either decays by the emission of neutrons and electromagnetic radiation to form a more stable isotope or fissions, emitting electromagnetic radiation, neutrons, and assemblies of protons and neutrons that form nuclei of other elements. The Periodic Table, shown in Fig. 2.1, classifies elements based on increasing atomic number and a periodic grouping of elements having similar chemical characteristics. The manner in which elements interact chemically varies periodically depending on the energy distribution of electrons in the atom. The basis for this grouping is the manner in which additional electrons join the atom as the atomic numbers of the elements increase. Quantum-mechanical laws that govern the behavior of electrons require that they reside in the vicinity of the nucleus in discrete spatial regions called orbitals. Each orbital corresponds to a specific energy state for electrons and is capable of accommodating two electrons. Electron orbitals can have a variety of spatial orientations, which gives a characteristic symmetry to the atom. Four quantum numbers, arising from solutions to the Schrödinger wave equation, governs the behavior of the electrons: the principal quantum number n, which can have any integer value from 1 to infinity; the azimuthal quantum number , which can have any integer

Materials Science for the Experimental Mechanist

value from 0 to (n − 1); the magnetic quantum number m  , which can have any integer value between − and +; and the spin quantum number m s which has values ±1/2. The Pauli exclusion principle states that no two electrons in a system can have the same four quantum numbers. As the number of electrons increases with increasing atomic number, orbitals are filled beginning with those having the lowest electron energy states and proceeding to the higher energy states. Elements with electrons in full, stable orbitals are chemically inert gases, which occupy the extreme right column of the periodic table (group 8). Electronegative elements, which occupy columns towards the right of

2.1 Structure of Materials

Part A 2.1

the periodic table, have nearly full orbitals and tend to interact with other atoms by accepting electrons to form a negatively charged entity called an anion. The negative charge arises since electrons join the originally neutral atom. Electropositive elements occupy columns towards the left on the periodic table and ionize by yielding electrons from their outer orbitals to form positively charged cations. Broadly speaking, elements are metals, metalloids, and nonmetals. The classification proceeds from the most electropositive elements on the left of the periodic table to the most electronegative elements on the right. A metal is a pure element. A metal that incorpo8A

1A 1

2

H

H

1s1 hydrogen

1s2 helium

1.008

2A

3A

4A

5A

6A

7A

3

4

5

6

7

8

9

4.003

10

Li

Be

B

C

N

O

F

Ne

[He]2s1 lithium

[He]2s2 beryllium

[He]2s22p1 boron

[He]2s22p2 carbon

[He]2s22p3 nitrogen

[He]2s22p4 oxygen

[He]2s22p5 fluorine

[He]2s22p6 neon

6.941

9.012

10.81

12.01

14.01

16.00

19.00

20.18

11

12

13

14

16

16

17

18

Na

Mg

Al

Si

P

S

Cl

Ar

[Ne]3s1 sodium

[Ne]3s2 magnesium

22.99

24.31

3B

4B

5B

6B

7B

19

20

21

22

23

24

25

26

27

K

Ca

Sc

Ti

V

Cr

Mn

Fe

[Ar]4s1 pollassium

[Ar]4s2 calcium

[Ar]4s23d1 scandium

[Ar]4s23d2 titanium

[Ar]4s23d3 vanadium

[Ar]4s13d5 chromium

[Ar]4s23d5 manganese

39.10

40.08

44.96

47.88

50.94

52.00

55.94

37

38

39

40

41

42

Rb

Sr

Y

Zr

Nb

[Kr]5s1 nubidium

[Kr]5s2 strontium

[Kr]5s24d1 yttrium

[Kr]5s24d2 zirconium

85.47

87.62

88.91

91.22

55

57

57

Cs

Ba

1

[Xe]6s casium

132.9

87

[Ne]3s23p1 aluminum

[Ne]3s23p2 silicon

[Ne]3s23p3 phosphorus

[Ne]3s23p4 sulfur

[Ne]3s23p5 chlorine

[Ne]3s23p6 argon

11B

12B

26.98

28.09

30.97

32.07

35.45

39.95

28

29

30

31

32

33

34

35

36

Co

Ni

Cu

Zn

Ga

Ge

As

Se

Br

Kr

[Ar]4s23d6 iron

[Ar]4s23d7 cobalt

[Ar]4s23d8 nickel

[Ar]4s13d10 copper

[Ar]4s23d10 zinc

55.85

58.93

58.69

63.55

65.39

69.72

72.58

74.92

78.96

79.90

43

44

45

46

47

48

49

50

51

52

53

52

Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Te

I

Xe

[Kr]5s14d4 niobium

[Kr]5s14d5 molybdenum

[Kr]5s24d5 technetium

[Kr]5s14d7 ruthenium

[Kr]5s14d8 rhodium

[Kr]4d10 palladium

[Kr]5s14d10 silver

[Kr]5s24d10 cadmium

92.91

95.94

(98)

101.1

102.9

106.4

107.9

112.4

114.8

118.7

121.8

127.6

126.9

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

La*

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

[Xe]6s2 barium

[Xe]6s25d1 lanthanum

[Xe]6s24f145d2 hafnium

[Xe]6s24f145d3 tantalum

[Xe]6s24f145d4 tungsten

[Xe]6s24f145d5 rhenium

[Xe]6s24f145d6 osmium

[Xe]6s24f145d7 iridium

[Xe]6s14f145d9 platinum

[Xe]6s14f145d10

[Xe]6s24f145d10

[Xe]6s24f145d106p1

[Xe]6s24f145d106p2

[Xe]6s24f145d106p3

[Xe]6s24f145d106p4

[Xe]6s24f145d106p5

[Xe]6s24f145d106p6

gold

mercury

thallium

lead

bismuth

polonium

astatine

radon

137.3

138.9

178.5

180.9

183.9

186.2

190.2

190.2

195.1

197.0

200.5

204.4

207.2

208.9

(209)

(210)

(222)

8B

[Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p3 [Ar]4s23d104p4 [Ar]4s23d104p5 [Ar]4s23d104p6 gallium germanium arsenic selenium bromine krypton

[Kr]5s24d105p1 [Kr]5s24d105p2 indium tin

83.80

[Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6 antimony tellurium iodine xenon

131.3

88

89

104

105

106

107

108

109

110

111

112

114

116

118

Ra

Ac ~

Rf

Db

Sg

Bh

Hs

Mt

Ds

Uuu

Uub

Uuq

Uuh

Uuo

1

[Rn] 7s francium

[Rn]7s2 radium

[Rn]7s26d1 actinium

(223)

(226)

(227)

(272)

(277)

(296)

(298)

(?)

Fr

58 Lanthanide series*

Actinide series ~

[Rn]7s25f146d2 [Rn]7s25f146d3 [Rn]7s25f146d4 [Rn]7s25f146d5 [Rn]7s25f146d6 [Rn]7s25f146d7 [Rn]7s15f146d9 rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium

(257)

59

(260)

60

(263)

61

(262)

62

(265)

63

(266)

64

(271)

65

66

67

68

69

70

71

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

[Xe]6s24f15d1 cerium

[Xe]6s24f3 praseodymium

[Xe]6s24f4 neodymium

[Xe]6s24f5 promethium

[Xe]6s24f6 samarium

[Xe]6s24f7 europium

[Xe]6s24f75d1 gadolinium

[Xe]6s24f9 terbium

[Xe]6s24f10 dysprosium

[Xe]6s24f11 holmium

[Xe]6s24f12 erbium

[Xe]6s24f13 thulium

[Xe]6s24f14 ytterbium

[Xe]6s24f145d1 lutetium

140.1

140.9

144.2

(147)

(150.4)

152.0

157.3

158.9

162.5

164.9

167.3

168.9

173.0

175.0

90

91

92

93

94

95

96

97

98

99

100

101

102

103

Th

Pa

U

Np

Pu

Am

Cm

Bk

Cf

Es

Fm

Md

No

Lr

[Rn]7s25f46d1 neptunium

[Rn]7s25f6 plutonium

[Rn]7s25f7 americium

[Rn]7s25f76d1 curium

[Rn]7s25f9 berkelium

[Rn]7s25f10 californium

[Xe]6s24f11 einsteinium

[Rn]7s25f12 fermium

[Rn]7s25f13 mendelevium

[Rn]7s25f14 nobelium

[Rn]7s25f146d1 lawrencium

(237)

(242)

(43)

(247)

(247)

(249)

(254)

(253)

(256)

(254)

(257)

[Rn]7s26d2 thorium

232.0

[Rn]7s25f26d1 [Rn]7s25f36d1 protactinium uranium

(231)

(238)

Liquids at room temperature

Gases at room temperature

Solids at room temperature

Fig. 2.1 The Periodic Table of the elements. Elements named in blue are liquids at room temperature. Elements named in red are

gases at room temperature. Elements named in black are solids at room temperature

19

20

Part A

Solid Mechanics Topics

Part A 2.1

rates atoms of other elements into its structure without changing its essential metallic character forms an alloy, which is not a metal since it is not a pure element. The major differences in materials have their origins in the nature of the bonds formed between atoms, which are determined by the manner in which electrons in the highest-energy orbitals interact with one another and by whether the centers of positive and negative charge of the atoms coincide. The work required to remove an ion from the substance in which it resides is a measure of the strength of these bonds. At suitable temperatures and pressures, all elements can exist in all states of matter, although in some cases this is very difficult to achieve experimentally. At ambient temperature and pressure, most elements are solids, some are gases, and a few are liquid. Primary Bonds Primary bonds are the strongest bonds that form among atoms. The manner in which electrons in the highest energy levels interact produces differences in the kinds of primary bonds. Valence electrons occupy the highest energy levels of atoms, called the valence levels. Valence electrons exhibit three basic types of behavior: atoms of electropositive elements yield their valence electrons relatively easily; atoms of electronegative elements readily accept electrons to fill their valence levels; and elements between these extremes can share electrons with neighboring atoms. The valence of an ion is the number of electrons yielded, accepted or shared by each atom in forming the ion. Valence is positive or negative according to whether the ion has a positive (cation) or negative (anion) charge. The behavior of valence electrons gives rise to three types of primary bonds: ionic, covalent and metallic. Ionic bonds occur between ions of strongly electropositive elements and strongly electronegative elements. Each atom of the electropositive element surrenders one or more electrons to one or more atoms of the electronegative element to form oppositely charged ions, which attract one another by the Coulomb force between opposite electrical charges. This exchange of electrons occurs in such a manner that the overall structure remains electrically neutral. To a good approximation, ions involved in ionic bonds behave as charged, essentially incompressible, spheres, which have no characteristic directionality. In contrast, covalent bonds involve sharing of valence electrons between neighboring atoms. This type of bonding occurs when the valence energy levels of the atoms are partially full, corresponding to valences in the vicinity of 4. These bonds

are strongly directional since the orbitals involved are typically nonspherical. In both ionic and covalent bonds nearest-neighbor ions are most strongly involved and the valence electrons are highly localized. Metallic bonds occur in strongly electropositive elements, which surrender their valence electrons to form a negatively charged electron gas or distribution of highly nonlocalized electrons that moves relatively freely throughout the substance. The positively charged ions repel one another but remain relatively stationary because the electron gas acting as glue holds them together. Metallic bonds are relatively nondirectional and the ions are approximately spherical. A major difference between the metallic bond and the ionic and covalent bonds is that it does not involve an exchange or sharing of electrons with nearest neighbors. The bonds in many substances closely approximate the pure bond types described above. However, mixtures of these archetypes occur frequently in nature, and a substance can show bonding characteristics that resemble more than one type. This hybrid bond situation occurs most often in substances that exhibit some characteristics of directional covalent bonds along with nondirectional metallic or ionic bonds. Secondary Bonds Some substances are composed of electrically neutral clusters of ions called molecules. Secondary bonds exist between molecules and are weaker than primary bonds. One type of secondary bond, the van der Waals bond, is due to the weak electrostatic interaction between molecules in which the instantaneous centers of positive and negative charge do not coincide. A molecule consisting of a single ion of an electropositive element and a single ion of an electronegative element, such as a molecule of HCl gas, is a simple example of a diatomic molecule. The center of negative charge of the system coincides with the nucleus of the chlorine ion, but the center of positive charge is displaced from the center of the ion because of the presence of the smaller, positively charged hydrogen ion (a single proton), which resides near the outer orbital of the chlorine ion. This results in the formation of an electrical dipole, which has a short-range attraction to similar dipoles, such as other HCl molecules, at distances of the order of the molecular dimensions. However, no long-range attraction exists since the overall charge of the molecule is zero. The example given is a permanent dipole formed by a spatial separation of centers of charge. A temporary dipole can occur when the instantaneous centers of charge separate because of the motion of electrons.

Materials Science for the Experimental Mechanist

2.1.2 Classification of Materials It is useful to categorize engineering materials in terms either of their functionality or the dominant type of atomic bonding present in the material. Since most materials perform several functions in a component, the classification scheme described in the following sections takes the latter approach. The nature and strength of atomic bonding influences not only the arrangement of atoms in space but also many physical properties such as electrical conductivity, thermal conductivity, and damping capacity. Ceramics Ceramic materials possess bonding that is primarily ionic with varying amounts of metallic or covalent character. The dominant features on the atomic scale are the localization of electrons in the vicinity of the ions and the relative incompressibility of atoms, leading to structures that are characterized by the packing of rigid spheres of various sizes. These materials typically have high melting points (> 1500 K), low thermal and electrical conductivities, high resistance to atmospheric corrosion, and low damping capacity. Mechanical properties of ceramics include high moduli of elasticity, high yield strength, high notch sensitivity, low ductility, low impact resistance, intermediate to low thermal shock resistance, and low fracture toughness. Applications that require resistance to extreme thermal, electrical or chemical environments, with the ability to absorb mechanical energy without catastrophic failure a secondary issue, typically employ ceramics. Metals Metallic materials include pure metals (elements) and alloys that exhibit primarily metallic bonding. The

free electron gas that permeates the lattice of ions causes these materials to exhibit high electrical and thermal conductivity. In addition they possess relatively high yield strengths, high moduli of elasticity, and melting points ranging from nearly room temperature to > 3200 K. Although generally malleable and ductile, they can exhibit extreme brittleness, depending on structure and temperature. One of the most useful features of metallic materials is their ability to be formed into complex shapes using a variety of thermomechanical processes, including melting and casting, hot working in the solid state, and a combination of cold working and annealing. All of these processes produce characteristic microstructures that lead to different combinations of physical and mechanical properties. Applications that require complex shapes having both strength and fracture resistance with moderate resistance to environmental degradation employ metallic materials. Metalloids Metalloids are elements in groups III–V of the periodic table and compounds formed from these elements. Covalent bonding dominates both the elements and compounds in this category. The name arises from the fact that they exhibit behavior intermediate between metals and ceramics. Many are semiconductors, that is, they exhibit an electrical conductivity lower than metals, but useable, which increases rather than decreases with temperature like metals. These materials exhibit high elastic moduli, relatively high melting points, low ductility, and poor formability. Commercially useful forms of these materials require processing by solidification directly from the molten state followed by solid-state treatments that do not involve significant deformation. Metalloids are useful in a variety of applications where sensitivity and response to electromagnetic radiation are important. Polymers Polymeric materials, also generically called plastics, are assemblies of complex molecules consisting of molecular structural units called mers that have a characteristic chemical composition and, often, a variety of spatial configurations. The assemblies of molecules generally take the form of long chains of mers held together by hydrogen bonds or networks of interconnected mers. Most structural polymers are made of mers with an organic basis, i. e., they contain carbon. They are characterized by relatively low strength, low thermal and electrical conductivity, low melting points, often high

21

Part A 2.1

The resulting attraction forms a weak bond at small distances and is typical of van der Waals bonds. The other secondary bond, the hydrogen bond, involves the single valence electron of hydrogen. In materials science and engineering, the most important type of hydrogen bond is that which occurs in polymers, which consist of long chains or networks of chemically identical units called mers. When the composition of a mer includes hydrogen, it is possible for the hydrogen atom to share its valence electron with identical mers in neighboring chains, so that the hydrogen atom is partly in one chain and partly in another. This sharing of the hydrogen atom creates a hydrogen bond between the chains. The bond is relatively weak but is an important factor in the behavior of polymeric materials.

2.1 Structure of Materials

22

Part A

Solid Mechanics Topics

Part A 2.1

ductility, and high formability by a variety of techniques. These materials are popular as electrical and thermal insulators and for structural applications that do not require high strength or exposure to high temperatures. Their principal advantages are relatively low cost, high formability, and resistance to most forms of atmospheric degradation. Composites Composite materials consist of those formed by intimate combinations of the other classes. Composites combine the advantages of two or more material classes by forming a hybrid material that exhibits certain desirable features of the constituents. Generally, one type of material predominates, forming a matrix containing a distribution of one or more other types on a microscale. A familiar example is glass-reinforced plastic (GRP), known by the commercial name of R . In this material, the high elastic modulus Fibreglass of the glass fibers (a ceramic) reinforces the toughness and formability of the polymeric matrix. Other classes of composites have metal matrices with ceramic dispersions (metal matrix composites, MMC), ceramic matrices with various types of additions (ceramic matrix composites, CMC), and polymeric matrices with metallic or ceramic additions. The latter, generically known as organic matrix composites (OMC) or polymer matrix composites (PMC), are important structural materials for aerospace applications.

2.1.3 Atomic Order Crystalline and Amorphous Materials The structure of materials at the atomic level can be highly ordered or nearly random, depending on the nature of the bonding and the thermomechanical history. Pure elements that exist in the solid state at ambient temperature and pressure always exhibit at least one form that is highly ordered in the sense that the surroundings of each atom are identical. Crystalline materials exhibit this locally ordered arrangement over large distances, creating long-range order. The formal definition of a crystal is a substance in which the structure surrounding each basis unit, an atom or molecule, is identical. That is, if one were able to observe the atomic or molecular arrangement from the vantage point of a single structural unit, the view would not depend on the location or orientation of the structural unit within the material. All metals and ceramic compounds and some polymeric materials have crystalline forms. Some ma-

terials can exhibit more than one crystalline form, called allotropes, depending on the temperature and pressure. It is this property of iron with small amounts of carbon dissolved that is the basis for the heat treatment of steel, which provides a wide range of properties. At the other extreme of atomic arrangement are amorphous materials. These materials can exhibit local order of structural units, but the arrangement a large number of such units is haphazard or random. There are two principal categories of amorphous structures: network structures and chain structures. The molecules of network structures lie at the nodes of an irregular network, like a badly constructed jungle gym. Nevertheless, the network has a high degree of connectivity and if the molecules are not particularly mobile, the network can be very stable. This type of structure is characteristic of most glasses. Materials possessing this structure possess a relatively rigid mechanical response at low temperatures, but become more fluid and deformable at elevated temperatures. Frequently the transition between the relatively rigid, low-temperature form and the more fluid high-temperature form occurs over a narrow temperature range. By convention the midpoint of this transition range defines the glasstransition temperature. Linear chain structures are characteristic of polymeric materials made of long chains of mers. Relatively weak hydrogen bonds and/or van der Waals bonds hold these chains together. The chains can move past one another with varying degrees of difficulty depending on the geometry of the molecular arrangement along the chain and the temperature. An individual chain can possess short-range order, but the collections of chains that comprise the substance sprawl haphazardly, like a bowl of spaghetti. Under certain conditions of formation, however, the chains can arrange themselves into a pattern with long-range order, giving rise to crystalline forms of polymeric materials. In addition, some elements, specific to the particular polymer, can bond with adjacent chains, creating a three-dimensional network structure. The addition of sulfur to natural rubber in the R is an example. process called Vulcanizing Materials that can exist in both the crystalline and amorphous states can also have intermediate, metastable structures in which these states coexist. Glass that has devitrified has microscopic crystalline regions dispersed in an amorphous network matrix. Combinations of heat treatment and mechanical deformation can alter the relative amounts of these structures, and the overall properties of the material.

Materials Science for the Experimental Mechanist

metric compound contains ions exactly in the ratios that produce electrical neutrality of the substance. In binary (two-component) compounds, the ratio of the number of ions of each kind present is the inverse of the ratio of the absolute values of their valences. For example, Na2 O has two sodium atoms for each oxygen atom. Since the valence of sodium is +1 and that of oxygen is −2, the 2 : 1 ratio of sodium to oxygen ions produces electrical neutrality of the structure.

Cubic

Simple

Body-centered

Tetragonal

Simple

Face-centered

Monoclinic

Body-centered

Simple

End-centered

Orthohombic

Simple Rhombohedral

Body-centered Hexagonal

Fig. 2.2 Bravais lattices and crystal systems

Face-centered

End-centered Triclinic

23

Part A 2.1

Crystal Structures of Elements and Compounds Because ionic bonds require ions of at least two elements, either metallic or covalent bonds join ions of pure elements in the solid state, although the condensed forms of highly electronegative elements and the inert gases exhibit weak short-range bonding typical of van der Waals bonds. Chemical compounds, which can exhibit ionic bonding as well as the other types of strong bonds, form when atoms of two or more elements combine in specific ratios. A stoichio-

2.1 Structure of Materials

24

Part A

Solid Mechanics Topics

Part A 2.1

In the solid state, patterns of atoms and molecules form lattices, which are three-dimensional arrays of points having the property that the surroundings of each lattice point are identical to those of any other lattice point. There are only 14 unique lattices, the Bravais lattices, shown in Fig. 2.2. Each lattice possesses three non-coplanar, non-collinear axes and a characteristic, unique array of lattice points occupied by structural units, which can be individual atoms or identical clusters of atoms, depending on the nature of the substance. The relative lengths of the repeat distance of lattice points along each axis and the angles that the axes make with one another define the seven crystal systems. Figure 2.2 also shows the crystal system for each of the Bravais lattices. Each of the illustrations in Fig. 2.2 represents the unit cell for the lattice, which is the smallest arrangement of lattice points that possesses the geometric characteristics of the extended structure. Repeating one of the figures in Fig. 2.2 indefinitely throughout space with an appropriately chosen structural unit at each lattice point defines a crystal structure. Lattice parameters include the angles between coordinate axes, if variable, and the dimensions of the unit cell, which contains one or more lattice points. To determine the number of points associated with a unit cell, count 1/8 for each corner point, 1/2 for each point on a face, and 1 for each point entirely within the cell. A primitive unit cell contains only one lattice point (one at each corner). The coordination number Z is the number of nearest neighbors to a lattice point. One of the most important characteristics of crystal lattices is symmetry, the property by which certain rigid-body motions bring the lattice into an equivalent configuration indistinguishable from the initial configuration. Symmetry operations occur by rotations about an axis, reflections across a plane or a combination of rotations, and translations along an axis. For example, a plane across which the structure is a mirror image of that on the opposite side is a mirror plane. An axis about which a rotation of 2π/n brings the lattice into coincidence forms an n-fold axis of symmetry. This characteristic of crystals has profound implications on certain physical properties. The geometry of the lattice provides a natural coordinate system for describing directions and planes using the axes of the unit cell as coordinate axes and the lattice parameters as units of measure. Principal crystallographic axes and directions are those parallel to the edges of the unit cell. A vector connecting two lattice points defines a lattice direction. A set of

three integers having no common factor that are in the same ratio as the direction cosines, relative to the coordinate axes, of such a vector characterizes the lattice direction. Square brackets, e.g., [100], denote specific crystallographic directions, while the same three integers enclosed by carats, e.g., 100, describe families of directions. Directions are crystallographically equivalent if they possess an identical arrangement of lattice points. Families of directions in the cubic crystal system are crystallographically equivalent, but those in noncubic crystals may not be because of differences in the lattice parameters. The Miller indices, another set of three integers determined in a different manner, specify crystallographic planes. The notation arose from the observation by 19th century crystallographers on naturally occurring crystals that the reciprocals of the intercepts of crystal faces with the principal crystallographic axes occurred in the ratios of small, whole numbers. To determine the Miller indices of a plane, first obtain the intercepts of the plane with each of the principal crystallographic directions. Then take the reciprocals of these intercepts and find the three smallest integers with no common factor that have the same ratios to one another as the reciprocals of the intercepts. Enclosed in parentheses, these are the Miller indices of the plane. For example, the (120) plane has intercepts of 1, 1/2, and ∞, in units of the lattice parameters, along the three principal crystallographic directions. Families of planes are those having the same three integers in different permutations, including negatives, as their Miller indices. Braces enclose the Miller indices of families, e.g., {120}. Crystallographically equivalent planes have the same density and distribution of lattice points. In cubic crystals, families of planes are crystallographically equivalent. Although there are examples of all of the crystal structures in naturally occurring materials, a relatively few suffice to describe common engineering materials. All metals are either body-centered cubic (bcc), facecentered cubic (fcc) or hexagonal close-packed (hcp). The latter two structures consist of different stacking sequences of closely packed planes containing identical spheres or ellipsoids, representing the positive ions in the metallic lattice. Figure 2.3 shows a plane of spheres packed as closely as possible in a plane. An identical plane fitted as compactly as possible on top of or below this plane occupies one of two possible locations, corresponding to the depressions between the spheres. These locations correspond to the upright and inverted triangular spaces between spheres in the figure. The same option exists when placing a third

Materials Science for the Experimental Mechanist

B-layer C-layer

Fig. 2.3 Plane of close-packed spheres

c

a

Fig. 2.4 Hexagonal close-packed unit cell

which is not a close-packed structure. Figure 2.6 shows the unit cell of this structure. The structure has a coordination number of eight and the unit cell contains two atoms. The density of a crystalline material follows from its crystal structure and the dimensions of its unit cell. By definition, density is mass per unit volume. For a unit cell this becomes the number of atoms in a unit cell n times the mass of the atom, divided by the cell volume Ω: ρ=

nA . Ω N0

25

Part A 2.1

identical plane on the second plane, but now two distinct situations arise depending on whether the third plane is exactly over the first or displaced from it in the other possible stacking location. In the first case, when the first and third planes are directly over one another, the stacking sequence is characteristic of hexagonal close-packed structures and is indicated ABAB. . . The close-packed, or basal, planes are normal to an axis of sixfold symmetry. Figure 2.4 shows the conventional unit cell for the hcp structure. Based on the hexagonal cell of the Bravais lattice, this unit cell contains two atoms. The c/a ratio is the height of the cell divided by the length of the side of the regular hexagon forming the base.√If the ions are perfect spheres, this ratio is 1.6333 = (8/3). In this instance, the coordination number of the structure is 12. However, most metals that exhibit this structure have c/a ratios different from this ideal value, indicating that oblate or prolate ellipsoids are more accurate than spheres as models for the atoms. Consequently, the coordination number is a hybrid quantity consisting of six atoms in the basal plane and six atoms at nearly the same distances in adjacent basal planes. Nevertheless, the conventional value for the coordination number of the hcp structure is 12 regardless of the c/a ratio. When the third plane in a close-packed structure occurs in an orientation that is not directly above the first, the stacking produces a face-centered cubic (fcc) structure. The sequence ABCABC. . . represents this stacking. The {111} planes are close-packed in this structure, the coordination number is 12, and the unit cell contains four atoms, as shown in Fig. 2.5. The third crystal structure typical of metallic elements and alloys is the body-centered cubic structure,

2.1 Structure of Materials

(2.1)

The mass of an atom is the atomic weight, A, divided by Avogadro’s number, N0 = 6.023 × 1023 , which is the number of atoms or molecules in one gram-atomic or gram-molecular, respectively, weight of a substance. The (8 − N) rule classifies crystal structures of elements that bond principally by covalent bonds, where N (≥ 4) is the number of the element’s group in the Periodic Table. The rule states that the element forms a crystal structure characterized by a coordination number of (8 − N). Thus, silicon in group 4 forms a crystal

Fig. 2.5 Face-centered cubic unit cell and {111} plane

26

Part A

Solid Mechanics Topics

Part A 2.1

Fig. 2.6 Unit cell of bcc structure

that has four nearest neighbors. Crystal structures based on this rule can be quite complex. Ionic compounds are composed of cations and anions formed from two or more kinds of atoms. These compounds form crystal structures based on two principles: (1) the entire structure must be electrically neutral and (2) the ions can be regarded as relatively rigid spheres of differing sizes forming a three-dimensional structure based on the efficient packing of different size spheres. The radius ratio rule is the principle that describes the crystal structures of many ionic compounds. This rule specifies the coordination number Z which gives the most efficient packing for spheres whose radius ratios are in a particular range. The radius ratio R employed for this calculation is the radius of the cation to that of the anion. Table 2.1 illustrates the relationship of this ratio to the resulting lattice geometry. Defects in Crystals The structures described in the previous sections are idealized descriptions that accurately characterize the arrangement of the vast majority of atomic and molecular arrangements in materials. Within any substance there will exist irregularities or defects in the structure that profoundly influence many of the properties of the material. These defects can exist on the electronic, atomic or molecular scale, depending on the substance. A common classification scheme for atomic and molecular defects in materials utilizes their dimensionality. Point defects have dimensions comparable to an atom or molecule. Line defects have an appreciable

extent along a linear path in the material, but essentially atomic dimensions in directions normal to that path. Surface defects have appreciable extent on a surface in two dimensions, but essentially atomic extent normal to the surface. Surface defects are regions of high local atomic disorder. This characteristic makes them prone to higher chemical activity than the regions that they bound. In addition, the local disorder associated with the boundaries has different mechanical properties than the bulk. These two features of surface defects cause them to be extremely important in affecting the mechanical and often the chemical properties of materials. Volume defects occur over volumes of several tens to several millions of atoms. The scale of these defects is generally at the mesoscale and above. The simplest kind of point defect occurs when an atom of a pure substance is missing from a lattice site. The vacant lattice site is a vacancy. Atoms of an element that leave their lattice sites and attempt to share a lattice site with another atom form interstitialcies. Clearly, interstitialcies are closely associated with vacancies. This combination typically occurs under conditions of neutron radiation, and the associated damage is a limiting factor to the use of materials in such environments. A lattice site occupied by an atom different from the host is a substitutional impurity, creating a substitutional solid solution with the host as solvent and the impurity as solute. The concentration of such impurities that a host element can tolerate is dependent in large part on the relative sizes of the impurity and the host atoms. Typically, solid solubility is extremely limited if the atomic sizes of solute and solvent differ by more than 15%. Interstitial solid solutions occur when smaller solvent atoms occupy some of the spaces (interstices) between solute atoms. Only five elements – C, N, O, H, and B – can be interstitial solutes in metals, principally because of the relative sizes of the atoms involved. The systematic or random insertion into linear polymers of mers having different chemical compositions forms copolymers. Since the substituent mers constitute a disruption in the basic polymer chain, they are substi-

Table 2.1 Radius ratios and crystal lattice geometry Z

R

2 3 4 6 8

0.000–0.155 0.155–0.225 0.225–0.414 0.414–0.732 0.732–1.000

Exact range √ 0 to 23 3 − 1 √ √ 2 1 3 √3 − 1 to √ 2 6−1 1 2−1 2 6 − 1 to √ √ 2 − 1 to 3 − 1 √ 3 − 1 to 1

Lattice geometry Linear Trigonal planar Tetrahedral Octahedral Cubic

Materials Science for the Experimental Mechanist

the region of the crystal that has experienced slip from that which has not. This boundary forms a linear crystal defect called a dislocation. As a dislocation passes over its slip plane, one part of the crystal moves relative to the other by a lattice vector, the Burgers vector, which is generally, but not always, one atomic spacing in the direction of the highest density of atoms. Dislocations are an important source of internal stress in crystalline materials and their motion is the principal mechanism of permanent deformation. Further information on the behavior of dislocations can be found in standard references [2.5, 6]. Homogeneous crystalline materials generally consist of an aggregate of grains, which are microscopic crystalline regions having differing orientations. That is, the principal crystallographic directions in each grain have different orientations relative to an external coordinate system than corresponding directions in neighboring grains. Grain boundaries, which are surfaces across which grain orientations change discontinuously, separate the grains. Since the grain boundaries contain most of the atomic disorder that accommodates the orientation change, they are surface defects. Grain boundaries permeate the material, forming a three-dimensional network that has a shape and topology determined by its thermomechanical history. Close-packed crystal structures, such as facecentered cubic and hexagonal close-packed, exhibit another form of surface defect called a stacking fault. This occurs when local conditions of deformation or crystal growth produce a stacking sequence that is locally different from that for the crystal structure. For example, suppose the stacking sequence of closepacked planes in the sequence ABC|ABC. . . is locally disrupted across the surface indicated by the vertical line so that to the left of the line the crystal is shifted to cause A planes to assume B stacking, B planes to assume C stacking and C planes to assume A stacking. Then the stacking sequence would be ABC|BCA. . . This is topologically equivalent to the removal of a plane of A stacking, creating a three-layer hcp structure while preserving the face-centered cubic structure on either side of the fault plane. Such a configuration is an intrinsic stacking fault and is associated with the formation and motion of certain kinds of dislocations in face-centered cubic crystals. Materials can also consist of an aggregate of regions having differing chemical composition and crystal structure from one another. Each region that is chemically and physically distinct and separated from the rest of the system by a boundary is called a phase. Each

27

Part A 2.1

tutional point defects in the same sense as solute atoms in a solid solution. The semiconducting properties of metalloids are due to the thermal excitation of electrons from filled valence bands into sparsely populated conduction bands. In a pure substance, this behavior is intrinsic semiconductivity. The addition of small amounts of impurities (of the order of one impurity atom per million host atoms) that have a different chemical valence significantly modifies the electrical conductivity of many such substances. These substitutional impurities, which are electronic defects as well as lattice defects, contribute additional electronic energy states that can contribute to conductivity. This is extrinsic semiconductivity. A donor impurity has more valence electrons than the host does, and contributes electron energy states. Acceptor impurities have a lower valence than the host, leaving holes, which can contribute to the transport of electrical charge, in the valence energy band. The holes are defects in the electronic structure having equal and opposite charges to the electrons. Point defects in ionic compounds must maintain electrical neutrality. One or more defects possessing equal and opposite charges must be present to cancel any local charge caused by the addition or removal of ions from lattice sites. One such defect is the Frenkel defect, a cation vacancy associated with a nearby cation interstitial. The excess charge created by the interstitial cation balances the charge deficiency due to the vacancy. In principle, such defects are possible with anions replacing cations, but the relatively large size of anions generally precludes the existence of anion interstitials. The other important type of point defect in ionic materials is the Schottky defect, which is a cation vacancy–anion vacancy pair where the ions have equal and opposite charges. Clearly, the removal of an electrically neutral unit maintains local electrical neutrality. Both types of defects can affect electrical and mechanical properties of the substance. Permanent deformation that preserves the number of lattice sites in crystalline materials generally occurs by slip, also called glide, which is the motion of one part of the crystal relative to the other across a densely populated atomic plane, the slip plane, in a densely populated atomic direction, the slip direction. For energetic reasons this motion does not occur simultaneously across the entire crystal, but commences at a free surface or region of high internal stress and propagates over the atomic plane until it either passes out of the crystal or encounters a barrier to further motion. At any instant during this propagation, a boundary separates

2.1 Structure of Materials

28

Part A

Solid Mechanics Topics

Part A 2.1

phase can contain a network of grain boundaries. However, the interphase boundaries that separate the phases are surface defects that differ from grain boundaries, since the regions they separate differ in not only spatial orientation and chemical composition but also, in most cases, in crystal structure. A multiphase material has a system of interphase boundaries whose extent and topology depend not only on the thermomechanical history of the material but also on the relative amounts of the phases present. Cracks and voids caused by solidification are examples of volume defects. These often arise during processing and can be significant sources of mechanical weakness and vulnerability to environmental attack of the material. Gases formed during fission of nuclear fuel can also form internal voids during operation, causing swelling and distortion on a macroscopic scale.

2.1.4 Equilibrium and Kinetics Thermodynamic Principles The overall composition and processing history determine the arrangements of groups of atoms in the microstructure. The resulting structure of the material determines its response to many conditions of measurement. Concepts of thermodynamic equilibrium, the rate of approach to equilibrium, and the response of the material to its physical and mechanical environment govern relationships that connect composition, processing, and microstructure. A thermodynamic system is any collection of items separated from their surroundings by a real or imaginary interface, which can be permeable or impermeable to the exchange of energy. Thermodynamics is the study of the laws that govern

1. the exchange of energy between the system and its surroundings 2. the energy content of such a system 3. the capacity of the system to do work and 4. the direction of heat flow in the system. The sum of the kinetic and potential energies of all components of the system is the internal energy U, which measures the energy content of a system. The first law of thermodynamics relates the rate of change of internal energy of a system to the rate of heat exchange with the surroundings and the rate of work done by external forces U˙ = Q˙ − P ,

(2.2)

where Q˙ represents the rate of heat exchanged with the surroundings, P is the power exerted by external forces, and the superposed dot indicates the derivative with respect to time. The second law of thermodynamics states that the rate of production of entropy S at an absolute temperature T , defined as Q˙ (2.3) , T must be ≥ 0. For a real system, changes in S refer to changes from a standard temperature and pressure. The internal energy less the product of the volume of the system V and the pressure on the system, P, is the heat content or enthalpy H. The relationship connecting these thermodynamic state variables is S˙ =

H = U + PV .

(2.4)

The definition of free energy, which is the capacity of the system to do work, differs when dealing with gaseous and condensed systems of matter. The Gibbs free energy G employed for condensed systems is related to other thermodynamic variables by G = H − TS .

(2.5)

The following discussion designates free energy by F without reference to the state of the system. Changes in the free energy of a system occur when variables that define the state of the system do work on the system. Gradients in the free energy of a system with respect to these state variables are generalized thermodynamic forces characterized by adjectives that describe the nature of the variable – chemical, thermal, mechanical, electrical, etc. – giving rise to the force. Absolute equilibrium exists when these forces sum vectorially to zero, i. e., the free energy of the system is an absolute minimum. A system in equilibrium with one or more type of force, while not being in equilibrium with others, is in a state of partial equilibrium. Equilibria can be stable, metastable or unstable. If small changes in the thermodynamic forces tend to alter the state of the system from its equilibrium condition, the equilibrium is unstable. If such changes tend to restore the system to its equilibrium state, the equilibrium is stable if the original state represents an absolute minimum in the free energy of the system or metastable if the minimum is local. These concepts are illustrated in Fig. 2.7, which depicts various states of mechanical equilibrium. Although materials are rarely in thermodynamic equilibrium with their surroundings during engineering applications, the extent of deviation from its equilibrium state determines the propensity of a structure to change

Materials Science for the Experimental Mechanist

with time. Since changes in structure cause changes in properties, the suitability of a material for a particular engineering application may also change.

2.1 Structure of Materials

29

Force ΔF *

P = exp(−ΔF ∗ /kT ) ,

(2.6)

when ΔF ∗ is expressed per atom or mole of the substance. In (2.6) k is Boltzmann’s constant, the universal gas constant per atom or molecule. The Arrhenius equation expresses the rate of change of many chemical reactions and solid-state processes: r = ν0 exp(−ΔF ∗ /kT ,

m n (a)

Phase Diagrams The structure of engineering materials at the microscale generally consists of regions that are physically and chemically distinct from one another, separated from the rest of the system by interfaces. These regions are called phases. The spatial distribution and, to a limited extent, the chemical composition of phases can be altered by thermomechanical processing prior to the end use of the material. Sometimes service conditions

m y

(b)

l

m

(c)

x

Fig. 2.7 (a) Metastable, (b) unstable, and (c) stable mechanical equilibrium states for a brick (after Barrett et al. [2.7])

can cause changes in the microstructure of a material, usually with accompanying changes in properties. Knowledge of the phases present in a material at various temperatures, pressures, and compositions is essential to the prediction of its engineering properties and to assessing its propensity for change under service conditions. The phase diagram, a map of the thermodynamically stable phases present at various temperatures, pressures, and compositions, presents this information in a graphical form. The number of degrees of freedom f (≥ 0) of a system refers to the number of environmental and composition variables that can be changed without changing the number of phases in equilibrium. The phase rule relates f to the number of distinct chemical species or components C and the number of phases present Φ: f = C −Φ+2 ,

(2.7)

in which the rate of change r is the product of the frequency of attempts at change ν0 and P is the probability of a successful attempt. Figure 2.8 illustrates the process schematically for the case of a thermally activated process assisted by external stress.

ΔF

l

(2.8)

ΔF (1-3) ΔF * (1) ΔF

(1-2)

(3)

(2) A*

A(1-2)

A0

Fig. 2.8 Free energy (F) versus slipped area ( A0 ) for deformation processes

Part A 2.1

Rate of Approach to Equilibrium In order for a system that is in metastable equilibrium in its current state (state 1) to attain the (stable or metastable) equilibrium state with the next lowest free energy (state 2), passage through an unstable equilibrium state (state 3) whose free energy is higher than that of states 1 or 2 is generally required. State 3 is an activated state and constitutes a barrier to the transition of the system from state 1 to the lower free energy of state 2. The free energy difference between states 1 and state 3 ΔF (1−3) must be supplied by internal and external sources in order to effect the transition from the metastable state to the activated state, from which subsequent transition to the next lower energy state is assumed to be spontaneous. The nature of the barrier determines the free energy difference. The free energy ΔF (1−3) contains both thermal energy and work done by external fields during activation. The difference between ΔF (1−3) and the work done by external sources is called the free energy of activation ΔF ∗ . The Boltzmann factor P is the probability that the system will change its state from state 1 to state 3

30

Part A

Solid Mechanics Topics

Part A 2.1

where 2 refers to temperature and pressure. The degrees of freedom in phase diagrams of condensed systems at constant pressure (isobaric diagrams) is given by a form of equation (2.6) in which 2 is replaced by 1, since the pressure is fixed. Either the weight percent or atomic percent of the components specifies the composition of a multicomponent system. In many engineering applications, weight percent is preferred as a more practical guide to the relative amounts of components. In scientific studies of the nature of diagrams formed from elements having similar chemical characteristics, atomic fraction is preferred. For an ncomponent system, the weight percent of component i can be calculated from its atomic percent by the relation (a/o)i (at.wt.)i × 100% , (2.9) (w/o)i = n  (a/o) j (at.wt.) j j=1

where (a/o) and (w/o) are the atomic percent and weight percent, respectively, and at.wt. refers to the atomic weight of the subscripted substance. The summation extends over all components. A similar expression relates the atomic percent of a component to the weight percents and atomic weights of the components. It is impossible to represent a phase diagram in three dimensions for a material consisting of more than three components. More commonly, diagrams that show the stable phases for a two-component (binary) system at one atmosphere pressure and various temperatures are used as guides to predict the stable phases. This permits a two-dimensional map with temperature as the ordinate and composition as the abscissa to represent the possible structures. ASM International offers an extensive compilation of binary phase diagrams of metals [2.8]. A similar compilation is available for ceramic systems [2.9]. Three-component (ternary) phase diagrams are more difficult to represent since they require two independent compositional variables in addition to temperature. The most common three-dimensional representation is a prism in which the component compositions are plotted on an equilateral triangle base and the temperature on an axis normal to the base. Ternary diagrams of many metallic systems have been determined are available as a compilation similar to that for the binary systems [2.10]. Phase diagrams not only give information on the number and composition of phases present in equilibrium, but also provide the data necessary to calculate the relative amounts of phases present at any temperature. Figure 2.9 illustrates this principle, the inverse lever law, using the silver–copper binary diagram as an example.

From this diagram we note that an alloy of 28.1 w/o Cu solidifies at 779.1 ◦ C into a solid consisting of two solid phases: a silver-rich phase α containing 8.8 w/o Cu and a copper-rich phase β containing 92.0 w/o Cu. This alloy, which passes directly from the liquid state to the solid state without going through a region of solid and liquid in equilibrium, is a eutectic. To determine how much of each phase is present in the eutectic alloy at the solidification temperature, we note that, while the overall composition of the alloy c0 is 28.1 w/o Cu, it is a mixture of copper- and silver-rich phases, each having a known composition. Performing a mass balance on the amount of copper in the alloy yields the result:   cβ − c0  × 100% (%)α =  cβ − cα (92 − 28.1) × 100% = 76.8% (2.10) = (92 − 8.8) for the weight percentage of the silver-rich phase in the alloy. The inverse lever law gets its name from the fact that the numerator in (2.10) is the distance on the composition axis from the overall alloy composition to the composition of the copper-rich phase cβ on the opposite side of the diagram from the silver-rich phase whose percentage is to be determined. The denominator is the entire distance between the compositions of the copperrich and silver-rich phases. The calculation is analogous to balancing the relative amounts of the phases on a lever whose fulcrum is at the overall alloy composition. While the numerical result shown illustrates the calculation for the eutectic alloy and temperature, the principle applies to any constant temperature as long as we choose the appropriate overall alloy composition Temperature (°C) 1200 1100 Liquid 1000 961.93 °C 900 800 8.8 % 28.1 % 700 (Ag) 600 500 400 300 200 Ag 10 20 30 40 50

1084.87 °C

(Cu) 779.1°C 92 %

60

70

80 90 Cu wt % copper

Fig. 2.9 Silver–copper phase diagram at 1 atm pressure

Materials Science for the Experimental Mechanist

and the compositions of the phases in equilibrium at that temperature.

2.1.5 Observation and Characterization of Structure

Metallographic techniques Metallographic approaches to studying the structure of materials are either surface techniques or transmission techniques. Surface techniques rely on images formed by the reflection of a source of illumination by a prepared surface. The wavelength of the illuminating radiation and the optical properties of the observation system determine the resolution of the technique. Optical metallography employs a reflecting microscope and illumination by visible light to examine the polished and etched surface of a specimen either cut from bulk material or prepared on the surface of a sample, taking care not to introduce damage during the preparation. Consequently, the examination is generally a destructive technique. Magnifications up to approximately 1200 × are possible with careful specimen preparation. Properly chosen etches reveal the arrangement of grain boundaries, phases and defects intersected by the plane of polish. Contrast arises from the differing reflectivities of the constituents of the microstructure. Interpretation of micrographs to relate the observed structure to the properties and behavior of the material requires experience and knowledge of the composition and thermomechanical history of the material. Scanning electron microscopy (SEM) employs a beam of electrons as an illuminating source [2.11]. The de Broglie wavelength of an electron is λ = h/ p, where h is Planck’s constant and p is the momentum of the electron. The de Broglie wavelength of electrons varies with the accelerating voltage V since for an electron eV = p2 /2m, where e and m are the charge and mass, respectively, of the electron. Electromagnetic lenses use the charge on the electron beam to condense, focus, and magnify it. Magnifications ob-

tained by this technique range from those characteristic of optical microscopy to nearly 100 times the best resolution obtained from optical techniques. Contrast among microconstituents arises because of their differing scattering powers for electrons. Since electron optics permits a much greater depth of field than is possible with visible light, SEM is the observation tool of choice for examining surfaces with a high degree of spatial irregularity, such as those produced by fracture. While the images are nearly always two-dimensional sections through three-dimensional structures, stereo microscopy is widely used in the examination of fracture surfaces. Although this method can employ an appreciable range of wavelengths, it is necessary to operate in a vacuum and to provide a means of dissipating any electrical charge on the specimen induced by the electron beam. In recent years, the development of special environmental chambers has permitted the observation of nonconductive materials at atmospheric pressure. Transmission metallographic techniques produce images that are the projection of the content of the irradiated volume on the plane of observation. Clearly, such techniques rely on the transparency of the material to the illumination. This requires the preparation of thin sections of the material, followed by treatments to improve the contrast of structural features, then observed by transmission microscopy. Visible light generally suffices as the illuminating source for biological materials. Since most engineering materials are not transparent to visible light, observation by transmitted radiation is less common for these materials. The following section discusses both of the principal exceptions, transmission electron microscopy (TEM) and x-ray radiography. Information obtained from metallographic observations serves a variety of purposes: to determine the microstructural state of a material at various stages of processing, to reveal the condition of material in a failed engineering part, and to obtain a semiquantitative estimate of some types of material behavior. All cases where the images represent plane sections require analysis by stereological techniques to obtain quantitative information that for relating structure to properties [2.12]. For a microstructure containing two or more microconstituents, such as phases, inclusions or internal cavities, the simplest parametric characterization is the volume fraction VV (n) of the n-th microconstituent. It is easily shown that this is also equal to the area fraction A A (n) of the n-th constituent on an observation plane, which can easily be measured on a polished metallographic specimen [2.12].

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Part A 2.1

In order to implement in the selection and design of materials and processes the axiom that structure at various levels determines the properties of materials, it is necessary to have means of examining, measuring, and describing quantitatively the structure at various scales. The information so obtained then leads to the development, validation, and verification of models that describe the behavior of engineering structures. The following sections describe two types of characterization techniques that are useful for this purpose.

2.1 Structure of Materials

32

Part A

Solid Mechanics Topics

Part A 2.1

Another common measurement of microstructure is the mean linear intercept Λ(t) or its reciprocal PL (t), the intercept density, of a test line parallel to the unit vector t with the traces of internal boundary surfaces on an observation plane. The intercept density is the number of intersections per unit length of the test line with the feature of interest. Since microstructures are generally anisotropic PL (t) depends on the orientation of the test line. The volume average of all measurements is an average intercept density, PL ; its reciprocal is the mean linear intercept Λ. The grain size refers to the mean linear intercept of test lines with grain boundaries. When employing this terminology, remember that the measurement is actually the mean distance between grain boundaries in the plane of measurement and may not apply to the three-dimensional structure. In addition, the concept of size implies an assumption of shape, which is not included in the measurement. Nevertheless Λ remains the most widely used quantitative characterization of microstructure in the materials literature. Similar measurements can be made of the intersections of test lines with interphase boundaries and internal cavities, or pores. The associated mean linear intercepts are phase diameters or pore diameters, respectively. The intercept density for a particular class of boundaries also measures the total boundary area per unit volume SV  through the fundamental equation of stereology: SV  = 2 PL  .

(2.11)

This quantity is important when studying properties influenced by phenomena that occur at internal boundaries. Diffraction Techniques In an irradiated array of atoms, each atom acts as a scattering center, both absorbing and re-radiating the incident radiation as spherical wavelets having the same wavelength as the incident wave. When the wavelength of the incident radiation is comparable to the spacing between atoms, interference effects can occur among the scattered wavelets. A crystalline material acts as a regular array of scattering centers, and produces a pattern of scattered radiation that is characteristic of the array and the wavelength of the radiation. Amorphous materials also produce scattering, but the lack of long-range atomic order precludes the development of patterns of diffracted beams. Diffracted rays occur by the constructive interference of radiation reflected from atomic planes. The model proposed by W. H. Bragg and W. L. Bragg, illustrated in Fig. 2.10, describes the phenomenon.

Radiation of wavelength λ is incident at an angle θ on parallel atomic planes separated by a distance d. Scattered radiation from adjacent planes interferes constructively, producing a scattered beam of radiation, when the path difference between rays scattered from adjacent planes is an integral number n of wavelengths. The relationship describing this phenomenon is Bragg’s law, nλ = 2d · sin θ ,

(2.12)

which is the basis for all measurements of the atomic dimensions of crystals. X-rays, a highly energetic electromagnetic radiation having a wavelength the same order of magnitude as the spacing between atoms in crystals, inspired the original derivation of (2.12) [2.13, 14]. If the crystal is sufficiently thin to permit diffracted beams to penetrate it, transmission diffraction patterns occur on the side of the crystal opposite to the incident radiation. However, even if the diffracted beams cannot penetrate the crystal, back-reflection diffraction patterns occur on the same side as the incident beam. Crystal structure analysis employs both types of patterns. One method of identifying unknown materials employs analysis of diffraction patterns formed by radiation of known wavelength to determine the lattice parameter and crystal structure of an unknown substance, which is then compared with a database of known substances for identification [2.15, 16]. Both incident and diffracted X-radiation can penetrate substantial thicknesses of many materials, rendering this diffraction technique useful for determining the condition of material in the interior of an engineering part. Lattice distortion giving rise to internal stress, averaged over the irradiated volume, can be determined by comparing the interplanar spacings of a material containing internal stress with those of a material free of internal stress [2.17]. This technique of x-ray stress analysis is extremely valuable in deter-

θ

θ θ d

Fig. 2.10 Derivation of Bragg’s law

Materials Science for the Experimental Mechanist

trons and neutrons, which have de Broglie wavelengths comparable to the interatomic spacing of the crystals. Electron beams are employed in both transmission and back-reflection modes to reveal information on the crystal structure and defect content of materials. The fact that electron beams possess electrical charge makes it possible to employ electromagnetic lenses to condense, focus, and magnify them. Transmission electron microscopy (TEM) uses diffracted electron beams transmitted through material and subsequently magnified to reveal features of atomic dimensions [2.21]. Since electron beams are highly absorbed by most materials, specimens examined in TEM are only a few thousand atoms thick. Nevertheless, this technique reveals much useful information about the nature and behavior of dislocations, grain boundaries and other atomic-scale defects. Orientation imaging microscopy (OIM) [2.22] employs back-reflection electron diffraction patterns to form maps called pole figures by determining the orientation of individual grains. These maps describe the orientation distribution of grains in a material, which is a major cause of the anisotropy of many bulk properties. The greater penetrating power of neutron beams reveals information from greater thicknesses of material than x-rays [2.23]. Since these beams are not electrically charged, their magnification by electromagnetic lenses is not possible. Therefore, their principal use is for measurements of changes in lattice spacings due to internal stresses, and not for forming images. In addition to revealing the structure of materials, many advanced techniques reveal not only the structure but also the chemistry of a substance at the microscale. While techniques such as the three-dimensional (3-D) atom probe, Auger electron spectroscopy and imaging, Z-contrast microscopy, and many others complement those for structural examination and often employ similar physical principles; a complete survey is beyond the scope of this review.

2.2 Properties of Materials Material properties are attributes that relate applied fields to response fields induced in the material. Applied and induced fields can be of any tensorial rank [2.24, 25]. Fields are conjugate to one another if the applied field does an incremental amount of work on the system by causing an incremental change in the induced field. External fields and the associated field variables include temperature T , stress τ, and electrical (E) and

magnetic (H) fields; their respective conjugate fields are entropy S, strain ε, electrical displacement D, and magnetic induction B. Cross-effects occur when changes in an applied field cause changes in induced fields that are not conjugate to the applied field. Consider a system subjected to a cyclic change of applied field variables such that each variable changes arbitrarily while the others are constant, then the pro-

33

Part A 2.2

mining residual stresses in heat-treated steel parts, for example. It has the advantage that lattice strain, which is the source of internal stress, is the direct result of the measurement. This is in contrast to the calculation of stress from measurements of strain on the external surface of a material, which include components due to both lattice strain and the stress-free, permanent strain caused by dislocation motion. When stress is calculated from such surface measurements, the assumption that dislocation motion is absent or negligible is implicit [2.18]. Diffraction contrast from scattered x-rays forms the basis for x-ray topography of crystals, useful in the study of the defect structure of highly perfect single crystals used in electronic applications [2.19]. Magnified diffraction spots from crystals reveal images of the structure of the material with a resolution comparable to the wavelength of the radiation. Small changes in the orientation and spacing of atomic planes due to the presence of internal defects change the diffraction conditions locally, creating contrast in the image of the diffracted beam. The availability of high-intensity xray sources from synchrotrons has greatly increased the applicability and usefulness of this technique. Radiography also forms images by the interaction of material with x-rays, but in this case, the contrast mechanism is by the differential absorption of radiation caused by varying thicknesses of material throughout an irradiated section as well as by regions having differing densities, which affects the ability of material to absorb x-rays [2.20]. Radiography can reveal macroscopic defects and heterogeneous distributions of phases in materials. Since contrast is not due to diffraction by the structure, this process cannot provide information on its crystal structure or the state of lattice strain in the material. The principles of diffraction apply not only to electromagnetic radiation but also to the scattering of highly energetic subatomic particles, such as elec-

2.2 Properties of Materials

34

Part A

Solid Mechanics Topics

Part A 2.2

cess reverses until the variables have their initial values. If such a closed cycle occurs under conditions such that the system is continuously in equilibrium with its surroundings, the thermodynamic states at the beginning and the end of the process will be the same and no net work is done. Properties that relate conjugate fields in this case are equilibrium properties and depend only on the thermodynamic state of a system, not its history. A caloric equation of state relates conjugate fields, which permits the description of the relationship between applied and induced fields in terms of appropriate derivatives of a thermodynamic potential. If processes that occur during the cycle dissipate energy, they produce entropy and the system is not in the same state in the initial and final conditions. Although the applied field variables are the same at the beginning and end of the cycle, the conjugate induced variables do not return to their original values. The change of thermodynamic state occurs because of irreversible changes to the structure of the material caused, at least in part, by changes in the internal structure of the system that occur during the cyclic change. Associated material properties are dissipative and exhibit hysteresis. Dissipative properties are dependent on time and the history of the system as well as its current thermodynamic state. However, in some cases it is possible to relate these properties to appropriate derivatives of a complex dissipative potential [2.24]. In general, equilibrium properties are structure insensitive, while dissipative properties are structure sensitive. The search for additional state variables that depend on the internal structure of the material is a subject of much continuing research. Transport processes are dynamic processes that cause the movement of matter or energy from one part of the system to another. These produce a flux of matter or energy occurring in response to a conjugate thermodynamic force, defined as the gradient of a thermodynamic field [2.26]. Attributes that connect fluxes and forces are generally structure-insensitive material properties.

2.2.1 The Continuum Approximation Continuity The complex nature of real materials requires approximation of their structure by a mathematical concept that forms a basis for calculations of the behavior of engineering components. This concept, the continuous medium, replaces discrete arrays of atoms and molecules with a continuous distribution of matter. Fields and properties defined at every mathematical

point in the medium are continuous, with continuous derivatives, except at a finite number of surfaces separating regions of continuity [2.24]. The process that defines properties at a point in the medium consists of taking the limit of the volume average of the property over increasingly smaller volumes. For sufficiently large volumes the average will be independent of sample size. However, as the volume sampled decreases below a critical range, the volume average will exhibit a dependence on sample size, which is a function of the nature of the property being measured and the material. The smallest sample volume that exhibits no size dependence defines the continuum limit for the material property. The continuum approximation assigns the value of the size-independent volume average of the property to a point at the centroid of the volume defined by the continuum limit. It follows that, when the size of a sample being tested approaches the continuum limits of relevant properties, results can be quite different from those predicted by a continuum model of the material. This limit depends on the material, its structural state, and the property being measured, so that all need to be specified when assigning a value to the continuum limit. The continuum approximation employs the concept of the material particle to link the discrete nature of real materials to the continuum. By the argument above, the material attributes of such a particle must be associated with a finite volume, the local value of the continuum limit for the attributes. As noted earlier, continuum limits may differ for different properties; consequently the effective size associated with a material particle will depend on the property associated with the continuum. Coordinate systems based on material particles, in contrast to those based on the crystal lattice, must necessarily be continuous by the assumption of global continuity of the medium. Despite the inherent differences in concepts of material structure, it is possible to develop models of many material properties based on atomic or molecular characteristics that interface well with the continuum model at the scale of the continuum limit. These limits can only be determined experimentally or by the comparison of properties calculated from physically based models of atomic arrays of various sizes. Such models form essential links between continuum mechanics and the knowledge of structure provided by materials science. Homogeneity The continuous medium used for calculation of the behavior of engineering structures also possesses the

Materials Science for the Experimental Mechanist

Isotropy The usual model of a homogeneous continuous medium also assumes that properties of the material are isotropic, i. e., they are independent of the direction of measurement. This assumption generally results in the minimum number of independent parameters that describe the property and simplifies the math-

ematics involved in analysis. However, isotropy is infrequently observed in real materials, and neglect of the anisotropic nature of material properties can lead to serious errors in estimates of the behavior of engineering structures. Properties of materials possess symmetry elements related to the structure of the material. The following Gedanken experiment illustrates the meaning of symmetry of a property [2.25]. Measure the property relative to some fixed set of coordinate axes. Then operate on the material with a symmetry operation. If the value of the property is unchanged, then it possesses the symmetry of the operation. Neumann’s principle states that the symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal. Notice that the principle does not require that the crystal and the property have the same symmetry elements, just that the symmetry of the property contains the symmetry elements of the crystal. Since an isotropic property has the same value regardless of the direction of measurement, it contains all possible point groups. As mentioned above, bulk properties of polycrystalline and multiphase materials are appropriate averages of the properties of their microconstituents, taking account of the distribution of composition, size, and orientation. A common source of bulk anisotropy in a material is the microscopic variation in orientation of its grains or microconstituents. An orientation distribution function (ODF) describes the orientation distribution of microscopic features relative to an external coordinate system. Depending on the feature being measured, quantitative stereology or x-ray diffraction analyses provide the information required to determine various types of ODFs. Used in conjunction with the properties of individual crystals of constituent phases, ODFs can describe the anisotropy of many bulk properties, such as the elastic anisotropy of metal sheet formed by rolling.

2.2.2 Equilibrium Properties Although the definition of equilibrium properties is phenomenological, without reference to the structure of the material, values of the properties depend on the structure of materials though models described in treatises on condensed-matter physics and materials science [2.27]. Properties described in the following sections are all first order in the sense that they arise from an assumed linear relationship between applied and induced fields. However, most properties have measurable de-

35

Part A 2.2

property of homogeneity, that is, properties at a material point do not depend on the location of the point in the medium. While this assumption is reasonably accurate for single-phase materials of uniform composition, it no longer applies to multiphase materials at the microscale. In the latter case, properties of the bulk depend on appropriate averages of the properties of the constituent phases, which include parameters such as the volume fraction of each phase, the average size of the particles of each phase, and in some cases the orientation of the phases relative to one another and to externally applied fields. Higher order models of multiphase materials treat each phase as a distinct continuum with properties distinct from other phases. Physical fields within the body are also influenced by the nature, orientation, and distribution of the phases present. Average values of induced internal fields determined from the external dimensions of a body and fields applied to the surfaces may not accurately reflect the actual conditions at points in the interior of the body due to different responses from constituent phases within the material. Material properties determined from the bulk response to these mean fields may differ significantly from similar properties measured on homogeneous specimens of the constituent phases. Measurements of local fields rely on techniques that sample small, finite volumes to determine local material response, then inferring the values of the fields producing the response. Since a quantitative determination of variables that determine the bulk properties of multiphase materials is often difficult, time-consuming, and expensive, experimental measurements on bulk specimens typically determine these properties. It is important to realize that applying the results of such measurements implicitly assumes that differences in structure of the material at the microscale have a negligible effect on the resulting properties. Structural variability due to inhomogeneity can be taken into account experimentally by determining properties of the same material with a range of structures and assigning a quantitative value to the uncertainty in the results.

2.2 Properties of Materials

36

Part A

Solid Mechanics Topics

Part A 2.2

pendencies on other applied fields. Such field-induced changes cause second-order effects, such as optical birefringence and the electro-optical effect, where firstorder properties depend on strain and electrical field, respectively. For compactness, this review will not consider these effects, although many of them can provide the basis for useful measurements in experimental mechanics. The usual description of equilibrium properties employs conjugate effects. In principle, any change in an applied field produces changes in all induced fields, although some of these effects may be quite small. Figure 2.11, adapted from Nye [2.25] and Cady [2.28], illustrates the interrelationship among first-order effects caused by applied and induced fields. Applied field variables appear at the vertices of the outer tetrahedron, while the corresponding vertices of the inner tetrahedron represent the conjugate induced field variables. Lines connecting the conjugate variables represent principal, or direct, effects. Specific Heat For a reversible change at constant volume, an incremental change in temperature dT produces a change of entropy dS:  ε C (2.13) dT , dS = T

where C ε is the heat capacity at constant strain (volume) and T is the absolute temperature. To maintain constant volume during a change in temperature, it is necessary to vary the external tractions on the surface of the body (or the pressure, in the case of a gas). When measurements are made at constant stress (or pressure), the volume varies. The heat capacity under these conditions is C τ . Specific heat is an extensive, isotropic property.

τ H B

D E

ε

S T

Fig. 2.11 Relationships among applied and induced fields

Electrical Permittivity A change in electrical field dE produces a change in electrical displacement dD:

dDi = κij dE j ,

(2.14)

where the suffixes, which range from 1 to 3, represent the components of the vectors D and E, represented by bold symbols, along reference coordinate axes and summation of repeated suffixes over the range is implied here and subsequently. The second-rank tensor κ is the electrical permittivity, which is in general anisotropic. Magnetic Permeability A change in magnetic field dH produces a corresponding change in magnetic induction or flux density dB:

dBi = μij dH j ,

(2.15)

where μ is the second-rank magnetic permeability tensor, which is in general anisotropic. Equations (2.14) and (2.15) apply only to the components of D and B that are due to the applied electrical or magnetic field, respectively. Some substances possess residual electrical or magnetic moments at zero fields. The integral forms of (2.14) and (2.15) must account for these effects. Hookean Elasticity An incremental change in stress dτ produces a corresponding change in strain dε according to:

dεij = sijk dτk ,

(2.16)

where s is the elastic compliance tensor. In this case, the applied and induced fields are second-rank tensors, so the compliances form a fourth-rank tensor that exhibits the symmetry of the material. The elastic constant, or elastic stiffness, tensor c, is the inverse of s in the sense that cijmn smnkλ = 1/2(δik δ jλ + δiλ δ jk ). Equation (2.16) is Cauchy’s generalization of Hooke’s law for a general state of stress. A material that obeys (2.16) is an ideal Hookean elastic material. Most engineering calculations involving elastic behavior assume not only Hooke’s law but also isotropy of the elastic constants. This reduces the number of independent elastic constants to two, i. e. cijmn = λδij δmn + μ(δim δ jn + δin δ jm ), where λ and μ are the Lamé constants. Various combinations of these constants are also employed, such as Young’s modulus, the shear modulus, the bulk modulus and Poisson’s ratio, but only two are independent in an isotropic material. The Chapter on continuum mechanics in this work lists additional relationships among the isotropic elastic constants [2.29].

Materials Science for the Experimental Mechanist

Cross-Effects All conjugate fields can be included in the description of the thermodynamic state. Define a function Φ such that

Φ = U − τij εij − E  D − Bk Hk − TS .

(2.17)

Then using the definition of work done on the system by reversible changes in the conjugate variables and the first and second laws of thermodynamics, we have dΦ = −εij dτij − D dE  − Bk dHk − S dT . (2.18) Since Φ = Φ(τ, E, H, T),     ∂Φ ∂Φ dτij + dE  dΦ = ∂τij E,H,T ∂E  τ,H,T     ∂Φ ∂Φ + dHk + dT ∂Hk E,τ,T ∂T E,H,τ

the coefficients of the form    2  ∂εij ∂ Φ = − ∂τij ∂T E,H ∂T σ,E,H   ∂S = = βijE,H ∂τij T,E,H

(2.22)

τ

ΔS = βij τij + (C /T )ΔT ,

(2.23)

where sT is the isothermal compliance tensor. Extension of this argument shows that the full matrix of matera)

Heat of deformation

Strain

Entropy

Elasticity

Heat capacity

(2.19)

Piezocaloric effect

Stress

it follows that   ∂Φ = −εij , ∂τij E,H,T   ∂Φ = −Bk , ∂Hk E,τ,T



 ∂Φ = −D , ∂E  τ,H,T   ∂Φ = −S . ∂T E,H,τ (2.20)

Further differentiation of (2.19) and (2.20) with respect to the independent variables gives relationships among

(2.21)

for the thermal expansion coefficient tensor β at constant electrical and magnetic field. Thermoelasticity illustrates cross-effects that occur among equilibrium properties. Consider the section of the tetrahedron containing only the applied field variables stress and temperature and the associated induced field variables, strain and entropy, illustrated in Fig. 2.12. In this illustration, the line connecting stress and strain represents the elastic relationships and the line connecting entropy and temperature denotes the heat capacity. The indirect, or cross-effects are the heat of deformation, the piezocaloric effect, thermal expansion, and thermal pressure, as shown in the figure. Ten independent variables specify the thermodynamic substate of the material. These can be the nine stress components and temperature, nine strain components and temperature, or either of the preceding with entropy replacing temperature. Specification of ten independent variables determines the ten dependent variables. Nye [2.25] shows that the relationships connecting the effects become: T τk + βij ΔT , εij = sijk

c s σ

Thermal expansion

Temperature

Thermal pressure

b)

f

ε α

S α

–f

37

Part A 2.2

While isotropy is a convenient approximation for mathematical calculations, all real materials exhibit elastic anisotropy. Single crystals of crystalline materials display symmetry of the elastic coefficients that reflects the symmetry of the crystal lattice at the microscale [2.25]. The single-crystal elastic constants are often expressed in a contracted notation in which the two suffixes of stress and strain components become a single suffix ranging from 1 to 6. The independent terms of the corresponding elastic constant tensor become a 6 × 6 array, which is not a tensor. This notation is convenient for performing engineering calculations, but when the elastic constants are not isotropic, changes in the components of the elastic constant matrix due to changes in the coordinate system of the problem must be calculated from the full, fourth-rank tensor description of the constants [2.25]. Anisotropic crystalline materials can display a lower elastic symmetry at the mesoscale than at the single-crystal scale due to preferred orientation of the crystallites introduced during processing.

2.2 Properties of Materials

T/C C/T T

Fig. 2.12a,b Equilibrium properties relevant to thermoelastic behavior: (a) quantity and (b) symbol

38

Part A

Solid Mechanics Topics

Part A 2.2

ial coefficients defined by relationships similar to (2.21) and written in the form of (2.22) is symmetric. The atomic basis for the linear relationships in (2.22) arises from the nature of the forces between atoms. Although the internal energy of a material is in reality a complex, many-body problem that is beyond the capabilities of even modern computers to solve, summation over all atoms of the bond energies of nearest-neighbor pairs offers a physically reasonable and computationally manageable alternative. The energy of an atom pair exhibits a long-range attraction and a short-range repulsion, resulting in a minimum energy for the pair at a distance equal to or somewhat greater than the interatomic spacing in the material. In the simplest case, a diatomic molecule of oppositely charged ions, the Coulomb force between unlike charges provides the attractive force and the mutual repulsion that arises when outer shells of electrons begin to overlap is the repulsive force. The minimum energy of the pair is called the dissociation energy, and is the energy required to separate the two ions to an infinite spacing at absolute zero. For the example cited the energy is radially symmetric, so that the energy–distance relationship is independent of the relative orientation of the ions, and the energy minimum occurs at the equilibrium interionic spacing. Figure 2.13 illustrates this behavior schematically. It follows from (2.16)–(2.20) that sijk = −

∂2Φ , ∂τij ∂τk

(2.24)

which can be expressed in terms of the strain relative to the equilibrium spacing by (2.16). Thus the elastic compliances are proportional to the curvature of the energy-distance curve of ionic pairs, measured at the equilibrium interionic distance. Extension of this model to condensed systems of atoms having more complex bonds leads to modification of the details of the relationships. The elastic constants of single crystals of single-phase cubic alloys can be estimated by an extension of this concept [2.30]. The energy–bond length relationship shown in Fig. 2.13 applies only at absolute zero. At finite temperatures, the energy of the pair is increased above the minimum by an amount proportional to kT . This implies that for any finite temperature T there exist two values of the interionic spacing with the same energy. The observed spacing at T is the average of these two. Most materials exhibit a curve that is asymmetric about r0 , the equilibrium spacing at T = 0 causing the mean spacing to change with temperature. Generally the na-

Lattice energy (eV) 15 12 9

Repulsive energy

6 3 0 –3 –6 Minimum energy

–9

Attractive energy

–12 –15

0

0.1

0.2

r0

0.4

0.5 0.6 0.7 0.8 Interionic spacing (nm)

Fig. 2.13 Energy–bond length relationship for an ionic

solid

ture of this asymmetry is such that the mean spacing increases with temperature, as illustrated in Fig. 2.13, leading to a positive coefficient of thermal expansion. Different values of the elastic compliances result from measurements under isothermal (ΔT = 0) or adiabatic (ΔS = 0) conditions. The adiabatic compliance tensor sS is determined by setting ΔS = 0 and solving for ΔT in the second of (2.22), eliminating ΔT from the first equation and defining sS as the resulting coefficients of σ. The relationships among the isothermal and adiabatic quantities are  τ C S T − sijk = −βij βk (2.25) . sijk T Since the coefficients of the thermal expansion tensor are positive for most materials, adiabatic compliances are typically smaller than isothermal compliances. Nye [2.25] gives expressions such as (2.25) relating differences between various properties with different sets of applied field variables being constant. This reasoning emphasizes the fact that the definitions of these properties include specification of the measurement conditions.

2.2.3 Dissipative Properties Equilibrium processes are reversible and conservative, whereas nonequilibrium processes are irreversible and nonconservative. In the latter cases, work done on the system dissipates in the form of heat, generating entropy. This introduces history and rate dependence in the response of the system to external fields and influ-

Materials Science for the Experimental Mechanist

depends on the conditions of measurement. Engineering standards establish boundaries between these regimes that form the basis for design calculations. For example the limit of applicability of the equations of elasticity in engineering calculations can be defined as the stress at which strain no longer obeys Hooke’s law (proportional limit) or the more easily measured offset yield stress, defined as the stress required to produce some small but easily measurable permanent strain. The boundaries are always compromises between the accuracy with which they can be measured and the intended uses of the information. Viscoelasticity When stresses or strains are independent variables applied periodically with time, the response of the corresponding dependent variable is out of phase with the applied field. This causes a delay in response, manifested by an induced field that has the same frequency, but is out of phase with the applied field. Consider a uniaxial stress τ(t) = τ0 exp(iωt) applied with frequency ω that causes a strain ε(t) = ε0 exp[i(ωt − δ)]. The complex compliance S∗ is defined as

S∗ =

ε0 ε (t) = exp (−iδ) . τ (t) τ0

(2.26)

The storage compliance S is the real component of S∗ , and the loss compliance S

is the imaginary component. These are related to the loss angle, δ, by tan(δ) = S

/S . The complex stiffness or modulus is the reciprocal of S∗ . Applying this concept to other states of stress and strain permits determination of all components of the viscoelastic constant tensor. The complex elastic constants and the loss angle are functions of the frequency of the applied field as can be noted by analysis of the above example using the theory of damped oscillations. Consider an experiment in which the component of total strain, ε0 , in a particular direction remains constant while the corresponding stress component is measured on a plane normal to the same axis. Let instantaneous and relaxed values of stress be τ0 and τ∞ , respectively. The dependence of stress on time t has the form: τ (t) = τ∞ + (τ0 − τ∞ ) exp (−t/ξε ) ,

(2.27)

if a single relaxation mechanism dominates the process. Each mechanism of relaxation possesses a characteristic relaxation time, which is the mathematical manifestation of the physical process causing the delay in response of the induced field to the applied field. The

39

Part A 2.2

ences the phenomenological material coefficients that relate applied and induced fields. The definition of properties associated with dissipative processes is formally similar to that for equilibrium properties with the caveat that history dependence affects the properties and their values under various measurement conditions. The following sections employ mechanical properties to illustrate mathematical descriptions of dissipative processes and definitions of associated properties. While these illustrations are most germane to typical problems in experimental mechanics, the mathematical descriptions of phenomena are similar for other processes relating other pairs of conjugate variables. Naturally, the physical interpretation of associated properties will vary with the particular conjugate phenomena investigated. The idealized behavior described by (2.16), is nondissipative. Real materials typically exhibit a timedependent response of the dependent variable to the independent variable, which is generally expressed as a relationship between the time rate of change of the dependent variable and the independent variable. Since the dependent variable does not instantaneously assume its final value in response to changes in the independent variable, these are not in phase during a cyclic process, and energy is dissipated during a closed cycle of deformation. In a static experiment viscoelastic, or anelastic, behavior is characterized by an induced strain field that exhibits not only an instantaneous response to a constant applied stress field, but also continues to increase until it reaches its final, higher, value over a period of time. Similarly, the stress field induced in response to an applied strain field relaxes from an initial value to a lower, constant value after a sufficiently long time. This leads to the definition of relaxed and unrelaxed elastic coefficients in relations of the form of (2.16), depending on whether the instantaneous or final values of the induced variables are employed. Chapter 29 on the characterization of viscoelasticity materials presents a comprehensive treatment of the viscoelastic behavior of materials. Viscoplastic behavior occurs when the material does not return to its original state on the restoration of the initial boundary conditions. Classical continuum plasticity is the limiting case in which time-dependent behavior is sufficiently small to be neglected. Many comprehensive treatments of the mathematics and phenomenology of these models are available to the interested reader; see [2.24]. The transition from reversible, dissipative behavior to irreversible, dissipative behavior is difficult to establish unequivocally and

2.2 Properties of Materials

40

Part A

Solid Mechanics Topics

Part A 2.2

value of t for which (τ0 − τ∞ ) exp(−t/ξε ) reaches 1/ e of its final value defines the relaxation time at constant strain, ξε . The ratio of ε0 to τ(t) is the viscoelastic compliance S(t). Clearly, this ratio depends on time during the relaxation process. Relaxed and unrelaxed compliances S0 and S∞ follow from dividing ε0 by τ0 and τ∞ , respectively. Viscoelastic stiffnesses, relaxed and unrelaxed elastic stiffnesses, or moduli, are the reciprocals of the corresponding compliances. A similar experiment in which the applied stress is kept constant and the strain increases with time to a limiting value results in the definition of a relaxation time at constant stress ξσ which is generally different in magnitude from ξε . A single relaxation time for viscoelastic effects often refers to the geometric mean of relaxation times determined in experiments where stress and strain are, respectively, the independent variables. Zener develops the theory of superposition of a distribution of a spectrum of relaxation times to the description of material response [2.30]. While the example given applies to uniaxial normal stress and the resulting normal strain, the concept applies to other states of stress or strain to obtain components of the full elastic constant and compliance tensors. The physical phenomena that cause relaxation in a material depend on its atomic and molecular structure. For example, the individual molecular chains in a linear polymer do not respond instantaneously to an applied stress field because of their mutual interference with relative motion. Over time, thermal motion of the molecules assisted by the local stress overcomes interference and a configuration in equilibrium with the applied stress field results. Upon removal of the stress, the material relaxes towards its initial configuration. Depending on the magnitude of the deformation and the extent of mutual interference of molecules, it may not be possible for the material to achieve its original state. In this case, the residual deformation is permanent and the deformation is viscoplastic. The book by Ferry [2.31] gives more detail on the theory and data on the viscoelastic properties of polymers. Mechanisms for energy dissipation in polymeric materials occur at the scale of longchain molecules, while those in metals and ceramics occur at the scales of atoms and ions [2.30]. Consequently, relaxation times for polymers tend to be longer than for metals and ceramics, leading to higher damping capacities in the former materials. Damping phenomena are relatively more important to engineering applications of polymeric materials that of metals and ceramics.

Viscoplasticity A cyclic deformation process that does not return the material to its initial state, even after relaxation, produces permanent, or plastic, deformation. This occurs when processes in the material that respond to a change in independent variable, stress or strain, result in irreversible changes that prevent the material from assuming its initial state after restoration of the independent variable to its original value. The physical mechanisms responsible for irreversible behavior in materials are extensively studied and modeled at the microscale. The natures of all such mechanisms are such that time-dependent behavior results on a macroscale. Notwithstanding a reasonably detailed understanding of the microscopic processes involved in permanent deformation, the application of this knowledge to the description of behavior at the macroscale is one of the most challenging research areas in the mechanics of materials. Plasticity in amorphous materials involves permanent rearrangement of the linear molecular or network structure of the substance. The fundamental process of deformation is the stretching, distorting or breaking of individual atomic or molecular bonds and reforming them with different neighbors. A volume change may or may not accompany these processes. Changes typically occur in a manner that is correlated with the local stress state and often involve a combination of thermal activation with stress. Since amorphous structures do not possess long-range atomic or molecular order, it is not appropriate to describe this process in terms of the presence or propagation of structural defects. Permanent deformation of these materials is still best described in terms of phenomenological theories. Chapter 29 reviews some phenomenological approaches to viscoplasticity in which differences of behavior among materials are contained in material parameters appearing in the theories, but not specifically linked to atomic structure or mechanisms. This section presents a formally similar treatment, but introduces material parameters for crystalline materials that are based on the atomic and defect structure of the deforming material. Representative phenomenological approaches to viscoplasticity in which differences of behavior among materials are contained in material parameters appearing in the theories, but not specifically linked to atomic structure or mechanisms. As noted earlier, in crystalline materials permanent deformation at the microscale below half the absolute melting temperature occurs primarily by slip, the relative displacement of material across crystallographic

Materials Science for the Experimental Mechanist

shape coordinate system, embedded in the body, having nodes at material points and deforming congruently with the body as a whole. If no dislocations remain in the body after deformation, lattice coordinates and shape coordinates are related by a one-to-one mapping. While deformation of the shape coordinate system is always compatible (unless voids or cracks form during deformation), deformation of the lattice coordinate system is not compatible if dislocations remain in the body. Following the convention employed in Chap. 1, define a total, or shape, distortion as the deformation gradient FS which maps a reference vector in the initial state dX to its counterpart in the final state dx using shape coordinates. The lattice distortion FL , relates dx to its counterpart expressed in lattice coordinates dζ while the dislocation distortion FD relates dζ to dX [2.32]. Then dx = FL dζ = FL FD dX = FS dX, leading to the multiplicative decomposition FS = FL FD .

(2.28)

By the assumption of continuity FS is compatible, while FL and FD can be compatible or incompatible, depending on the nature of the deformation process. If either is incompatible, the other must be also and the incompatibilities must be equal and opposite. To facilitate understanding the connection between dislocation deformation and macroscale changes in shape, define a reference state consisting of a body free of external tractions and having a zero or selfequilibrating internal stress field. Consider the transition of a small volume of material δV bounded by a surface δS from the reference state to a deformed state by a two-step thought process [2.33, 34]: 1. remove δV from the reference material, replacing any tractions exerted by the rest of the body with tractions on the surface of δV so that its state remains unchanged, and give δV an arbitrary permanent change in shape δV ( dζ = FD dX), 2. replace δV in the original location of δV by applying the necessary tractions to its surface, δS , then remove the tractions, allowing δV to relax to its new, deformed configuration δV

bounded by δS

( dx = FL dζ). Two possibilities exist for step 1: dislocations may or may not be introduced into δV by the permanent change in shape. Each leads to a different situation in δV

after step 2. Case I refers to the situation where no dislocations are introduced in δV and FD is compatible within δV resulting in a stress-free distortion.

41

Part A 2.2

planes. Relative motion occurs such that the local structure of the material is preserved, that is, nearestneighbor distances are unchanged and the material remains crystalline. Motion does not occur simultaneously across an entire slip plane, but sequentially, so that the atomic distortion caused by the motion is localized near boundaries between regions where relative displacement has occurred and regions where it has yet to occur. These boundaries are dislocations, which are the principal source of internal stress in materials. For crystalline regions permanently deformed entirely by slip, dislocations pass out of the region and vanish at free surfaces or accumulate at internal surfaces that bound the region. Characterization of a dislocation requires specification of the slip vector, or Burgers vector b, which is the vector describing the relative displacement of material across the slip plane, and the line direction, a unit vector, t, tangent to the boundary, pointing in an arbitrarily chosen positive direction. Conventionally the senses of b and t are chosen such that the unit normal to the slip plane n = (b × t)/|b| points towards the region of lattice expansion associated with the dislocation. While motion of dislocations in their slip planes conserves lattice sites and produces no volume change, motion of dislocations normal to their slip planes, known as climb, creates or destroys vacant lattice sites, producing a permanent change in volume. Generally this nonconservative motion is associated with deformation at temperatures above one-half of the absolute melting point of the material, while conservative motion with no volume change dominates at lower temperatures. Permanent deformation by dislocation motion results in one of two final configurations. If dislocations pass entirely through the body, the shape is permanently changed, but no (additional) residual stress is introduced in the deformed material. In this case the total deformation is equal to the deformation due to dislocations, or dislocation deformation, both of which are compatible in the usual sense of finite deformation theory [2.24]. In the second case, dislocations remain in the material, causing additional deformation of the crystal lattice accompanied by residual stresses. Relating the latter scenario to conventional measures of deformation based on changes in the external dimensions of a body requires recognition of the fact that deformation of the crystal lattice may not be congruent with the deformation of the body as a whole. It is necessary to describe the process with reference to two coordinate systems: the lattice coordinate system, based on the crystal lattice in which dislocation deformation is measured, and the

2.2 Properties of Materials

42

Part A

Solid Mechanics Topics

Part A 2.2

Since a permanent change of shape has occurred, when δV is reintroduced into the body any misfit must be accommodated by dislocations distributed on δS

after relaxation. The resulting compatible FL within δV

is composed of a component due to these dislocations and a component due to surface tractions arising from the constraint of the rest of the body. This corresponds to Eshelby’s misfitting inclusion problem [2.33]. Case II refers to the situation where dislocations remain in δV

after step 1 and FD is not compatible. After reinsertion within δV

FL now consists of an incompatible distortion due to the dislocations within δV

in addition to the compatible distortion due to the boundary conditions on δS

. The implications of this decomposition become evident if δV

is considered as a representative volume element (RVE) or continuum limit for a continuum theory of plasticity. Including dislocations in δV

requires a decomposition of shape distortion in which both the dislocation and lattice components are incompatible. Figure 2.14 illustrates the relationships among the various components of deformation discussed above and the familiar elastic and plastic distortion components employed in continuum mechanics Chap. 1. Referring to the previous discussion of the decomposition of distortion, case I applies to the situation when the dislocation distortion coincides with the plastic distortion FP and the lattice distortion is entirely due to boundary conditions on δV

. Both distortions are compatible except on the boundary δS where dislocations have accumulated. Case II corresponds to the situation when dislocations that remain in δV cause an incompatible lattice distortion FLD that has an incompatibility equal and opposite to FD . The remaining compatible component of the lattice distortion FLC is Deformed state

Reference state FS Fp FD

FL

F LC = F e

F LD

Fig. 2.14 Decomposition of shape (total) distortion (after [2.35])

due to tractions on the boundary of δV and is formally equal to the conventional elastic distortion Fe . The multiplicative decompositions now become FS = FL FD = (FLC )(FLD FD ) = Fe Fp ,

(2.29)

where we have identified Fe , the conventional elastic distortion, with FLC and the conventional compatible plastic distortion Fp with the product FLD FD . Continuum plasticity employs the decomposition of total distortion into the two compatible distortion components on the extreme right-hand side (RHS) of (2.29), which does not permit the inclusion of dislocations explicitly in the theory and restricts its application to situations in which permanent distortion is uniform throughout the deformed region of material. Crystal plasticity modifies this model by invoking the geometry of crystalline slip to allow deformed regions of the material to experience differing amounts of distortion in response to the local stress field. By the arguments above the boundaries between these regions must contain dislocations sufficient to make the local shape distortion compatible, although the presence of these dislocations is not included explicitly in the theory. However, this approach often employs dislocation theory to construct constitutive relations at the microscale, which permits the local rate of distortion to differ throughout the material in response to the local state of stress [2.36]. For a viscoplastic material the rate of plastic distortion depends on the applied stress through a flow law of the form [2.37] ε˙ˆ ijD = Φ(σˆ ∗ , π1 · · · π K , Θ)σˆ ij∗ ,

(2.30)

in which the carat indicates the deviatoric component of the tensor Φ is a scalar function πi are internal material parameters dependent on structure and Θ is the absolute temperature. The effect of internal structure on mechanical behavior enters through its contribution to the viscoplastic potential Φ. The rate dependence of mechanical response originates with the resistance of the material to dislocation motion. Obstacles to dislocation motion are primarily the intrinsic resistance of the lattice (Peierls stress) and the stress fields of other defects and dislocations. Dislocations move over their slip planes by overcoming the resistance due to these obstacles under the influence of the local stress field and thermal activation, as illustrated schematically in Fig. 2.8. Observable deformation occurs by the collective motion of large numbers of dislocations over an obstacle array with a spectrum of strengths.

Materials Science for the Experimental Mechanist

The kinematics of deformation relate to dislocation motion through the geometry of the microscopic process. The dislocation distortion rate due to dislocation motion on N slip systems is ˙ D = LD = F

N  

b(k) ⊗ (ρ(k) × v(k) ) ,

(2.31)

where the superposed dot refers to differentiation with respect to time. For the k-th slip system, ρ(k) is a vector whose length is a measure of the instantaneous length of mobile dislocations per unit volume having Burgers vector b(k) and whose direction is determined by the edge-screw character of the distribution of line directions, while v(k) is a vector normal to ρ(k) that specifies the mean velocity of the mobile dislocation configuration [2.35]. For typical deformation processes, the lattice distortion and lattice distortion rate are small compared to the dislocation distortion rate so that (2.31) is approximately equal to the total or shape distortion rate. Employing a single thermally activated deformation process due to a single dislocation mechanism, Hartley [2.35] demonstrated how the microscale parameters describing the process can lead to an interpretation of the viscoplastic potential in terms of parameters determined by the model of the dislocation–barrier interaction. Similar models can be constructed for creep, or high-temperature deformation under constant stress, where the primary mechanism of dislocation motion is climb, or motion normal to the glide plane. Constructing a model of this process that can be inserted into a continuum description of deformation includes several challenging components. First, models must be constructed of the interaction of individual dislocations with various obstacles at the scale of individual obstacles. These models must then be scaled up to include the interaction of a distribution of dislocations with a distribution of obstacles on a slip plane. The interaction of dislocation motion on the various slip planes in a crystal follows at the next scale, and finally the distribution of crystals and their associate slip planes throughout the material must be established. At each level, appropriate statistical models of the microscopic quantities must be constructed and related to the adjacent scales of structure. This process of multiscale modeling has received much attention in the past decade. Attempts to relate previous knowledge of dislocation theory and viscoplasticity have benefited considerable from advances in the computational speed and capacity of supercomputers. Present capabilities still fall short of permitting engineering properties

to be calculated from inputs based entirely on microscopic models of dislocation processes. Nevertheless, considerable progress has been made in this area and the inclusion of microscale models of deformation processes in engineering design codes promises to add a significant component of material design to the process in the future.

2.2.4 Transport Properties of Materials When a system is not in the same state throughout, gradients of free energy exist, causing fluxes of energy that tend to eliminate the gradients. The free energy gradients are thermodynamic forces and the resulting fluxes can include matter, energy or electric charge depending on the nature of the force. Although the description of the relationship between forces and fluxes is called irreversible thermodynamics [2.26], it is important to recognize that systems in which such forces occur are not in thermodynamic equilibrium [2.26]. Phenomenological equations similar to those defining equilibrium and dissipative properties define properties that relate forces and fluxes. Thermal Transport A difference in temperature between various parts of a system causes heat to flow. In this case, the thermodynamic force is the temperature gradient. A vector represents the heat flux. The magnitude of the vector is equal to the flux of heat flowing across unit area normal to the vector and the direction of the vector is the direction of heat flow. In general, the heat flux vector need not be parallel to the temperature gradient. The dependence of the heat flux on the temperature gradient defines the thermal conductivity, k, by the relationship

qi = −kij T j ,

(2.32)

where the suffix j refers to partial differentiation with respect to the spatial variable x j . The negative sign specifies that heat flows from regions of higher temperature to regions of lower temperature. The thermal resistivity tensor r is the inverse of k; both are symmetric second-rank tensors. Principal values of the conductivity tensor are positive for all known materials. The conductivity ellipsoid is the quadric surface [2.24] k () = kij i  j

(2.33)

such that the thermal conductivity k() in the direction of the unit vector  is the reciprocal of the square root

43

Part A 2.2

k=1

2.2 Properties of Materials

44

Part A

Solid Mechanics Topics

of the radius vector from the origin to the surface in the direction of . By Neumann’s principle, the symmetry elements of the conductivity tensor contain those of the point group of any crystalline material that they describe. Isotropic thermal conductivity tensors describe a material that is isotropic with respect to heat flow.

Part A 2.2

Electrical Charge Transport The treatment of electrical conductivity in anisotropic materials is formally identical to that for thermal conductivity. In the former case, the flux is electrical charge instead of heat and the corresponding force is the gradient of electrical potential ϕ I = −E i . The expression

ji = −σik ϕk = σik E k

(2.34)

defines the electrical conductivity tensor σ in terms of the current density j and E. Equation (2.34) is Ohm’s law for a material exhibiting general anisotropy with respect to electrical conductivity. The electrical resistivity tensor ρ is the inverse of σ. All comments in the previous section about the symmetry properties and representation of the thermal conductivity and resistivity tensors apply to the corresponding tensors of electrical properties. The preceding discussion describes the conduction of heat and electrical charge as independent processes. However, when both occur together, they are coupled in a fashion similar to that describing equilibrium properties. For a description of the phenomenon of thermoelectricity and the associated material properties, the interested reader can consult [2.25, Chap. 12]. Mass Transport Engineering materials produced by ordinary manufacturing processes are never in their equilibrium state. A typical example is the segregation of components in an alloy prepared by solidification from the liquid state. The segregation is due to nonuniform cooling and compositional differences between the liquid and the first solids that form on solidification. It is generally desirable to eliminate this segregation to form a material with more uniform properties than the as-cast structure exhibits. This modification is possible because of the tendency of the material to remove spatial variations in composition under appropriate conditions by diffusion, a solid-state mass transport process. Two types of diffusion occur in materials. Stochastic, or random, diffusion occurs because of the random thermal motion of atoms or molecules in a substance. In the absence of any force biasing the random motion, the mean position of the diffusing entity will not change

with time. However, its root-mean-square (RMS) displacement from its origin will not vanish. This means that at any instant t after measurement begins, the displacement of the entity from its starting point may not be zero. In three dimensions, the RMS displacement u(t) is √ u(t) = 6Dt , (2.35) where D is the diffusion coefficient for the diffusing species. The diffusion coefficient depends strongly on temperature and the mechanism by which the diffusing entity moves through the material. Elementary arguments based on a model of diffusion as a thermally activated process lead to the expression of the Arrhenius form for the diffusion coefficient [2.38]   −ΔH ∗ (2.36) , D = D0 exp kT where ΔH ∗ is the enthalpy of activation and D0 is a factor containing the square of the distance moved in a single jump of the diffusing entity, the frequency of jumps, a geometric factor, and a term related to the entropy of activation. Self-diffusion is synonymous with stochastic diffusion. Thermodynamic forces existing in materials bias stochastic diffusion, causing chemical diffusion, a net flux of matter that tends to eliminate gradients in the chemical potential of each component. For the purpose of describing diffusion, the chemical potential μ(i) of component i in a system is the partial derivative of the Gibbs free energy of the system with respect to the number of atoms or molecules of component i present, holding constant the temperature, stress state, and number of atoms of all other components. Multicomponent systems can exhibit fluxes of each component due to gradients in the chemical potentials of all components. In chemical diffusion, the force conjugate to the flux of each component is the gradient of the chemical potential of that component. Gradients of other fields, such as temperature and electrical potential, can also cause fluxes of the components. The description of diffusion including these cross effects is formally similar to that for thermoelectricity, giving rise to the definition of the thermal diffusion coefficient, electrostatic diffusion coefficient, and related properties. The following section presents a brief treatment of the principal equations and concepts in chemical diffusion in a two-component system. Consider a binary system of components A and B in which equal and opposite fluxes of the components J (K ) , where K takes on the values A and B, occur

Materials Science for the Experimental Mechanist

parallel to a characteristic direction xi . Fick’s first law assumes that the flux of each component is proportional to the concentration gradient and occurs in the opposite direction Ji(A) = − D˜

∂NA , ∂xi

(2.37)

Property Change Through Heat Treatment One of the most useful features of many structural materials, particularly metals and alloys, is the ability to alter the microstructure so that selected physical and mechanical properties can be significantly changed without substantially affecting the external shape. This change is usually accomplished through heat treatment, which is the exposure of a material to an elevated temperature, generally somewhere between the proposed service temperature and the melting point of the material and often in a protective atmosphere to prevent adverse interaction with the environment. The extent of the exposure must be determined empirically to effect desired changes of properties without undesirable side effects. During the exposure internal stresses may be relieved by

the nucleation and growth of new, stress-free grains (recrystallization), preceded by the mutual annihilation of pre-existing dislocations (recovery); metastable phases present may transform to more stable forms and compositional variations at the microscale can be diminished by interdiffusion of components of the material. The three examples below illustrate such changes. Many structural metals and alloys are prepared for use by a sequence of processes beginning with the solidification of an ingot from a molten source, followed by several stages of substantial permanent deformation, often alternated with heat treatments, to arrive at a final form suitable for the fabrication of engineering structures. The deformation generally begins at elevated temperatures (above 0.5Tm , where Tm is the absolute melting temperature) where the deformation can be accomplished at lower loads. This process also breaks up the grain structure of the cast ingot, which is highly anisotropic. The elevated temperatures at which deformation is conducted promote mass transport of alloying elements, which promotes compositional homogeneity by diminishing local variations in the material. Since interaction with the environment, resulting in undesirable consequences such as surface roughness, accompanies deformation at elevated temperatures, final processing occurs at near-ambient temperatures. This low-temperature deformation introduces work-hardening, which may occur to such an extent that additional processing is not possible without damage to the material. In such cases, intermediate heat treatments that cause recovery and recrystallization in the material provide a means for lowering the flow stress to a point where additional cold work can be accomplished without fracture of the material. Changes in the internal structure of materials by altering the number, amount, and composition of phases present is a common result of heat treatment. Both the hardening of steel and precipitation hardening of aluminum alloys are examples of this process. The hardening of steel results from the fact that iron undergoes a transformation of crystal structure from body-centered cubic to face-centered cubic on heating from ambient temperature to above 910 ◦ C. When carbon is added, the bcc form of iron, ferrite, has a lower solubility limit than the fcc form, austenite. Thus an alloy that may be single phase at an elevated temperature will tend to become two-phase at ambient temperature as the structure transforms from austenite to ferrite. If the alloy is cooled slowly through the transforma-

45

Part A 2.2

where NA is the atomic fraction of component A, and D˜ is the coefficient of chemical diffusion, also known as the interdiffusion coefficient. The interdiffusion coefficient contains the thermodynamic factor relating the concentration to the chemical potential of component A. In many systems, experiments show that the fluxes of two species are not equal and opposite due to the accompanying flux of the point defects responsible for diffusion at the nanoscale. Although the gradient in defect concentration is generally negligible relative to that of the other diffusing species, the defect flux can be similar in magnitude to the fluxes of the other diffusing species. The phenomenological treatment of this effect describes the flux of each component in terms of a relation similar to (2.37) with the interdiffusion coefficient replaced by an intrinsic diffusion coefficient D(K) which contains the thermodynamic factor relating the chemical potential to the concentration of component K. Neglecting the defect concentration relative to those of the atomic or molecular species requires that N (A) + N (B) = 1, and the gradients are equal and opposite. Then (2.32) describes the flux of either species with the interdiffusion coefficient replaced by (N (A) D(B) + N (B) D(A) ). For additional information on experimental and theoretical treatment of this topic, the interested reader can consult the excellent book by Glicksman [2.38].

2.2 Properties of Materials

46

Part A

Solid Mechanics Topics

Part A 2.2

tion temperature range, this occurs by the formation of a second phase, Fe3 C or cementite, which appears in the form of an eutectoid, pearlite, consisting of alternating lamellae of cementite and ferrite. However if the alloy is cooled rapidly (quenched) from the higher temperature, the precipitation has insufficient time to occur and the excess carbon is trapped in the bcc lattice, distorting it to form a metastable, bodycentered tetragonal structure (martensite). Martensite is considerably harder than ferrite or pearlite, so the resulting alloy has substantially higher yield strength that a material of identical composition that has been slowly cooled from the same elevated temperature. Since martensite is often too brittle for use in the as-quenched condition, subsequent heat treatments below the transformation temperature can further alter the structure and increase toughness by reducing internal stresses without significantly affecting the yield strength. Precipitation hardening has similarities to the heat treatment of steel, but does not require a phase change of the pure element that is the base of the alloy system. This process is most commonly employed to strengthen aluminum alloys. The effect exploits the fact that the solubility of most alloying elements in a metal increases with increasing temperature. Thus, in the appropriate range of composition, an alloy that is single phase at an elevated temperature may have an equilibrium structure consisting of two or more phases at ambient temperature. Heat treatment to cause precipitation hardening consists of heating such an alloy to a temperature in the single-phase region, followed by quenching to ambient temperature to retain the singlephase structure, then reheating to a lower temperature in the multiphase region and holding for an empirically determined time (aging) to permit the structure to approach the equilibrium phase structure. In the early stage of aging metastable proto-precipitates are formed, which strain the lattice of the matrix material and increase the yield stress. As aging progresses and precipitates grow from these nuclei, internal stresses diminish, causing the yield stress to decrease, approaching the value of the as-quenched, single-phase solid solution. Processing of these materials is designed to conduct the aging treatment at temperatures and times that optimize the combination of strength, ductility, and toughness required for intended applications. Service temperatures for precipitation hardened alloys must be sufficiently below the aging temperatures so that additional unintended changes in properties do not occur during use.

2.2.5 Measurement Principles for Material Properties The measured properties of materials and the use of these measurements in various experimental mechanics applications require an appreciation of the limitations and dependencies described in the previous sections. The physical foundations of phenomenological relationships connecting structure and properties of materials comprise the preceding sections of this chapter. This connection leads to the following general principles, which call attention to the need for appropriate qualification of experimental measurements involving the properties of materials. Isolation In conducting experiments where measurements of conjugate fields provide data for the calculation of material properties, either isolate the system from additional fields that can affect the results or hold all but the conjugate fields constant. Without isolation or a constant environment of other fields, the measured properties will include cross-effects giving an incorrect value for the property associated with the conjugate fields. The same principle applies when specifically measuring cross-effects. Structure Consistency When measuring a material property or using the value of a material property in a measurement, ensure that samples to which the property applies are structurally identical. This is especially true of structure-sensitive properties, such as yield strength. However, variations in microstructure due to varying conditions of preparation can also affect other properties, such as electrical resistivity, sufficiently to cause inconsistencies in measurements with different samples. Quantitative Specification of Structure When measuring a material property or employing a material as an element in a measurement system, describe the composition and microstructural condition of the material in sufficient quantitative detail that subsequent similar experiments can employ structurally comparable material. Failure to observe this principle will introduce an unaccountable element of variability in the results of apparently comparable experiments. Consistency of Assumptions In analyzing results of measurements involving materials, ensure that the conditions of measurement

Materials Science for the Experimental Mechanist

satisfy the assumptions employed in the calculation. For example, the calculation of internal stress from measured values of strain must employ lattice strain or acknowledge the fact that the use of measured values of total strain implicitly assume that no plastic deforma-

References

tion occurs during the measurements. Only diffraction techniques measure lattice strain. Other techniques that use surface strains must satisfy the conditions under which these strains equal lattice strains, i. e., negligible dislocation motion.

2.2 2.3 2.4

2.5 2.6 2.7

2.8

2.9

2.10

2.11

2.12 2.13 2.14 2.15

2.16

2.17

J.F. Shackelford: Introduction to Materials Science for Engineers, 6th edn. (Pearson-Prentice Hall, New York 2004) D.W. Callister: Materials Science and Engineering An Introduction, 6th edn. (Wiley, New York 2003) D.R. Askeland: The Science and Engineering of Materials, 2nd edn. (Thomson, Stanford 1992) M.F. Ashby, D.R. Jones: Engineering Materials 1: An Introduction to Their Properties and Applications, 2nd edn. (Butterworth Heinemann, Oxford 2000) J.P. Hirth, J. Lothe: Theory of Dislocations, 2nd edn. (Wiley, New York 1982) F.R.N. Nabarro, J.P. Hirth (Eds.): Dislocations in Solids, Vol. 12 (Elsevier, Amsterdam 2004) C.S. Barrett, W.D. Nix, A. Tetleman: Principles of Engineering Materials (Prentice-Hall, Englewood Cliffs 1973) T.B. Massalski, H. Okamoto, P.R. Subramanian, L. Kacprzak: Binary Alloy Phase Diagrams, 2nd edn. (ASM Int., Materials Park 1990) R.S. Roth, T. Negas, L.P. Cook: Phase Diagrams for Ceramists, Vol. 5 (American Ceramic Society, Inc., 1983) P. Villers, A. Prince, H. Okamoto: Handbook of Ternary Alloy Phase Diagrams (ASM Int., Materials Park 1995) J. Goldstein, D.E. Newbury, P. Echlin, D.C. Joy, A.D. Romig Jr., C.E. Lyman, C. Fiori, E. Lifshin: Scanning Electron Microscopy and X-Ray Microanalysis, 2nd edn. (Springer, New York 1992) R.T. DeHoff, F.N. Rhines: Quantitative Microscopy (McGraw-Hill, New York 1968) B.D. Cullity: Elements of X-ray Diffraction, 2nd edn. (Addison-Wesley, Reading 1978) B.E. Warren: X-ray Diffraction (General, New York 1990) D.K. Smith, R. Jenkins: The powder diffraction file: Past, present and future, Rigaku J. 6(2), 3–14 (1989) W.I.F. David, K. Shankland, L.B. McCusker, C. Baerlocher (Eds.): Structure Determination from Powder Diffraction Data, International Union of Crystallography (Oxford Univ. Press, New York 2002) I.C. Noyan, J.B. Cohen, B. Ilschner, N.J. Grant: Residual Stress Measurement by Diffraction and Interpretation (Springer, New York 1989)

2.18 2.19

2.20

2.21

2.22

2.23

2.24

2.25 2.26 2.27 2.28

2.29 2.30 2.31 2.32 2.33 2.34

2.35

C.S. Hartley: International Conference on Experimental Mechanics, ICEM12-12th (Italy 2004) D.K. Bowen, B.K. Tanner: High Resolution X-ray Diffractometry and Topography (Taylor Francis, New York 1998) L. Cartz: Nondestructive Testing Radiography, Ultrasonics, Liquid Penetrant, Magnetic Particle, Eddy Current (ASM Int., Materials Park 1995) D.B. Williams, C.B. Carter: Transmission electron microscopy: a textbook for materials science (Kluwer Academic, New York, 1996) B.L. Adams, S.I. Wright, K. Kunze: Orientation imaging: The emergence of a new microscopy, Met. Mat. Trans. A 24, 819–831 (1993) J.A. James, J.R. Santistban, L. Edwards, M.R. Daymond: A virtual laboratory for neutron and synchrotron strain scanning, Physica B 350(1–3), 743–746 (2004) L.E. Malvern: Introduction to the Mechanics of a Continuous Medium (Prentice-Hall, Englewood Cliffs 1969) J.F. Nye: Physical Properties of Crystals (Clarendon, Oxford 1985) S.M. DeGroot: Thermodynamics of Irreversible Processes (North-Holland, Amsterdam 1951) M.P. Marder: Condensed Matter Physics (Wiley, New York 2000) W.G. Cady: Piezoelectricity: An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals (Dover, New York 1964) C.S. Hartley: Single crystal elastic moduli of disordered cubic alloys, Acta Mater. 51, 1373–1391 (2003) C. Zener: Elasticity and Anelasticity of Metals (Univ. Chicago Press, Chicago 1948) J.D. Ferry: Viscoelastic Properties of Polymers, 3rd edn. (Wiley, New York 1980) B. A. Bilby: Continuous Distributions of Dislocations, Progress in Solid Mechanics (1960) J.D. Eshelby: The continuum theory of lattice defects, Solid State Phys. 3, 79–144 (1956) E. Kröner: Continuum theory of defects. In: Les Houches, Session XXXV, 1980 – Physics of Defects, ed. by R. Balian, M. Kleman, J.-P. Poirer (NorthHolland, Amsterdam 1981) pp. 282–315 C.S. Hartley: A method for linking thermally activated dislocation mechanisms of yielding with

Part A 2

References 2.1

47

48

Part A

Solid Mechanics Topics

2.36

continuum plasticity theory, Philos. Mag. 83, 3783– 3808 (2003) R. Asaro: Crystal Plasticity, J. Appl. Mech. 50, 921–934 (1983)

2.37 2.38

J. Lubliner: Plasticity Theory (Macmillan, New York 1990) M.E. Glicksman: Diffusion in Solids (Wiley, New York 2000)

Part A 2

49

Mechanics of 3. Mechanics of Polymers: Viscoelasticity

Wolfgang G. Knauss, Igor Emri, Hongbing Lu

3.1

3.2

Historical Background ........................... 3.1.1 The Building Blocks of the Theory of Viscoelasticity ..........................

49

Linear Viscoelasticity ............................. 3.2.1 A Simple Linear Concept: Response to a Step-Function Input .............. 3.2.2 Specific Constitutive Responses (Isotropic Solids) .......................... 3.2.3 Mathematical Representation of the Relaxation and Creep Functions 3.2.4 General Constitutive Law for Linear and Isotropic Solid: Poisson Effect .. 3.2.5 Spectral and Functional Representations ...........................

51

3.2.6 Special Stress or Strain Histories Related to Material Characterization 3.2.7 Dissipation Under Cyclical Deformation............ 3.2.8 Temperature Effects ...................... 3.2.9 The Effect of Pressure on Viscoelastic Behavior of Rubbery Solids ......................... 3.2.10 The Effect of Moisture and Solvents on Viscoelastic Behavior................ 3.3

3.4

50

51

Measurements and Methods .................. 3.3.1 Laboratory Concerns ..................... 3.3.2 Volumetric (Bulk) Response ........... 3.3.3 The CEM Measuring System ............ 3.3.4 Nano/Microindentation for Measurements of Viscoelastic Properties of Small Amounts of Material......... 3.3.5 Photoviscoelasticity ...................... Nonlinearly Viscoelastic Material Characterization ................................... 3.4.1 Visual Assessment of Nonlinear Behavior................... 3.4.2 Characterization of Nonlinearly Viscoelastic Behavior Under Biaxial Stress States ............

56 63 63

68 69 69 70 71 74

76 83 84 84

85

53

3.5

Closing Remarks ...................................

89

53 55

3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known.............. 3.6.1 Further Reading ...........................

90 90

55

References ..................................................

92

3.1 Historical Background During the past five decades the use of polymers has seen a tremendous rise in engineering applications. This growing acceptance of a variety of polymer-based de-

signs derives in part from the ease with which these materials can be formed into virtually any shape, and in part because of their generally excellent performance

Part A 3

With the heavy influx of polymers into engineering designs their special, deformation-rate-sensitive properties require particular attention. Although we often refer to them as time-dependent materials, their properties really do not depend on time, but time histories factor prominently in the responses of polymeric components or structures. Structural responses involving time-dependent materials cannot be assessed by simply substituting time-dependent modulus functions for their elastic counterparts. The outline provided here is intended to provide guidance to the experimentally inclined researcher who is not thoroughly familiar with how these materials behave, but needs to be aware of these materials because laboratory life and applications today invariably involve their use.

50

Part A

Solid Mechanics Topics

Part A 3.1

in otherwise normally corrosive environments. This recent emergence is driven by our evolving capabilities during the last seven decades to synthesize polymers in great variety and to address their processing into useful shapes. Historically polymers have played a significant role in human developments, as illustrated by the introductory comments in [3.1]. Of great consequence for the survival or dominance of tribes or nations was the development of animal-derived adhesives for the construction of high-performance bows, starting with the American Indian of the Northwest through the developments by the Tartars and leading to the extraordinary military exploits of the Turks in the latter Middle Ages [3.2]. In principle, these very old methods of producing weaponry continue to aid today in the construction of modern aerospace structures. While the current technology still uses principles exploited by our ancestors many years ago, the advent of the synthetic polymers has provided a plethora of properties available for a vast range of different engineering designs. This range of properties is, indeed, so large that empirical methods are no longer sufficient to effect reliable engineering developments but must now be supported by optimum analytical methods to aid in the design process. One characteristic of polymers is their relative sensitivity to load exposure for extended periods of time or to the rate of deformations imposed on them. This behavior is usually and widely combined under the concept of viscoelastic behavior, though it is sometimes characterized as representing fading memory of the material. These time-sensitive characteristics typically extend over many decades of the time scale and characteristically set polymers apart from the normal engineering metals. While the strain-rate sensitivity [3.3] and the time dependence of failure in metals [3.4] are recognized and creep as well as creep rupture [3.5–10] of metals is well documented, one finds that the incorporation of rate-dependent material properties into models of time-dependent crack growth – other than fatigue of intrinsically rate-insensitive materials – still stands on a relatively weak foundation. Metallic glasses (i.e., amorphous metals) are relatively newcomers to the pool of engineering materials. Their physical properties are at the beginning of exploration, but it is already becoming clear through initial studies [3.11, 12] that their amorphous structure endows them with properties many of which closely resemble those of amorphous polymers. While these developments are essentially in their infancy at this time it is

well to bear in mind that certain parts of the following exposition are also applicable to these materials. Because the emphasis in this volume is placed on experimental methods, rather than on stress analysis methods, only a cursory review of the linearized theory of viscoelasticity is included. For the reader’s educational benefit a number of books and papers have been listed in the Further Reading section, which can serve as resources for a more in-depth treatment. This review of material description and analysis is thus guided by particular deformation histories as a background for measurements addressing material characterization to be used in engineering design applications. Although the nonlinearly viscoelastic characteristic of these materials are not well understood in a general, three-dimensional setting, we include some reference to these characteristics in the hope that this understanding will assist the experimentalist with properly interpreting laboratory measurements.

3.1.1 The Building Blocks of the Theory of Viscoelasticity Forces are subject to the laws of Newtonian mechanics, and are, accordingly, governed by the classical laws of motion. While relativistic effects have been studied in connection with deforming solids, such concerns are suppressed in the present context. Many texts deal with Newtonian mechanics to various degrees of sophistication so that only a statement of the necessary terminology is required for the present purposes. In the interest of brevity we thus dispense with a detailed presentation of the analysis of stress and of the analysis of strain, except for summarizing notational conventions and defining certain variables commonly understood in the context of the linear theory of elasticity. We adhere to the common notation of the Greek letters τ and ε denoting stress and strain, respectively. Repeated indices on components imply summation; identical subscripts (e.g., τ11 ) denote normal components and different ones shear (e.g., τ12 ). The dilatational components of stress, τii , are often written as σkk , with the strain complement being εkk . Because the viscoelastic constitutive description is readily expressed in terms of deviatoric and dilatational components, it is necessary to recall the components Sij of the deviatoric stress as 1 Sij = τij − τkk · δij , 3

(3.1)

where δij denotes the Kronecker operator. Similarly, the corresponding deviatoric strain e is written in compo-

Mechanics of Polymers: Viscoelasticity

nent form as 1 eij = εij − εkk · δij . (3.2) 3 For further definitions and derivations of measures of stress or strain the reader is referred to typical texts.

3.2 Linear Viscoelasticity

51

The remaining building block of the theory consists of the constitutive behavior, which differentiates viscoelastic materials from elastic ones. The next section is devoted to a brief definition of linearly viscoelastic material behavior.

3.2 Linear Viscoelasticity face (boundary) of a viscoelastic solid. Specification of such a quantity under uniaxial relaxation is not particularly useful, except to note that in the limit of short (glassy) response its value is a limit constant, and also under long-term conditions when the equilibrium (or rubbery) modulus is effective, in which case the Poisson’s ratio is very close to 0.5 (incompressibility).

3.2.1 A Simple Linear Concept: Response to a Step-Function Input It is convenient for instructional purposes to consider that the stress can be described, so that the strain follows from the stress. The reverse may hold with equal validity. In general, of course, neither may be prescribed a priori, and a general connection relates them. The structure of the linear theory must be completely symmetric in the sense that the mathematical formulation applies to these relations regardless of which variable is considered the prescribed or the derived one. For introductory purposes we shall use, therefore, the concept of a cause c(t) (input) and an effect e(t) (output) that are connected by a functional relationship. The latter must be linear with respect to (a) the amplitude (additivity with respect to magnitude) and (b) time in the sense that they obey additivity independent of time. It is primarily a matter of convenience that the cause-and-effect relation is typically expressed with the aid of a step-function cause. Other representations are

Table 3.1 Nomenclature for viscoelastic material functions Type of loading Shear

Bulk

Uniaxial extension

μ(t) J(t) μ (ω) μ (ω) J  (ω) J  (ω)

K (t) M(t) K  (ω) K  (ω) M  (ω) M  (ω)

E(t) D(t) E  (ω) E  (ω) D (ω) D (ω)

Mode Quasistatic

Relaxation Creep Strain prescribed

Harmonic Stress prescribed

Storage Loss Storage Loss

Part A 3.2

The framework for describing linearly viscoelastic material behavior, as used effectively for engineering applications, is phenomenological. It is based mathematically on either an integral or differential formulation with the material representation described realistically in numerical (tabular) or functional form(s). The fundamental equations governing the linearized theory of viscoelasticity are the same as those for the linearized theory of elasticity, except that the generalized Hooke’s law of elasticity is replaced by a constitutive description that is sensitive to the material’s (past) history of loading or deformation. It will be the purpose of the immediately subsequent subsections to summarize this formalism of material description in preparation for various forms of material characterization. Little or no reference is made to general solution methods for viscoelastic boundary value problems. For this purpose the reader is referred to the few texts available as listed in Sect. 3.6.1. Rather than repeating the theory as already outlined closely in [3.13] we summarize below the concepts and equations most necessary for experimental work; if necessary, the reader may consult the initially cited reference(s) (Sect. 3.6.1) for a more expansive treatment. In brief, the viscoelastic material functions of first-order interest are given in Table 3.1. Note the absence of a generic viscoelastic Poisson function, because that particular response is a functional of the deformation or stress history applied to the sur-

52

Part A

Solid Mechanics Topics

Part A 3.2

feasible and we shall address a common one (steadystate harmonic) later on as a special case. For now, let E(t, t1 ) represent a time-dependent effect that results from a step cause c(t1 ) = h(t1 ) of unit amplitude imposed at time t1 ; h(t1 ) denotes the Heaviside step function applied at time t1 . For the present we are concerned only with nonaging materials, i. e. with materials, the intrinsic properties of which do not change with time. (With this definition in mind it is clear that the nomenclature timedependent materials in place of viscoelastic materials is really a misnomer; but that terminology is widely used, nevertheless.) We can assert then that for a non-aging material any linearity of operation, or relation between an effect and its cause, requires satisfaction of Postulate (a): proportionality with respect to amplitude, and Postulate (b): additivity of effects independent of the time sequence, when the corresponding causes are added, regardless of the respective application times. Condition (a) states that, if the cause c(0) elicits E(t, 0), then a cause of different amplitude, say c1 (0) ≡ α · h(0), with α a constant, elicits a response α · E(t, 0). Under the non-aging restriction this relation is to be independent of the time when the cause starts to act, so that c1 (t1 ) ≡ α · h(t1 ) → αE(t, t1 ); t > t1 also holds. This means simply that the response–effect relation shown in the upper part of Fig. 3.1 holds also for a different time t2 , which occurs later in time than t1 .

Condition (b) entails then that, if two causes c1 (t1 ) ≡ α1 · h(t1 ) and c2 (t2 ) ≡ α2 · h(t2 ), imposed at different times t1 and, t2 act jointly, then their corresponding effects α1 · E(t, t1 ) and α2 · E(t, t2 ) is their sum while observing their proper time sequence. Let the common time scale start at t = 0; then the combined effect, say e(t), is expressed by c(t) ≡ c1 (t1 ) + c2 (t2 ) → e(t) = α1 · E(t − t1 ) + α1 · E(t − t2 ) . (3.3)

Specifically, here the first response does not start until the time t1 is reached, and the response due to the second cause is not experienced until time t2 , as illustrated in Fig. 3.1. Having established the addition process for two causes and their responses, the extension to an arbitrary number of discrete step causes is clearly recognized as a corresponding sum for the collective effects e(tn ), up to time t, in the generalized form of (3.3), namely  (3.4) e(t) = αn E(t − tn ) ; (tn < t) . This result may be further generalized for causes represented by a continuous cause function of time, say c(t). To this end consider a continuously varying function c(t) decomposed into an initially discrete approximation of steps of finite (small) amplitudes. With the intent of ultimately proceeding to the limit of infinitesimal steps, note that the amplitude of an individual step amplitude at, say, time τn is given by   (3.5) αn → Δc(τn ) = (Δc/Δτ) Δτ , τn

Stress

which, when substituted into (3.4), leads to    e(t) = E(t − tn )(Δc/Δτ) Δτ .

Strain

τn

t1

Time

Stress

t1

Time

Strain

In the limit n → ∞ (Δτ → dτ), the sum  Δc  e(t) = lim E(t − tn )  Δτ , Δτ τn

(3.7)

passes over into the integral 2

t

2 1

(3.6)

e(t) =

1

E(t − τ)

dc(τ) dτ . dτ

(3.8)

0

t1

t2

Time

t1

t2

Time

Fig. 3.1 Additivity of prescribed stress steps and corres-

ponding addition of responses

Inasmuch as this expression can contain the effect of a step-function contribution at zero time of magnitude c(0), this fact can be expressed explicitly through the

Mechanics of Polymers: Viscoelasticity

alternate notation

3.2 Linear Viscoelasticity

53

effect, one obtains the inverse relation(s) t

e(t) = c(0)E(t) +

E(t − τ)

0+

dc(τ) dτ , dτ

(3.9)

3.2.2 Specific Constitutive Responses (Isotropic Solids) For illustrative purposes and to keep the discussion within limits, the following considerations are limited to isotropic materials. Recalling that the stress and strain states may be decomposed into shear and dilatational contributions (deviatoric and dilatational components), we deal first with the shear response followed by the volumetric part. Thermal characterization will then be dealt with subsequently. Shear Response Let τ denote any shear stress component and ε its corresponding shear strain. Consider ε to be the cause and τ its effect. Denote the material characteristic E(t) for unit step excitation from Sect. 3.2.1 in the present shear context by μ(t). This function will be henceforth identified as the relaxation modulus in shear (for an isotropic material). It follows then from (3.8) and (3.9) that t dε(ξ) (3.10) dξ τ(t) = 2 μ(t − ξ) dξ 0

t = 2ε(0)μ(t) + 2 0+

μ(t − ξ)

dε(ξ) dξ . dξ

(3.11)

The factor of 2 in the shear response is consistent with elasticity theory, inasmuch as in the limits of short- and long-term behavior all viscoelasticity relations must revert to the elastic counterparts. If one interchanges the cause and effect by letting the shear stress represent the cause, and the strain the

t J(t − ξ)

dτ dξ dξ

(3.12)

0

t

1 1 = τ(0)J(t) + 2 2

J(t − ξ) 0+

dτ dξ , dξ

(3.13)

where now the function E ≡ J(t) is called the shear creep compliance, which represents the creep response of the material in shear under application of a step shear stress of unit magnitude as the cause. Bulk or Dilatation Response Let εii (t) represent the first strain invariant and σ jj (t) the corresponding stress invariant. The latter is recognized as three times the pressure P(t), i. e., σ jj (t) ≡ 3P(t). In completely analogous fashion to (3.12) and (3.13) the bulk behavior, governed by the bulk relaxation modulus K (t) ≡ E(t), is represented by

t σ jj = 3

K (t − ξ)

dεii (ξ) dξ dξ

(3.14)

0

t = 3εii (0)K (t) + 3

K (t − ξ)

0+

dεii (ξ) dξii . dξ (3.15)

Similarly, one writes the inverse relation as 1 εii = 3

t M(t − ξ)

dσ jj (ξ) dξ dξ

(3.16)

0

1 1 = σ jj (0)M(t) + 3 3

t M(t − ξ) 0+

dσ jj (ξ) dξii , dξ (3.17)

where the function M(t) ≡ E(t) represents now the dilatational creep compliance (or bulk creep compliance); in physical terms, this is the time-dependent fractional volume change resulting from the imposition of a unit step pressure.

3.2.3 Mathematical Representation of the Relaxation and Creep Functions Various mathematical forms have been suggested and used to represent the material property functions

Part A 3.2

where the lower integral limit 0+ merely indicates that the integration starts at infinitesimally positive time so as to exclude the discontinuity at zero. Alternatively, the same result follows from observing that for a step discontinuity in c(t) the derivative in (3.5) is represented by the Dirac delta function δ(t). In fact, this latter remark holds for any jump discontinuity in c(t) at any time, after and including any at t = 0. In mathematical terms this form is recognized as a convolution integral, which in the context of the dynamic (vibration) response of linear systems is also known as the Duhamel integral.

1 ε(t) = 2

54

Part A

Solid Mechanics Topics

Part A 3.2

analytically. Preferred forms have evolved, with precision being balanced against ease of mathematical use or a minimum number of parameters required. All viscoelastic material functions possess the common characteristic that they vary monotonically with time: relaxation functions decreasing and creep functions increasing monotonically. A second characteristic of realistic material behavior is that time is (almost) invariably measured in terms of (base 10) logarithmic units of time. Thus changes in viscoelastic response may appear to be minor when considered as a function of the real time, but substantial if viewed against a logarithmic time scale. Early representations of viscoelastic responses were closely allied with (simple) mechanical analog models (Kelvin, Voigt) or their derivatives. Without delving into the details of this evolutionary process, their generalization to broader time frames led to the spectral representation of viscoelastic properties, so that it is useful to present only the rudiments of that development. The building blocks of the analog models are the Maxwell and the Voigt models illustrated in Fig. 3.2a,b. In this modeling a mechanical force F corresponds to the shear stress τ and, similarly, a displacement/deflection δ corresponds to a strain ε. Under a stepwise applied deformation of magnitude ε0 – separating the force-application points in the Maxwell model – the stress (force) abates or relaxes by the relation τ(t) = ε0 μm exp(−t/ξ) , a)

b)

F

(3.18)

c)

F

μ μ

η

η

μ1

η1

μ2

η2

μ3

η3

F

F F

d) μ∞

F

μ1

μ2

μ3

μj

η1

η2

η3

ηj

F

where ξ = ηm /μm is called the (single) relaxation time. Similarly, applying a step stress (force) of magnitude to the Voigt element engenders a time-dependent separation (strain) of the force-application points described by τ0 [1 − exp(−t/ς)] , (3.19) ε(t) = μν where ς = ην /μν is now called the retardation time since it governs the rate of retarded or delayed motion. Note that this representation is used for illustration purposes here and that the retardation time for the Voigt material is not necessarily meant to be equal to the relaxation time of the Maxwell solid. It can also be easily shown that this is not true for a standard linear solid either. By inductive reasoning, that statement holds for arbitrarily complex analog models. The relaxation modulus and creep compliance commensurate with (3.18) (Maxwell model) and (3.19) (Voigt model) for the Wiechert and Kelvin models (Fig. 3.2c,d) are, respectively  μn exp(−t/ξn ) (3.20) μ(t) = μ∞ + n

and J(t) = Jg +



Jn [1 − exp(−t/ςn )] + η0 t ,

where Jg and η0 arise from letting η1 → 0 (the first Voigt element degenerates to a spring) and μn → 0 (the last Voigt element degenerates to a dashpot). These series representations with exponentials are often referred to as Prony series. As the number of relaxation times increases indefinitely, the generalization of the expression for the shear relaxation modulus, becomes ∞ dξ (3.22) , μ(t) = μ∞ + H (ξ) exp(−t/ξ) ξ 0

where the function H (ξ) is called the distribution function of the relaxation times, or relaxation spectrum, for short; the creep counterpart presents itself with the help of the retardation spectrum L(ζ ) as ∞

μj

ηj

J(t) = Jg +

L(ζ )[1 − exp(−t/ζ )] 0

F

Fig. 3.2a–d Mechanical analogue models: (a) Maxwell, (b) Voigt, (c) Wiechert, and (d) Kelvin

(3.21)

n

dζ + ηt , ζ (3.23)

Note that although the relaxation times ξ and the retardation times ζ do not, strictly speaking, extend over the range from zero to infinity, the integration limits are so

Mechanics of Polymers: Viscoelasticity

3.2 Linear Viscoelasticity

assigned for convenience since the functions H and L can always be chosen to be zero in the corresponding part of the infinite range.

3.2.5 Spectral and Functional Representations

3.2.4 General Constitutive Law for Linear and Isotropic Solid: Poisson Effect

A discrete relaxation spectrum in the form  H (ξ) = μn ξn δ(ξn ) ,

One combines the shear and bulk behavior exemplified in (3.13), (3.16) and (3.19), (3.20) into the general stress–strain relation  t  2 ∂εkk K (t − τ) − μ(t − τ) dτ σij (t) = δij 3 ∂τ t +2 0−

μ(t − τ)

∂εij dτ , ∂τ

(3.24)

where δij is again the Kronecker delta. Poisson Contraction A recurring and important parameter in linear elasticity is Poisson’s ratio. It characterizes the contraction/expansion behavior of the solid in a uniaxial stress state, and is an almost essential parameter for deriving other material constants such as the Young’s, shear or bulk modulus from each other. For viscoelastic solids the equivalent behavior cannot in general be characterized by a constant; instead, the material equivalent to the elastic Poisson’s ratio is also a time-dependent function, which is a functional of the stress (strain) history imposed on a uniaxially stressed material sample. This time-dependent function covers basically the same (long) time scale as the other viscoelastic responses and is typically measured in terms of 10–20 decades of time at any one temperature. However, compared to these other functions, its value changes usually from a maximum value of 0.35 or 0.4 at the short end of the time spectrum to 0.5 for the long time frame. Several approximations are useful. In the nearglassy domain (short times) its value can be taken as a constant equal to that derived from measurements well below the glass-transition temperature. In the long time range for essentially rubber-like behavior the approximation of 0.5 is appropriate, though not if one wishes to convert shear or Young’s data to bulk behavior, in which case small deviations from this value can play a very significant role. If knowledge in the range between the (near-)glassy and (near-)rubbery domain are required, neither of the two limit constants are strictly appropriate and careful measurements are required [3.14–16].

(3.25)

where δ(ξn ) represents the Dirac delta function, clearly leads to the series representation (3.20) and can trace the modulus function arbitrarily well by choosing the number of terms in the series to be sufficiently large; a choice of numbers of terms equal to or larger than twice the number of decades of the transition is often desirable. For a history of procedures to determine the coefficients μn see the works by Hopkins and Hamming [3.17], Schapery [3.18], Clauser and Knauss [3.19], Hedstrom et al. [3.20], Emri and Tschoegl [3.21–26], and Emri et al. [3.27, 28], all of which battle the ill-conditioned nature of the numerical determination process. This fact may result in physically inadmissible, negative values (energy generation), though the overall response function may be rendered very well. A more recent development that largely circumvents such problems, is based on the trust region concept [3.29], which has been incorporated into MATLAB, thus providing a relatively fast and readily available procedure. The numerical determination of these coefficients occurs through an ill-conditioned integral or matrix and is not free of potentially large errors in the coefficients, including physically inadmissible negative values, though the overall response function may be rendered very well. Although expressions as given in (3.22) and (3.23) render complete descriptions of the relaxation or creep behavior once H (ξ) or L(ξ) are determined for any material in general, simple approximate representations can fulfill a useful purpose. Thus, the special function   μ0 − μ∞ ξ0 n exp(−ξ0 /ξ) (3.26) H (ξ) = Γ (n) ξ with the four parameters μ0 , μ∞ , ξ0 , and n representing material constants, where Γ (n) is the gamma function, leads to the power-law representation for the relaxation response μ(t) = μ∞ +

μ0 − μ∞ . (1 + t/ξ0 )n

(3.27)

This equation is represented in Fig. 3.3 for the parameter values μ∞ = 102 , μ0 = 105 , ξ0 = 10−4 , and n = 0.35. It follows quickly from (3.22) and the figure that μ0 represents the modulus as t → 0, and μ∞

Part A 3.2

0−

55

56

Part A

Solid Mechanics Topics

3.2.6 Special Stress or Strain Histories Related to Material Characterization

log modulus 6

For the purposes of measuring viscoelastic properties in the laboratory we consider several examples in terms of shear states of stress and strain. Extensional or compression properties follow totally analogous descriptions.

5 4 3 2

Part A 3.2

1 –10

–5

0

5

10 log t

Fig. 3.3 Example of the power-law representation of a relaxation modulus

its behavior as μ(t → ∞); ξ0 locates the central part of the transition region and n the (negative) slope. It bears pointing out that, while this functional representation conveys the generally observed behavior of the relaxation phenomenon, it usually serves only in an approximate manner: the short- and long-term modulus limits along with the position along the log-time axis and the slope in the mid-section can be readily adjusted through the four material parameters, but it is usually a matter of luck (and rarely possible) to also represent the proper curvature in the transitions from short- and long-term behavior. Nevertheless, functions of the type (3.26) or (3.27) can be very useful in capturing the essential features of a problem. With respect to fracture Schapery draws heavily on the simplified power-law representation. An alternative representation of one-dimensional viscoelastic behavior (shear or extension), though not accessed through a distribution function of the type described above, is the so-called stretch exponential formulation; it is often used in the polymer physics community and was introduced for torsional relaxation by Kohlrausch [3.30] and reintroduced for dielectric studies by Williams and Watts [3.31]. It is, therefore, often referred to as the KWW representation. In the case of relaxation behavior it takes the form (with the addition of the long-term equilibrium modulus μ∞ ),

(3.28) μ(t) = μ∞ + μ0 exp −(t/ξ0 )β . Further observations and references relating to this representation are delineated in [3.13].

Unidimensional Stress State We call a stress or strain state unidimensional when it involves only one controlled or primary displacement or stress component, as in pure shear or unidirectional extension/compression. Typical engineering characterizations of materials occur by means of uniaxial (tension) tests. We insert here a cautionary note with respect to laboratory practices. In contrast to working with metallic specimens, clamping polymers typically introduces complications that are not necessarily totally resolvable in terms of linear viscoelasticity. For example, clamping a tensile specimen in a standard test machine with serrated compression claps introduces a nonlinear material response such that, during the course of a test, relaxation or creep may occur under the clamps. Sometimes an effort is made to alleviate this problem by gluing metal tabs to the end of specimens, only to introduce the potential of the glue line to contribute to the overall relaxation or deformation. If the contribution of the glue line to the deformation is judged to be small, an estimate of its effect may be derived with the help of linear viscoelasticity, and this should be stated in reporting the data. For rate-insensitive materials the pertinent property is Young’s modulus E. For viscoelastic solids this constant is supplanted by the uniaxial relaxation modulus E(t) and its inverse, the uniaxial creep compliance D(t). Although the general constitutive relation (3.24) can be written for the uniaxial stress state (σ11 (t) = σ0 (t), say, σ22 = σ33 = 0), the resulting relation for the uniaxial stress is an integral equation for the stress or strain ε11 (t), involving the relaxation moduli in shear and dilatation. In view of the difficulties associated with determining the bulk response, it is not customary to follow this interconversion path, but to work directly with the uniaxial relaxation modulus E(t) and/or its inverse, the uniaxial creep compliance D(t). Thus, if σ11 (t) is the uniaxial stress and ε11 (t) the corresponding strain, one writes, similar to (3.10) and (3.12),

t σ11 (t) = ε11 (0)E(t) + 0+

E(t − ξ)

dε11 (ξ) dξ dξ

Mechanics of Polymers: Viscoelasticity

ε

ε

ε0

t0

tions to generate a ramp

+

t0

t

t0

t

t

modulus (3.22) together with the convolution relation (3.10) to render, with ε˙ (t) = const = ε˙ 0 and ε(0) = 0, the general result t

0+

We insert here a cautionary note with respect to laboratory practices: In contrast to working with metallic specimens, clamping polymers typically introduces complications that are not necessarily totally resolvable in terms of linear viscoelasticity. For example, clamping a tensile specimen in a standard test machine with serrated compression clamps introduces nonlinear material response such that during the course of a test relaxation or creep may occur under the clamps. Sometimes an effort is made to alleviate this problem by gluing metal tabs to the end of specimens, only to introduce the potential of the glue-line to contribute to the overall relaxation or deformation. If the contribution of the glue-line to the deformation is judged to be small an estimate of its effect may be derived with the help of linear viscoelasticity, and such should be stated in reporting the data. Constant-Strain-Rate History. A common test method

for material characterization involves the prescription of a constant deformation rate such that the strain increases linearly with time (small deformations). Without loss of generality we make use of a shear strain history in the form ε(t) = ε˙ 0 t (≡ 0 for t ≤ 0, ε˙ 0 = const for t ≥ 0) and employ the general representation for the relaxation τ

µ (t)

Error

ε0

t0

t

t0

t

Fig. 3.5 Difference in relaxation response resulting from

step and ramp strain history

μ(t − ξ)˙ε0 dξ

τ(t) = 2 0

t  = 2˙ε0

∞ μ∞ +

0

0

t − ξ dς  H (ς) exp − du ς ς (3.29)

t = 2μ˙ε0

μ(u) du = 2˙ε0 t · 0

1 t

t μ(u) du 0

= 2ε(t)μ(t) (3.30) ¯ ; t −ξ ≡ u .  t Here μ(t) ¯ = 1t 0 μ(u) du is recognized as the relaxation modulus averaged over the past time (the time-averaged relaxation modulus). Ramp Strain History. A recurring question in viscoelas-

tic material characterization arises when step functions are called for analytically but cannot be supplied experimentally because equipment response is too slow or dynamic (inertial) equipment vibrations disturb the input signal: In such situations one needs to determine the error if the response to a ramp history is supplied instead of a step function with the ramp time being t0 . To provide an answer, take explicit recourse to postulate (b) in Sect. 3.2.1 in connection with (3.29)/(3.30) to evaluate (additively) the latter for the strain histories shown in Fig. 3.4. To arrive at an approximate result as a quantitative guide, let us use the power-law representation (3.27) for the relaxation modulus. Making use of Taylor series approximations of the resulting functions for t  0 one arrives at (the derivation is lengthy though straightforward)   n t0 /ξ0 τ(t) = μ(t) 1 + (3.31) +... 2ε0 2 (1 + t/ξ0 )

Part A 3.2

and the inverse relation as t dσ11 (ξ) ε11 (t) = σ11 (0)D(t) + D(t − ξ) dξ . dξ

ε

57

Fig. 3.4 Superposition of linear func-

ε

=

3.2 Linear Viscoelasticity

58

Part A

Solid Mechanics Topics

Part A 3.2

as long as μ∞ can be neglected relative to μ0 (usually on the order of 100–1000 times smaller). The derivation is lengthy though straightforward. The expression in the square brackets contains the time-dependent error by which the ramp response differs from the ideal relaxation modulus, as illustrated in Fig. 3.5, which tends to zero as time grows without limit beyond t0 . By way of example, if n = 1/2 and an error in the relaxation modulus of maximally 5% is acceptable, this condition can be met by recording data only for times larger than t/t0 = 5 − ς0 /t0 . Since ς0 /t0 is always positive the relaxation modulus is within about 5% of the ramp-induced measurement as long as one discounts data taken before 5t0 . To be on the safe side, one typically dismisses data for an initial time interval equal to ten times the ramp rise time. In case the time penalty for the dismissal of that time range is too severe, methods have been devised that allow for incorporation of this earlier ramp data as delineated in [3.32, 33]. On the other hand, the wide availability of computational power makes an additional data reduction scheme available: Using a Prony series (discrete spectrum) representation, one evaluates the constant-strain-rate response with the aid of (3.30), leaving the individual values of the spectral lines as unknowns. With regard to the relaxation times one has two options: (a) one leaves them also as unknowns, or (b) one fixes them such that they are one or two per decade apart over the whole range of the measurements. The second option (b) is the easier/faster one and provides essentially the same precision of representation as option (a). After this choice has been made, one fits the analytical expression with the aid of Matlab to the measurement results. Matlab will handle either cases (a) or (b). There may be issues involving possible dynamic overshoots in the rate-transition region, because a test machine is not able to (sufficiently faithfully) duplicate the rapid change in rate transition from constant to zero rate, unless the initial rate is very low. This discrepancy is, however, considerably smaller that that associated with replacing a ramp loading for a step history. Mixed Uniaxial Deformation/Stress Histories Material parameters from measured relaxation or creep data are typically extracted via Volterra integral equations of the first kind, i. e., of the type of (3.20) or (3.21). A problem arises because these equations are ill-posed in the sense that the determination of

the kernel (material) functions from modulus or creep data involving Volterra equations of the first kind can lead to sizeable errors, whether the functions are sought in closed form or chosen in spectral or discrete (Prony series) form [3.18, 19, 27]. On the other hand, Volterra equations of the second kind do not suffer from this mathematical inversion instability (well-posed problem). Accordingly, we briefly present an experimental arrangement that alleviates this inherent difficulty [3.28]. At the same time, this particular scheme also allows the simultaneous determination of both the relaxation and creep properties, thus circumventing the calculation of one from the other. In addition, the resulting data provides the possibility of a check on the linearity of the viscoelastic data through a standard evaluation of a convolution integral. Relaxation and/or creep functions can be determined from an experimental arrangement that incorporates a linearly elastic spring of spring constant ks as illustrated in Fig. 3.6, readily illustrated in terms of a tensile situation. The following is, however, subject to the assumption that the elastic deformations of the test frame and/or the load cell are small compared to those of the specimen and the deformation of the added spring. If the high stiffness of the material does not warrant that assumption it is necessary to determine the contribution of the testing machine and incorporate it into the stiffness ks . Similar relations apply for a shear stress/deformation arrangement. In the case of

lb

Δl b0 ls ls Δl Δl s0

Fig. 3.6 Arrangement for multiple material properties de-

termination via a single test

Mechanics of Polymers: Viscoelasticity

bulk/volume response the spring could be replaced by a compressible liquid, though this possibility has not been tested in the laboratory, to our knowledge. For a suddenly applied gross extension (compression) of the spring by an amount Δl = const, both the bar and the spring will change lengths according to Δlb (t) + Δls (t) = Δl ,

(3.32)

where the notation in Fig. 3.6 is employed (subscript ‘b’ refers to the bar and “s” to the spring). The correspondingly changing stress (force) in the bar is given by

(3.33)

which is also determined by Ab Fb (t) = lb

t E(t − ξ)

d [Δlb (ξ)] dξ dξ

0

+

Ab 0 Δl E(t) , lb b

(3.34)

which, together with (3.32), renders upon simple manipulation  Ab Δlb (t) εb (0)E(t) + Δl ks Δl t E(t − ξ)

+

 d [εb (ξ)] dξ = 1 . dξ

(3.35)

0

This is a Volterra integral equation of the second kind, as can be readily shown by the transformation of variables ξ = t − u; it is well behaved for determining the relaxation function E(t). By measuring Δlb (t) along with the other parameters in this equation, one determines the relaxation modulus E(t). Similarly, one can cast this force equilibrium equation in terms of the creep compliance of the material and the force in the spring as ks lb Fb (t) + Ab

 t D(t − ξ)

d [Fb (ξ)] dξ dξ

0

Time-Harmonic Deformation A frequently employed characterization of viscoelastic materials is achieved through sinusoidal strain histories of frequency ω. Historically, this type of material characterization refers to dynamic properties, because they are measured with moving parts as opposed to methods leading to quasi-static relaxation or creep. However, in the context of mechanics dynamic is reserved for situations involving inertia (wave) effects. For this reason, we replace in the sequel the traditional dynamic (properties) with harmonic, signifying sinusoidal. Whether one asks for the response from a strain history that varies with sin(ωt) or cos(ωt) may be accomplished by dealing with the (mathematically) complex counterpart

ε(t) = ε0 exp(iωt) · h(t)

0

(3.36)

It is clear then that, if both deformations and the stress in the bar are measured, both the relaxation modulus

(3.38)

so that after the final statement has been obtained one would be interested, correspondingly in either the real or the imaginary part of the result. Here h(t) is again the Heaviside step function, according to which the real part of the strain history represents a step at zero time with amplitude ε0 . The evaluation of the appropriate response may be accomplished with the general modulus representation so that substitution of (3.22) and (3.38) into (3.12) or (3.13) renders, after an interchange in the order of integration,   ∞ dς τ(t) = 2ε0 μ∞ + H (ς) exp(−t/ς) ς  t t −ξ H (ς) exp − ς 0 0  dς × exp(iωξ) dξ ς ∞

0

 + Fb (0)D(t) = ks Δl .

and the creep compliance can be determined and the determination of the Prony series parameters proceeds without difficulty [3.21–26] The additional inherent characteristic of this (hybrid) experimental–computational approach is that it may be used for determining the limit of linearly viscoelastic behavior of the material. By determining the two material functions of creep and relaxation simultaneously one can examine whether the determined functions satisfy the essential linearity constraint, see (3.62)–(3.64) t D(t − ξ)E(ξ) dξ = t . (3.37)

+ 2ε0 iω

59

Part A 3.2

Fb (t) = Fs (t) = kb (t)Δls (t) = ks [Δl − Δlb (t)] ,

3.2 Linear Viscoelasticity

60

Part A

Solid Mechanics Topics

t + 2ε0 iωμ∞

exp(iωξ) dξ ,

(3.39)

0

which ultimately leads to τ(t) = 2ε0 [μ(t) − μ∞ ] ∞ iωH (ς) − 2ε0 exp(−t/ς) dς 1 + iως 0

  ∞ iωH (ς) dς . + 2ε(t) μ∞ + 1 + iως

(3.40)

Part A 3.2

0

The first two terms are transient in nature and (eventually) die out, while the third term represents the steady-state response. For the interpretation of measurements it is important to appreciate the influence of the transient terms on the measurements. Even though a standard linear solid, represented by the spring–dashpot analog in Fig. 3.7 does not reflect the full spectral range of engineering materials, it provides a simple demonstration for the decay of the transient terms. Its relaxation modulus (in shear, for example) is given by μ(t) = μ∞ + μs exp(−t/ζ0 ) ,

(3.41)

where ζ0 denotes the relaxation time and μ∞ and μs are modulus parameters. Using the imaginary part of (3.40) corresponding to the start-up deformation history ε(t) = ε0 sin(ωt)h(t) one finds for the corresponding stress history ωζ0 μs τ(t) =R= (cos ωt + ωζ0 sin ωt) 2μ∞ ε0 μ∞ 1 + ω2 ζ02 ωζ0 μs e−t/ζn . (3.42) − μ∞ 1 + ω2 ζ02 F

μ

η

μ0 F

The last term is the transient. An exemplary presentation with μs /μ∞ = 5, ωζ0 = 1, and ζ0 = 20 is shown in Fig. 3.8. For longer relaxation times the decay lasts longer; for shorter ones the converse is true. One readily establishes that in this example the decay is (exponentially) complete after four to five times the relaxation time. The implication for real materials with very long relaxation times deserves extended attention. The expression for the standard linear solid can be generalized by replacing (3.41) with the corresponding Prony series representation. 1  ωζn μn τ(t) = (cos ωt + ωζn sin ωt) 2μ∞ ε0 μ∞ n 1 + ω2 ζn2 1  ωζn μn −t/ζn0 e . (3.43) − μ∞ n 1 + ω2 ζn2 Upon noting that the fractions in the last term sum do not exceed μn /2 one can bound the second sum by 1  ωζn μn −t/ζn0 e μ∞ n 1 + ω2 ζn2   1  1 μ(t) ≤ μn e−t/ζn0 = −1 . (3.44) 2μ∞ n 2 μ∞ This expression tends to zero only when t → ∞, a time frame that is, from an experimental point of view, too long in most instances. For relatively short times that fall into the transition range, the ratio of moduli is not small, as it can be on the order of 10 or 100, or even larger. There are, however, situations for which this error can be managed, and these correspond to those cases when the relaxation modulus changes very slowly during the time while sinusoidal measurements are being R 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 0

20

40

60

80

100

120 140 160 Normalized time

Fig. 3.7 Standard

Fig. 3.8 Transient start-up behavior of a standard linear

linear solid

solid under ε(t) = h(t) sin(ωt)

Mechanics of Polymers: Viscoelasticity

τ(t) = μ∞ + 2ε(t)

∞ 0

iωH (ς) dς . 1 + iως

Both the strain ε(t) and the right-hand side are complex numbers. One calls μ∗ (ω) ≡ μ∞ +

∞ 0

iωH (ς) dς 1 + iως

Stress:

(3.46)

τ (t) sin[ωt +Δ(ω)] 2ε0 μ(ω) Strain:

1

the complex modulus μ∗ = μ (ω) + iμ (ω) with its real and imaginary parts defined by ∞ (ως)2  μ (ω) = μ∞ + H (ς) dς (3.47) 1 + (ως)2 0

(the storage modulus) and ∞ ως μ (ω) = H (ς) dς 1 + (ως)2

(3.48)

0

(the loss modulus), respectively. Polar representation allows the shorthand notation μ∗ = μ(ω) exp[iΔ(ω)] ,

(3.49)

where

μ (ω) and μ (ω)  μ(ω) ≡ |μ∗ (ω)| = [μ (ω)]2 + [μ (ω)]2 , tan Δ(ω) =

(3.50)

so that, also μ (ω) = μ(ω) cos Δ(ω) and μ (ω) = μ(ω) sin Δ(ω) .

(3.51)

The complex stress response (3.45) can then be written, using (3.50), as τ(t) = 2ε(t)μ∗ (ω) = 2ε0 exp(iωt)μ(ω) exp[iΔ(ω)] , (3.52)

(3.45)

ε (t) = sin ωt ε0

t Δω

Fig. 3.9 Illustration of the frequency-dependent phase shift between the applied strain and the resulting stress

61

which may be separated into its real or imaginary part according to τ(t) = 2ε0 μ(ω) cos[ωt + Δ(ω)] and τ(t) = 2ε0 μ(ω) sin[ωt + Δ(ω)] .

(3.53)

Thus the effect of the viscoelastic material properties is to make the strain lag behind the stress (the strain is retarded) as illustrated in Fig. 3.9. It is easy to verify that the high- and low-frequency limits of the steadystate response are given by μ∗ (ω → ∞) = μ(t → 0) = μ0 , the glassy response, and μ∗ (ω → 0) = μ(t → ∞) = μ∞ , as the long-term or rubbery response (real). An Example for a Standard Linear Solid. For the stan-

dard linear solid (Fig. 3.7) the steady-state portion of the response (3.52) simplifies to ω2 ς02 , μ (ω) = μ∞ + μs 1 + ω2 ς02 ως0 μ (ω) = μs , (3.54) 1 + ω2 ς02 ως0 tan Δ(ω) = . (3.55) μ∞ /μs + (1 + μ∞ /μs )(ως0 )2

Part A 3.2

made. This situation arises when the material is near its glassy state or when it approaches rubbery behavior. As long as the modulus ratio can be considered nearly constant in the test period, the error simply offsets the test results by additive constant values that may be subtracted from the data. Clearly, that proposition does not hold when the material interrogation occurs around the middle of the transition range. There are many measurements being made with commercially available test equipment, when frequency scans or relatively short time blocks of different frequencies are applied to a test specimen at a set temperature, or while the specimen temperature is being changed continuously. In these situations the data reduction customarily does not recognize the transient nature of the measurements and caution is required so as not to interpret the results without further examination. Because viscoelastic materials dissipate energy, prolonged sinusoidal excitation generates rises in temperature. In view of the sensitivity of these materials to temperature changes as discussed in Sects. 3.2.7 and 3.2.8, care is in order not to allow such thermal build-up to occur unintentionally or not to take such changes into account at the time of test data evaluations. Consider now only the steady-state portion of (3.40) so that

3.2 Linear Viscoelasticity

62

Part A

Solid Mechanics Topics

which, upon using the transformation t − ξ = u, yields

Log of functions 3

τ(t) = iω 2ε(t)

2

∞

μ(u) e−iωu du = μ∗ (ω) .

If one recalls that the integral represents the Fourier transform F {μ(t), t → ω} of the modulus in the integrand one may write

1 0

μ∗ (ω) = iωF {μ(t), t → ω} –1

Part A 3.2

–2 –1.5

(3.59)

−∞

(3.60)

along with the inverse, –1

–0.5

0

0.5

1

1.5 2 2.5 Log frequency

Fig. 3.10 Steady-state response of a standard linear solid to

sinusoidal excitation, (μ0 = 1, μ = 100, μ/η = ς0 = 0.1). Symbols: μ (ω); short dash: μ (ω); long dash: tan Δ(ω)

While this material model is usually not suitable for representing real solids (its time frame is far too short), this simple analog model represents all the proper limit responses possessed by a real material, in that it has short-term (μ0 + μs , glassy), long-term (μ∞ , rubbery) as well as transient response behavior as illustrated in Fig. 3.10. Note that, with only one relaxation time present, the transition time scale is on the order of at most two decades. The more general representation of the viscoelastic functions under sinusoidal excitation can also be interpreted as a Fourier transform of the relaxation or creep response. Complex Properties as Fourier Transforms. It is often

desirable to derive the harmonic properties from monotonic response behaviors (relaxation or creep). To effect this consider the strain excitation of (3.38), ε(t) = ε0 exp(iωt)h(t) ,

(3.56)

and substitute this into the convolution relation for the stress (3.11), t μ(t − ξ)

τ(t) = 2ε(0)μ(t) + 2 0−

dε(ξ) dξ , dξ

(3.57)

and restrict consideration to the steady-state response. In this case, the lower limit is at t → −∞ so that the integral may be written as t μ(t − ξ) eiωξ dξ ,

τ(t) = lim 2ε0 iω t→∞

−t

(3.58)

μ(t) =

1 2π

∞ −∞

μ∗ (ω) −iωt dω . e iω

(3.61)

Thus the relaxation modulus can be computed from the complex modulus by the last integral. Note also that because of (3.60) μ and μ are derivable from a single function, μ(t), so that they are not independent. Conversely, if one measures μ and μ in a laboratory they should obey a certain interrelation; a deviation in that respect may be construed either as unsatisfactory experimental work or as evidence of nonlinearly viscoelastic behavior. Relationships Among Properties In Sect. 3.2.2 exemplary functional representation of some properties has been described that are generic for the description of any viscoelastic property. On the other hand, the situation often arises that a particular function is determined experimentally relatively readily, but really its complementary function is needed. The particularly simple situation most often encountered is that the modulus is known, but the compliance is needed (or vice versa). This case will be dealt with first. Consider the case when the relaxation modulus (in shear), μ(t) is known, and the (shear) creep compliance J(t) is desired. Clearly, the modulus and the compliance cannot be independent material functions. In the linearly elastic case these relations lead to reciprocal relations between modulus and compliance. One refers to such relationships as inverse relations or functions. Analogous treatments hold for all other viscoelastic functions. Recall (3.10) or (3.11), which give the shear stress in terms of an arbitrary strain history. In the linearly elastic case these inverse relations lead to reciprocal relations between modulus and compliance. Recall also that the creep compliance is the strain history resulting from a step stress being imposed in a shear test. As a corollary, if the prescribed strain

Mechanics of Polymers: Viscoelasticity

history is the creep compliance, then a constant (step) stress history must evolve. Accordingly, substitution of the compliance J(t) into (3.10) must render the step stress of unit amplitude so that t h(t) = J(0)μ(t) +

μ(t − ξ)

dJ(ξ) dξ . dξ

(3.62)

0+ + → 0 , J(0+ )/μ(0+ ) = 1

t μ(t − ξ)J(ξ) dξ = t .

(3.63)

0

Note that this relation is completely symmetric in the sense that, also, t J(t − ξ)μ(ξ) dξ = t .

(3.64)

0

Similar relations hold for the uniaxial modulus E(t) and its creep compliance D(t), and for the bulk modulus K (t) and bulk compliance M(t). Interrelation for Complex Representation. Because

the so-called harmonic or complex material characterization is the result of prescribing a specific time history with the frequency as a single time-like (but constant) parameter, the interrelation between the complex modulus and the corresponding compliance is simple. It follows from equations (3.45) and (3.46) that 1 2ε0 eiωt 2ε(t) = ∗ = = J ∗ (ω) , τ(t) μ (ω) τ0 ei(ωt+Δ(ω))

(3.65)

where the function J ∗ (ω) is the complex shear compliance, with the component J  (ω) and imaginary component −J  (ω) related to the complex modulus by J ∗ (ω) = J  (ω) − iJ  (ω) = J(ω)eiγ (ω) =

1 e−iΔ(ω) = μ∗ (ω) μ(ω)

(3.66)

so that, clearly, 1 and γ (ω) = −Δ(ω) , μ(ω)  with J(ω) = [J  (ω)]2 + [J  (ω)]2 . J(ω) =

(3.67)

63

Thus in the frequency domain of the harmonic material description the interconnection between properties is purely algebraic. Corresponding relations for the bulk behavior follow readily from here.

3.2.7 Dissipation Under Cyclical Deformation In view of the immediately following discussion of the influence of temperature on the time dependence of viscoelastic materials we point out that general experience tells us that cyclical deformations engender heat dissipation with an attendant rise in temperature [3.34, 35]. How the heat generated in a viscoelastic solid as a function of the stress or strain amplitude is described in [3.13]. Here it suffices to point out that the heat generation is proportional to the magnitude of the imaginary part of the harmonic modulus or compliance. For this reason these (magnitudes of imaginary parts of the) properties are often referred to as the loss modulus or the loss compliance. We simply quote here a typical result for the energy w dissipated per cycle and unit volume, and refer the reader to [3.13] for a quick, but more detailed exposition:   ∞  2mπ m ε2m μ (3.68) . w/cycle = π T m=1

3.2.8 Temperature Effects Temperature is one of the most important environmental variables to affect polymers in engineering use, primarily because normal use conditions are relatively close to the material characteristic called the glasstransition temperature – or glass temperature for short. In parochial terms the glass temperature signifies the temperature at which the material changes from a stiff or hard material to a soft or compliant one. The major effect of the temperature, however perceived by the user, is through its influence on the creep or relaxation time scale of the material. Solids other than polymers also possess characteristic temperatures, such as the melting temperature in metals, while the melting temperature in the polymer context signifies specifically the melting of crystallites in (semi-)crystalline variants. Also, typical amorphous solids such as silicate glasses and amorphous metals exhibit distinct glass-transition temperatures; indeed, much of our understanding of glass-transition phenomena in polymers originated in understanding related phenomena in the context of silicate glasses.

Part A 3.2

Note that, as t so that at time 0+ an elastic result prevails. Upon integrating both sides of (3.62) with respect to time – or alternatively, using the Laplace transform – one readily arrives at the equivalent result; the uniaxial counterpart has already been cited effectively in (3.37).

3.2 Linear Viscoelasticity

64

Part A

Solid Mechanics Topics

Part A 3.2

The Entropic Contribution Among the long-chain polymers, elastomers possess a molecular structure that comes closest to our idealized understanding of molecular interaction. Elastomer is an alternative name for rubber, a cross-linked polymer that possesses a glass transition temperature which is distinctly below normal environmental conditions. Molecule segments are freely mobile relative to each other except for being pinned at the cross-link sites. The classical constitutive behavior under moderate deformations (up to about 100% strain in uniaxial tension) has been formulated by Treloar [3.36]. Because this constitutive formulation involves the entropy of a deformed rubber network, this temperature effect of the properties is usually called the entropic temperature effect. In the present context it suffices to quote his results in the form of the constitutive law for an incompressible solid. Of common interest is the dependence of the stress on the material property appropriate for uniaxial tension (in the 1-direction)   1 1 2 subject to λ1 λ2 λ3 = 1 , σ = NkT λ1 − 3 λ1 (3.69)

where λ1 , λ2 , and λ3 denote the (principal) stretch ratios of the deformation illustrated in Fig. 3.11 (though not shown for the condition λ1 λ2 λ3 = 1), the multiplicative factor consists of the number of chain segments between cross-links N, k is Boltzmann’s constant, and T is absolute temperature. Since for infinitesimal deformations λ1 = 1 + ε11 , one finds that NkT must equal the elastic Young’s modulus E ∞ . Thus the (small-strain) Young’s modulus is directly proportional to the absolute temperature, and this holds also for the shear modulus because, under the restriction/assumption of incompressibility the shear modulus μ∞ of the rubber

obeys μ∞ = 13 E ∞ . Thus μ∞ /T = Nk is a material constant, from which it follows that comparative moduli obtained at temperatures T and T0 are related by T μ∞|T = μ∞|T0 or equivalently T0 T E ∞|T = E ∞|T0 . (3.70) T0 If one takes into account that temperature changes affect also the dimensions of a test specimen by changing both its cross-sectional area and length, this is taken into account by modifying (3.71) to include the density ratio according to ρT μ∞|T0 or equivalently μ∞|T = ρ0 T0 ρT E ∞|T = E ∞|T0 , (3.70a) ρ0 T0 where ρ0 is the density at the reference temperature and ρ is that for the test conditions. To generate a master curve as discussed below it is therefore necessary to first multiply modulus data by the ratio of the absolute temperature T (or ρT , if the densities at the two temperatures are sufficiently different) at which the data was acquired, and the reference temperature T0 (or ρ0 T0 ). For compliance data one multiplies by the inverse density/temperature ratio. Time–Temperature Trade-Off Phenomenon A generally much more significant influence of temperature on the viscoelastic behavior is experienced in connection with the time scales under relaxation or creep. To set the proper stage we define first the notion of the glass-transition temperature Tg . To this end consider a measurement of the specific volume as a function Volume

λ1

λ3

λ2

B A Equilibrium line

Fig. 3.11 Deformation of a cube into a parallelepiped. The

unit cube sides have been stretched (contracted) orthogonally in length to the stretch ratios λ1 , λ2 , and λ3

Tg

Temperature

Fig. 3.12 Volume–temperature relation for amorphous solids (polymers)

Mechanics of Polymers: Viscoelasticity

log G (t) Experimental window

sensitive properties, at least for polymers. For ease of presentation we ignore first the entropic temperature effect discussed. The technological evolution of metallic glasses is relatively recent, so that a limited amount of data exist in this regard. However, new data on the applicability of the time–temperature trade-off in these materials have been supplied in [3.12]. Moreover, we limit ourselves to considerations above the glass-transition temperature, with discussion of behavior around or below that temperature range reserved for later amplification. Experimental constraints usually do not allow the full time range of relaxation to be measured at any one temperature. Instead, measurements can typically be made only within the time frame of a certain experimental window, as indicated in Fig. 3.13. This figure shows several (idealized) segments as resulting from different temperature environments at a fixed (usually atmospheric) pressure. A single curve may be constructed from these segments by shifting the temperature segments along the log-time axis (indicated by arrows) with respect to one obtained at a (reference) temperature chosen arbitrarily, to construct the master curve. This master curve is then accepted as the response of the material over the extended time range at the chosen reference temperature. Because this time– temperature trade-off has been deduced from physical measurements without the benefit of a time scale of unlimited extent, the assurance that this shift process is a physically acceptable or valid scheme can be derived only from the quality with which the shifting or

P = P0 T1

T0 = T3

⎛ σ 273 ⎛ ⎜ (psi) ⎝ ε0 T ⎝

log0 ⎜ 5

T3

Temperature (°C) –30.0 –25.0 –22.5 –22.0 –17.5 –15.0 –12.5 –7.5 –5.0 –2.5 5.0

T2 T4

log aT4

4

Master curve at T3

3

T5

ε0 = 0.05

T1 < T2 < ··· ωc . Next consider the carrier load to increase at a constant rate P˙0 . Differentiation of (3.119) with respect to time guarantees a positive loading rate as long as P˙0 ≥ ΔP0 ω. Additionally, the substitution √ of this inequality into (3.91) together with a(t) = RH(t) shows that the contact area will then also not decrease during the entire indentation history. Figure 3.28 shows a comparison of these developments with an application of (3.111) [3.96] under the assumption of a constant Poisson ratio: when measurements over a relatively short time (such as ≈ 250 s used in this study) are made, the Poisson’s ratio [3.15] does not change significantly for polymers in the glassy state, such as PMMA or polycarbonate (PC) and thus introduces negligible errors in the complex compliance data. To compute the complex modulus of PC and polymethyl methacrylate (PMMA) at 75 Hz, data were acquired continuously at this frequency for ≈ 125 s. In general the data increase correctly with time, and approach a nearly constant value for each material. These constant values are considered to represent the steady state and are quoted as the storage modulus. Also shown for comparison in Fig. 3.28 are data measured with the aid of conventional dynamic mechanical analysis (DMA) for the same batch of PC and PMMA. The uniaxial storage modulus of PC measured by DMA at 0.75 Hz is 2.29 GPa. However, the storage modulus computed using (3.111) is at least 40% higher than this

3.3 Measurements and Methods

84

Part A

Solid Mechanics Topics

ing temperatures are of the same order of magnitude. Because dynamic events occur in still shorter time frames wave mechanics typically little concern in this regard. The situation is quite different when soft or elastomeric polymers serve as photoelastic model materials. In that case quasistatic environments (around room temperature) typically involve only the long-term or rubbery behavior of the material with the stiffness measured in terms of the rubbery or long-term equilibrium modulus. On the other hand, when wave propagation phenomena are part of the investigation, the longest relaxation times (relaxation times that govern the tran-

sition to purely elastic behavior for the relatively very long times) are likely to be excited so that the output of the measurements must consider the effect of viscoelastic response. It is beyond the scope of this presentation to delineate the full details of the use of viscoelastically photoelastic material behavior, especially since during the past few years investigators have shown a strong inclination to use alternative tools. However, it seems useful to include a list of references from which the evolution of this topic as well as its current status may be explored. These are listed as a separate group in Sect. 3.6.1 under References on Photoviscoelasticity.

Part A 3.4

3.4 Nonlinearly Viscoelastic Material Characterization Viscoelastic materials are often employed under conditions fostering nonlinear behavior. In contrast to the mutual independence in the dilatational and deviatoric responses in a linearly viscoelastic material, the viscoelastic responses in different directions are coupled and must be investigated in multiaxial loading conditions. Most results published in the literature are restricted to investigating the viscoelastic behavior in the uniaxial stress or uniaxial strain states [3.101, 102] and few results are reported for time-dependent multiaxial behavior [3.103–105]. Note that BauwensCrowet’s study [3.104] incorporates the effect of pressure on the viscoelastic behavior but not in the data analysis, simply as a result of using uniaxial compression deformations. Along similar lines Knauss, Emri, and collaborators [3.106–110] provided a series of studies deriving nonlinear viscoelastic behavior from changes in the dilatation (free-volume change) which correlated well with experiments and gave at least a partial physical interpretation to the Schapery scheme [3.111, 112] for shifting linearly viscoelastic data in accordance with a stress or strain state. The studies became the precursors to investigate the effect of shear stresses or strains on nonlinear behavior as described below. Thus the proper interpretation of the uniaxial data as well as their generalization to multiaxial stress or deformation states is highly questionable. This remains true regardless of the fact that such data interpretation has been incorporated into commercially available computer codes. Certainly, there are very few engineering situations where structural material use is limited to uniaxial states. In this section we describe some aspects of nonlinearly viscoelastic behavior in

the multiaxial stress state. References regarding nonlinear behavior in the context of uniaxial deformations are too numerous to list here. The reader is advised to consult the following journals: the Journal of Polymer, Applied Polymer Science, Polymer Engineering and Science, the Journal of Materials Science and Mechanics of Time-Dependent Materials, to list the most prevalent ones.

3.4.1 Visual Assessment of Nonlinear Behavior Although it is clear that even under small deformations entailing linearly viscoelastic behavior the imposition of a constant strain rate on a tensile or compression specimen results in a stress response that is not linearly related to the deformation when the relation is established in real time (not on a logarithmic time scale). Because such responses that appear nonlinear on paper are not necessarily indicative of nonlinear constitutive behavior, it warrants a brief exposition of how nonlinear behavior is unequivocally separated from the linear type. To demonstrate the nonlinear behavior of a material, consider first isochronal behavior of a linearly viscoelastic material. Although isochronal behavior can be obtained for different deformation or stress histories, consider the case of shear creep under various stress levels σn , so that the corresponding strain is εn = J(t)σn for any time t. Consider an arbitrary but fixed time t ∗ , at which time the ratio εn /σn = J(t ∗ )

(3.121)

Mechanics of Polymers: Viscoelasticity

Shear stress (MPa) 16 14 t = 10s 12 t = 104 s 10

ial. These values are likely to be different for other materials.

3.4.2 Characterization of Nonlinearly Viscoelastic Behavior Under Biaxial Stress States In the following sections, we describe several ways of measuring nonlinearly viscoelastic behavior in multiaxial situations. Hollow Cylinder under Axial/Torsional Loading A hollow cylinder under axial/torsional loading conditions provides a vehicle for investigating the nonlinearly viscoelastic behavior under multiaxial loading conditions. Figure 3.30 shows a schematic diagram of a cylinder specimen. Dimensions can be prescribed corresponding to the load and deformation range of interest as allowed by an axial/torsional buckling analysis. Two examples are given here for specimen dimensions. With the use of a specimen with outer diameter of 22.23 mm, a wall thickness of 1.59 mm, and a test length of 88.9 mm, the ratio of the wall thickness to the radius is 0.14. These specimens can reach a surface shear strain on the order of 4.0–4.5%. With the use of an outer diameter of 25.15 mm, a thickness of 3.18 mm, and a test length of 76.2 mm, the ratio of the wall thickness to the radius is 0.29. This allows a maximum shear strain prior to buckling in the range of 8.5–12.5% based on elastic analysis. The actual maximum shear strain that can be achieved prior to buckling can be slightly different due to the viscoelastic effects involved in the material. Estimates of strains leading to buckling may be achieved by considering a Young’s modulus for the material that corresponds to the lowest value achieved at the test temperature and the time period of interest for the measurements.

8

x

6

Fig. 3.30 A thin-

4 2 y 0

0

0.01

0.02

0.03 Shear strain

Fig. 3.29 Isochronal shear stress–shear strain relation of PMMA at 80 ◦ C

85

D O z

walled cylinder specimen for combined tension/compression and torsion to generate biaxial stress states [3.113]

Part A 3.4

takes on a particular property value. At that time all possible values of σn and εn are related linearly: a plot of σ versus ε at time t ∗ renders a linear relation with slope 1/J(t ∗ ) and encompassing the origin. For different times t ∗ , straight lines with different slopes result and the linearly viscoelastic material can thus be characterized by a fan of straight lines emanating from the origin, the slope of each line corresponding to a different time t ∗ . The slopes decrease monotonically as these times increase. Each straight line is called a linear isochronal stress–strain relation for the particular time t ∗ . Figure 3.29 shows such an isochronal representation for PMMA. It is seen that, at short times and when the strains are small in creep (shear strain 0.005, shear stress 8 MPa), the material response is close to linearly viscoelastic where all the data stay within the linear fan formed, in this case, by the upper line derived from the shear creep compliance at 10 s, and the lower line corresponding to the compliance in shear at 104 s. At the higher stress levels and at longer times, when the strains are larger, material nonlinearity becomes pronounced, as data points in Fig. 3.29 are outside the linear fan emanating from the origin. In this isochronal plot, the deviation from linearly viscoelastic behavior begins at approximately 0.5% strain level. Isochronal data at other temperatures indicate also that the nonlinearity occurs at ≈ 0.5% strain and a shear stress of about 7.6 MPa at all temperatures for this mater-

3.4 Nonlinearly Viscoelastic Material Characterization

86

Part A

Solid Mechanics Topics

Part A 3.4

Application of Digital Image Correlation The use of strain gages for determining surface strains on a polymer specimen is fraught with problems, since strain gages tend to be much stiffer than the polymer undergoing time-dependent deformations, and, in addition, the potential for increases in local temperature due to the currents in the strain gage complicates definitive data evaluation. Digital image correlation (DIC, Chap. 20) thus offers a perfect tool, though that method is not directly applicable to cylindrical surfaces as typically employed. While we abstain from a detailed review of this method in this context (we refer the reader to [3.114] for particulars), here it is of interest for the completeness of presentation only to summarize this special application to cylindrical surface applications. The results are presented in the form of (apparent) creep compliances defined by 2ε(t)/τ0 . We emphasize again that the creep compliance for a linear material is a function that depends only on time but not on the applied stress. This no longer holds in the nonlinearly viscoelastic regime, but we adhere to the use of this ratio as a creep compliance for convenience. To use DIC for tracking axial, circumferential, and shear strains on a cylindrical surface, a speckle pattern is projected onto the specimen surface. While the same image acquisition system can be used as for flat images special allowance needs to be made for the motion of surface speckles on a cylindrical surface, the orientation of which is also not known a priori. If the focal length of the imaging device is long compared to the radius of the cylinder, an image can be considered as a projection of a cylinder onto an observation plane. Planar deformations can be determined using digital im-

age correlation techniques [3.115, 116], and corrected for curvature to determine the axial, circumferential, and shear strains [3.114]. To assure that the motion is properly interpreted in a cylindrical coordinate system – that the camera axis is effectively very well aligned with the cylinder axis and test frame orientation – the imaging system must also establish the axis of rotation. This can be achieved through offsetting the specimen against a darker background (Fig. 3.31) so as to ensure sufficient contrast between the specimen edge and background for identification of any inclination of the cylinder axis relative to the reference axis within the image recording system. The axis orientation is then also evaluated using the principles of digital image correlation. Without such a determination the parameters identifying the projection of the cylindrical surface onto a plane lead to uncontrollable errors in the data interpretation. The details of the relevant data manipulation can be found in the cited references. Specimen Preparation Specimens can be machined from solid cylinders or from tubes, though tubular specimens tend to have a different molecular orientation because of the extrusion process. Prior to machining, the cylinders need to be annealed at a temperature near the glass-transition temperature to remove residual stresses. The thinwalled cylinder samples need to be annealed again after machining to remove or reduce residual surface stresses possibly acquired during turning. To avoid excessive gravity deformations, annealing is best conducted in an oil bath. Any possible weight gain must be monitored with a balance possessing sufficient resolution. The weight gain should be low enough to avoid any effect of the oil on the viscoelastic behavior of materials. For testing in the glassy state, specimens must have about the same aging times; and the aging times should be at least a few days so that during measurements the aging time change is not significant (on a logarithmic scale). Because physical aging is such an important topic we devote further comments to it in the next subsection. Prior to experiments samples need to be kept in an environment with a constant relative humidity that is the same as the relative humidity during measurements. The relative humidity can be generated through a saturated salt solution in an enclosed container [3.118]. Physical Aging in Specimen Preparation. When an

Fig. 3.31 A typical speckle pattern on a cylinder surface

inclined with respect to the observation axis of the imaging system

amorphous polymer is cooled (continuously) from its melt state, its volume will deviate from its equilibrium state at the glass-transition temperature (Fig. 3.12).

Mechanics of Polymers: Viscoelasticity

87

the time required to attain equilibrium after quenching within practical limits. The value of t ∗ may have to be determined prior to commencing characterization tests. If the viscoelastic properties are investigated without paying attention to the aging process, characterization of polymers and their composites are not likely to generate repeatable results. For characterization of the long-term viscoelastic behavior through accelerated testing, physical aging effects have to be considered, in addition to time–temperature superposition and other mechanisms. An Example of Nonlinearly Viscoelastic Behavior under Combined Axial/Shear Stresses Figure 3.32 shows the creep response in pure shear for PMMA at 80 ◦ C [3.113, 117]. The axial force was controlled to be zero in these measurements. The plotted shear creep compliance was converted from the relaxation modulus in shear under infinitesimal deformation, representing the creep behavior in the linearly viscoelastic regime. For a material that behaves linearly viscoelastically, all curves should be coincident in this plot. In the case of data at these stress levels, the deformations in shear at higher stress levels are accelerated relative to the behavior at infinitesimal strains. This observation constitutes another criterion for separating linear from nonlinear response. To represent the nonlinear characteristics we draw on the isochronal representation discussed above. At any given time spanned in Fig. 3.32, there are five data points from creep under a pure shear stress, giving four sets of isochronal stress/strain data. Plotting these four log (shear creep compliance) (1/MPa) –2.2 –2.4

σ = 0, τ = 16 MPa σ = 0, τ = 14.7 MPa σ = 0, τ = 12.3 MPa σ = 0, τ = 9.4 MPa

T = 80°C

–2.6 –2.8 Inversion from µ(t)

–3 –3.2 0.5

1

1.5

2

2.5

3

3.5

4 4.5 log (time) (s)

Fig. 3.32 Shear creep compliance of PMMA at several levels of shear stress at 80 ◦ C (after [3.117])

Part A 3.4

Polymers have different viscoelastic characteristics depending on whether they are below the glass-transition temperature (Tg ), in the glass-transition region (in the neighborhood of Tg ), or in the rubbery state (above Tg ). In the rubbery state a polymer is in or near thermodynamic equilibrium, where long-range cooperative motions of long-chain molecules are dominant and result in translational movements of molecules. Below the glass-transition temperature, short-range motions in the form of side-chain motions and rotations of segments of the main chain (primarily in long-term behavior) are dominant. The glass-transition range depends on the cooling rate. After cooling a polymer initially in the rubbery state to an isothermal condition in the glassy state, the polymer enters a thermodynamically nonequilibrium, or metastable, state associated with a smaller density than an optimal condition (equilibrium) would allow. In the equilibrium state the density would increase continuously to its maximum value. If the temperature in the isothermal condition is near Tg , the density increase can occur in a relatively short time, but if the temperature is far below Tg this process occurs over a long period of time, on the order of days, weeks, or months. Prior to reaching the maximum density, as time evolves, depending on how long this process has taken place, the polymer possesses a different viscoelastic response. This phenomenon is called physical aging because no chemical changes occur. The time after quenching to an isothermal condition in the glassy state is called aging time. The viscoelastic functions (e.g., bulk and shear relaxation moduli) change during aging until that process is complete within practical time limitations. The effect of physical aging is similar to a continual decrease of the temperature and results in the reduction of the free volume that provides the space for the mobility of the polymer chain segments as the chain undergoes any rearrangement. There now exists a relatively large body of information on physical aging and the reader is referred to a number of representative publications, in which references in the open literature expand on this topic. References [3.38–44,119–122] showed that physical aging leads to an aging time factor multiplying the external time, analogous to the temperature-dependent multiplier (shift factor) for thermorheologically simple solids in the context of linear viscoelasticity theory. Elaborations of this theme for various materials have been offered to a large extent by McKenna and by Gates as well as their various collaborators Effects of physical aging can be pronounced before aging time reaches the value, say, t ∗ , which is

3.4 Nonlinearly Viscoelastic Material Characterization

88

Part A

Solid Mechanics Topics

log(axial creep compliance) (1/MPa) –2.2

A–A

Clamped with bolt A – A

Glued for θ > 40°

–2.4 Tension + torsion σ = 25.3 MPa, τ = 14.5 MPa

T = 50°C

–2.6 θ

–2.8 Compression + torsion σ = 25.3 MPa, τ = 14.5 MPa

–3

Part A 3.4

–3.2 0.5

A x y A

1

1.5

2

2.5

3

3.5

4 4.5 log (time) (s)

Fig. 3.33 Axial creep compliance of PMMA under tension/torsion and compression/torsion at 50 ◦ C

data points at each of the 16 fixed times, say, gives the isochronal stress–strain relation shown in Fig. 3.29. It is clear that the creep rate increases with an increase in applied shear stress, indicating nonlinear creep behavior in shear. We note that for isochronal behavior at strains above 0.5%, there exists a fan emanating from the shear strain 0.5% and a shear stress of 7.6 MPa. The corresponding fan center is considered to be the yield point, above which the creep rate is accelerTest section 10.16 R = 10.16

22.54

22.3

22.54

44.6

15.14 30.48

Fig. 3.34 Geometry of an Arcan specimen (all dimensions are in mm, thickness is 3 mm)

Fig. 3.35 Fixture for testing Arcan specimens

ated measurably. It is of interest to note that the creep process is more pronounced (accelerated) in tension/torsion than under compression/torsion as illustrated in Fig. 3.33 for 50 ◦ C. We have already observed that thin-walled cylinders tend to buckle under sufficiently high torsion and/or compression. A cylinder with an outer diameter of 25.15 mm, a thickness of 3.18 mm, and a test length of 76.2 mm would buckle at ≈ 5% shear strain under pure torsion. The use of thicker-walled cylinders would reduce the homogeneity of the stress and strain within the cylinder wall and lead to inaccuracy in the determination of stress or strain. Other techniques, such as testing with the Arcan specimen should, therefore, be used when the nonlinearly viscoelastic behavior at larger deformations is investigated. Use of the Arcan Specimen Arcan’s specimen [3.123–125] can be used for multibiaxial test with the use of a uniaxial material test system. Figure 3.34 shows an Arcan specimen, and Fig. 3.35 a corresponding test fixture. The loading axis can form different angles with respect to the specimen axis so that biaxial stress states can be generated in the region of uniform deformation in the middle of the specimen. When the loading axis of the fixture is aligned with the major specimen axis, this configura-

Mechanics of Polymers: Viscoelasticity

Fig. 3.36 Isochronal contours of creep strains under fixed biaxial loading. Each contour corresponds to a different time between 10 s and about 105 s. Ellipses correspond to linear response characteristics

3.5 Closing Remarks

89

Normal strain 0.025 0.02 0.015 0.01 0.005 0 – 0.005 – 0.01 – 0.015 – 0.02 – 0.025 –0.03

–0.02

–0.01

0

0.01

0.02 0.03 Normal strain

show ellipses (a/b = 2) that would correspond to totally linearly viscoelastic behavior.

3.5 Closing Remarks As was stated at the very beginning, today’s laboratory and general engineering environment is bound to involve polymers, whether of the rigid or the soft variety. The difference between these two derives merely from the value of their glass-transition temperature relative to the use temperature (usually room temperature). As illustrated in this chapter the linearized theory of viscoelasticity is well understood and formulated mathematically, even though its current application in engineering designs is usually not on a par with this understanding. A considerable degree of response estimation can be achieved with this knowledge, but a serious deficiency arises from the fact that when structural failures are of concern the linearized theory soon encounters limitations as nonlinear behavior is encountered. There is, today, no counterpart nonlinear viscoelastic material description that parallels the plasticity theory for metallic solids. Because the atomic structures of metals and polymers are fundamentally different, it would seem imprudent to characterize polymer nonlinear behavior along similar lines of physical reasoning and mathematical formulation, notwithstanding the fact that in uniaxial deformations permanent deformations in metals and polymers may appear to be similar. That similarity disappears as soon as temperature or extended

time scales follow an initially nonlinear deformation history. It is becoming clear already that the superposition of dilatational stresses or volumetric strains has a greater influence on nonlinear material response of polymers than is true for metals. Consequently it would seem questionable whether uniaxial tensile or compressive behavior would be a suitable method for assessing nonlinear polymer response, since that stress state involves both shear and bulk (volumetric) components. To support this observation one only needs to recall that very small amounts of volume change can have a highly disproportionate effect on the time dependence of the material, as delineated in Sects. 3.2.8, 3.2.9, and 3.2.10. This is well illustrated by the best known and large effect which a change in temperature has on the relaxation times, where dilatational strains are indeed very small compared to typical shear deformations; responses under pressure and with solvent swelling underscore this observation. The recent publication history for time-dependent material behavior exhibits an increasing number of papers dealing with nonlinear polymer behavior, indicating that efforts are underway to address this lack of understanding in the engineering profession. At the same time it is also becoming clear that the intrinsic time-dependent behavior of polymers is closely

Part A 3.5

tion induces shear forces applied to an Arcan specimen so that there is a pure shear zone in the central portion of the specimen. Other orientations allow the nonlinearly viscoelastic shear behavior to be characterized under loading conditions combining tension/shear, compression/shear, and pure shear. The data processing is illustrated using the results obtained by Knauss and Zhu [3.126,127] as an example. Figure 3.36 shows isochronal creep shear and normal strains at 80 ◦ C using an Arcan specimen under a nominal (maximum) shear stress of 19.3 MPa. At each fixed time, line segments connect points to form an isochronal strain contour. The innermost contour corresponds to a creep time of 10 s, and the outermost contour is the results from 0.8 × 105 s. For comparison purposes we also

90

Part A

Solid Mechanics Topics

connected to the molecular processes that are well represented by the linearly viscoelastic characterization of these solids. It is thus not unreasonable, in retrospect, to have devoted a chapter mostly to describing linearly

viscoelastic solids with the expectation that this knowledge provides a necessary if not sufficient background for dealing with future issues that need to be resolved in the laboratory.

3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known

Part A 3.6

Experimental work deals with a variety of situations/test configurations for which boundary value histories are not readily confined to a limited number of cases. For example, when deformations are to be measured by photoelasticity with the help of a viscoelastic, photosensitive material in a two-dimensional domain it may be important to know the local stress state very well, if knowing the stress state in a test configuration is an important prerequisite for resolving an engineering analysis problem. There exists a large class of elastic boundary value problems for which the distribution of stresses (or strains) turns out to be independent of the material properties. In such cases the effect of loading and material properties is expressed through a material-dependent factor and a load factor, both multiplying a function(s) that depends only on the spatial coordinates governing the distribution of stress or strain. It then follows that for the corresponding viscoelastic solution the distribution of stresses is also independent of the material properties and that the time dependence is formulated as the convolution of a material-dependent multiplicative function with the time-dependent load factored out from the spatial distribution function(s). In the experimental context beams and plates fall into this category, though even the simple plate configuration can involve Poisson’s ratio in its deformation field. More important is the class of simply connected two-dimensional domains for which the in-plane stress distribution is independent of the material properties. Consider first two-dimensional, quasistatic problems with (only) traction boundary conditions prescribed on a simply connected domain. For such problems the stress distribution of an elastic solid throughout the interior is independent of the material properties. The same situation prevails for multiply connected domains, provided the traction on each perforation is self-equilibrating. If the latter condition is not satisfied, then a history-dependent Poisson function enters the stress field description so that the stress distribution is, at best, only approximately independent of, or insensitive to, the material behavior. This topic is discussed in a slightly more detailed manner in [3.13].

3.6.1 Further Reading For general background it appears useful to identify the dominant publications to bolster one’s understanding of the theory of viscoelasticity. To that end we summarize here first, without comment or reference number, a list of publications in book or paper form, only one of which appears as an explicit reference in the text, namely the book by J.D. Ferry as part of the text development [3.45] with respect to the special topic of thermorheologically simple solids. 1. T. Alfrey: Mechanical Behavior of High Polymers (Interscience, New York, 1948) 2. B. Gross: Mathematical Structure of the Theories of Viscoelasticity (Herrmann, Paris, 1953) (re-issued 1968) 3. A.V. Tobolsky: Properties and Structure of Polymers (Wiley, New York, 1960) 4. M.E. Gurtin, E. Sternberg: On the linear theory of viscoelasticity, Arch. Rat. Mech. Anal. 11, 291–356 (1962) 5. F. Bueche: Physical Properties of Polymers (Interscience, New York, 1962) 6. M.L. Williams: The structural analysis of viscoelastic materials, AIAA J. 2, 785–809 (1964) 7. Flügge (Ed.): Encyclopedia of Physics VIa/3, M.J. Leitman, G.M.C. Fischer: The linear theory of viscoelasticity (Springer, Berlin, Heidelberg, 1973) 8. J.J. Aklonis, W.J. MacKnight: Introduction to Polymer Viscoelasticity (Wiley, New York 1983) 9. N.W. Tschoegl: The Phenomenological Theory of Linear Viscoelastic Behavior, an Introduction (Springer, Berlin, 1989) 10. A. Drozdov: Viscoelastic Structures, Mechanics of Growth and Aging (Academic, New York, 1998) 11. D.R. Bland: The Theory of Linear Viscoelasticity, Int. Ser. Mon. Pure Appl. Math. 10 (Pergamon, New York, 1960) 12. R.M. Christensen: Theory of Viscoelasticity: An Introduction (Academic, New York, 1971); see also R.M. Christensen: Theory of Viscoelasticity An Introduction, 2nd ed. (Dover, New York, 1982)

Mechanics of Polymers: Viscoelasticity

17. C.W. Folkes: Two systems for automatic reduction of time-dependent photomechanics data, Exp. Mech. 10, 64–71 (1970) 18. P.S. Theocaris: Phenomenological analysis of mechanical and optical behaviour of rheo-optically simple materials. In: Photoelastic Effect and its applications, ed. by J. Kestens (Springer, Berlin New York, 1975) pp. 146–152 19. B.D. Coleman, E.H. Dill: Photoviscoelasticity: Theory and practice. In: The Photoelastic Effect and its Applications, ed. by J. Kestens, (Springer, Berlin New York, 1975) pp. 455–505 20. M.A. Narbut: On the correspondence between dynamic stress states in an elastic body and in its photoviscoelastic model, Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron. (USSR) 1, 116–122 (1978) 21. R.J. Arenz, U. Soltész: Time-dependent optical characterization in the photoviscoelastic study of stress-waver propagation, Exp. Mech. 21, 227–233 (1981) 22. K.S. Kim, K.L. Dickerson, W.G. Knauss (Eds): Viscoelastic effect on dynamic crack propagation in Homalit 100. In: Workshop on Dynamic Fracture (California Institute of Technology, Pasadena, 1983) pp. 205–233 23. H. Weber: Ein nichtlineares Stoffgesetz für die ebene photoviskoelastische Spannungsanalyse, Rheol. Acta. 22, 114–122 (1983) 24. Y. Miyano, S. Nakamura, S. Sugimon, T.C. Woo: A simplified optical method for measuring residual stress by rapid cooling in a thermosetting resin strip, Exp. Mech. 26, 185–192 (1986) 25. A. Bakic: Practical execution of photoviscoelastic experiments, Oesterreichische Ingenieur- und Architekten-Zeitschrift, 131, 260–263 (1986) 26. K.S. Kim, K.L. Dickerson, W.G. Knauss: Viscoelastic behavior of opto-mechanical properties and its application to viscoelastic fracture studies, Int. J. Fract. 12, 265–283 (1987) 27. T. Kunio, Y. Miyano, S. Sugimori: Fundamentals of photoviscoelastic technique for analysis of time and temperature dependent stress and strain. In: Applied Stress Analysis, ed. by T.H. Hyde, E. Ollerton (Elsevier Applied Sciences, London, 1990) pp. 588– 597 28. K.-H. Laermann, C.Yuhai: On the measurement of the material response of linear photoviscoelastic polymers, Measurement, 279–286 (1993)

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References on Photoviscoelasticity 1. R.D. Mindlin: A mathematical theory of photoviscoelasticity, J. Appl. Phys. 29, 206–210 (1949) 2. R.S. Stein, S. Onogi, D.A. Keedy: The dynamic birefringence of high polymers, J. Polym. Sci. 57, 801–821 (1962) 3. C.W. Ferguson: Analysis of stress-wave propagation by photoviscoelastic techniques, J. Soc. Motion Pict. Telev. Eng. 73, 782–787 (1964) 4. P.S. Theocaris, D. Mylonas: Viscoelastic effects in birefringent coating, J. Appl. Mech. 29, 601–607 (1962) 5. M.L. Williams, R.J. Arenz: The engineering analysis of linear photoviscoelastic materials, Exp. Mech. 4, 249–262 (1964) 6. E.H. Dill: On phenomenological rheo-optical constitutive relations, J. Polym. Sci. Part C 5, 67–74 (1964) 7. C.L. Amba-Rao: Stress-strain-time-birefringence relations in photoelastic plastics with creep, J. Polym. Sci. Pt. C 5, 75–86 (1964) 8. B.E. Read: Dynamic birefringence of amorphous polymers, J. Polym. Sci. Pt. C 5, 87–100 (1964) 9. R.D. Andrews, T.J. Hammack: Temperature dependence of orientation birefringence of polymers in the glassy and rubbery states, J. Polym. Sci. Pt. C 5, 101–112 (1964) 10. R. Yamada, C. Hayashi, S. Onogi, M. Horio: Dynamic birefringence of several high polymers, J. Polym. Sci. Pt. C 5, 123–127 (1964) 11. K. Sasguri, R.S. Stain: Dynamic birefringence of polyolefins, J. Polym. Sci. Pt. C 5, 139–152 (1964) 12. D.G. Legrand, W.R. Haaf: Rheo-optical properties of polymers, J. Polym. Sci. Pt. C 5, 153–161 (1964) 13. I.M. Daniel: Experimental methods for dynamic stress analysis in viscoelastic materials, J. Appl. Mech. 32, 598–606 (1965) 14. I.M. Daniel: Quasistatic properties of a photoviscoelastic material, Exp. Mech. 5, 83–89 (1965) 15. A.J. Arenz, C.W. Ferguson, M.L. Williams: The mechanical and optical characterization of a Solithane 113 composition, Exp. Mech. 7, 183– 188 (1967) 16. H.F. Brinson: Mechanical, optical viscoelastic characterization of Hysol 4290: Time and temperature behavior of Hysol 4290 as obtained from creep tests in conjunction with the time-temperature superposition principle, Exp. Mech. 8, 561–566 (1968)

3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known

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29. S. Yoneyama, J. Gotoh, M. Takashi: Experimental analysis of rolling contact stresses in a viscoelastic strip, Exp. Mech. 40, 203–210 (2000) 30. A.I. Shyu, C.T. Isayev, T.I. Li: Photoviscoelastic behavior of amorphous polymers during transition

from the glassy to rubbery state, J. Polym. Sci. Pt. B Polym. Phys. 39, 2252–2262 (2001) 31. Y.-H. Zhao, J. Huang: Photoviscoelastic stress analysis of a plate with a central hole, Exp. Mech. 41, 312–18 (2001)

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W.G. Knauss: The mechanics of polymer fracture, Appl. Mech. Rev. 26, 1–17 (1973) C. Singer, E.J. Holmgard, A.R. Hall (Eds.): A History of Technology (Oxford University Press, New York 1954) J.M. Kelly: Strain rate sensitivity and yield point behavior in mild steel, Int. J. Solids Struct. 3, 521–532 (1967) H.H. Johnson, P.C. Paris: Subcritical flaw growth, Eng. Fract. Mech. 1, 3–45 (1968) I. Finnie: Stress analysis for creep and creeprupture. In: Appllied Mechanics Surveys, ed. by H.N. Abramson (Spartan Macmillan, New York 1966) pp. 373–383 F. Garofalo: Fundamentals of Creep and Creep Rupture in Metals (Macmillan, New York 1965) N.J. Grant, A.W. Mullendore (Eds.): Deformation and Fracture at Elevated Temperatures (MIT Press, Cambridge 1965) F.A. McClintock, A.S. Argon (Eds.): Mechanical Behavior of Materials (Addison-Wesley, Reading 1966) J.B. Conway: Numerical Methods for Creep and Rupture Analyses (Gordon-Breach, New York 1967) J.B. Conway, P.N. Flagella: Creep Rupture Data for the Refractory Metals at High Temperatures (Gordon-Breach, New York 1971) M. Tao: High Temperature Deformation of Vitreloy Bulk Metallic Glasses and Their Composite. Ph.D. Thesis (California Institute of Technology, Pasadena 2006) J. Lu, G. Ravichandran, W.L. Johnson: Deformation behavior of the Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 bulk metallic glass over a wide range of strain-rates and temperatures, Acta Mater. 51, 3429–3443 (2003) W.G. Knauss: Viscoelasticity and the timedependent fracture of polymers. In: Comprehensive Structural Integrity, Vol. 2, ed. by I. Milne, R.O. Ritchie, B. Karihaloo (Elsevier, Amsterdam 2003) W. Flügge: Viscoelasticity (Springer, Berlin 1975) H. Lu, X. Zhang, W.G. Knauss: Uniaxial, shear and Poisson relaxation and their conversion to bulk relaxation, Polym. Eng. Sci. 37, 1053–1064 (1997) N.W. Tschoegl, W.G. Knauss, I. Emri: Poisson’s ratio in linear viscoelasticity, a critical review, Mech. Time-Depend. Mater. 6, 3–51 (2002) I.L. Hopkins, R.W. Hamming: On creep and relaxation, J. Appl. Phys. 28, 906–909 (1957)

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R.A. Schapery: Approximate methods of transform inversion for viscoelastic stress analysis, Proc. 4th US Natl. Congr. Appl. Mech. (1962) pp. 1075–1085 J.F. Clauser, W.G. Knauss: On the numerical determination of relaxation and retardation spectra for linearly viscoelastic materials, Trans. Soc. Rheol. 12, 143–153 (1968) G.W. Hedstrom, L. Thigpen, B.P. Bonner, P.H. Worley: Regularization and inverse problems in viscoelasticity, J. Appl. Mech. 51, 121–12 (1984) I. Emri, N.W. Tschoegl: Generating line spectra from experimental responses. Part I: Relaxation modulus and creep compliance, Rheol. Acta. 32, 311–321 (1993) I. Emri, N.W. Tschoegl: Generating line spectra from experimental responses. Part IV: Application to experimental data, Rheol. Acta. 33, 60–70 (1994) I. Emri, N.W. Tschoegl: An iterative computer algorithm for generating line spectra from linear viscoelastic response functions, Int. J. Polym. Mater. 40, 55–79 (1998) N.W. Tschoegl, I. Emri: Generating line spectra from experimental responses. Part II. Storage and loss functions, Rheol. Acta. 32, 322–327 (1993) N.W. Tschoegl, I. Emri: Generating line spectra from experimental responses. Part III. Interconversion between relaxation and retardation behavior, Int. J. Polym. Mater. 18, 117–127 (1992) I. Emri, N.W. Tschoegl: Generating line spectra from experimental responses. Part V. Time-dependent viscosity, Rheol. Acta. 36, 303–306 (1997) I. Emri, B.S. von Bernstorff, R. Cvelbar, A. Nikonov: Re-examination of the approximate methods for interconversion between frequency- and timedependent material functions, J. Non-Newton. Fluid Mech. 129, 75–84 (2005) A. Nikonov, A.R. Davies, I. Emri: The determination of creep and relaxation functions from a single experiment, J. Rheol. 49, 1193–1211 (2005) M.A. Branch, T.F. Coleman, Y. Li: A subspace interior, and conjugate gradient method for large-scale bound-constrained minimization problems, Siam J. Sci. Comput. 21, 1–23 (1999) F. Kohlrausch: Experimental-Untersuchungen über die elastische Nachwirkung bei der Torsion, Ausdehnung und Biegung, Pogg. Ann. Phys. 8, 337 (1876)

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J. Bischoff, E. Catsiff, A.V. Tobolsky: Elastoviscous properties of amorphous polymers in the transition region 1, J. Am. Chem. Soc. 74, 3378–3381 (1952) F. Schwarzl, A.J. Staverman: Time-temperature dependence of linearly viscoelastic behavior, J. Appl. Phys. 23, 838–843 (1952) M.L. Williams, R.F. Landel, J.D. Ferry: Mechanical properties of substances of high molecular weight. 19. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids, J. Am. Chem. Soc. 77, 3701–3707 (1955) D.J. Plazek: Temperature dependence of the viscoelastic behavior of polystyrene, J. Phys. Chem. US 69, 3480–3487 (1965) D.J. Plazek: The temperature dependence of the viscoelastic behavior of poly(vinyl acetate), Polym. J. 12, 43–53 (1980) R.A. Fava (Ed.): Methods of experimental physics 16C. In: Viscoelastic and Steady-State Rheological Response, ed. by D.J. Plazek (Academic, New York 1979) L.W. Morland, E.H. Lee: Stress analysis for linear viscoelastic materials with temperature variation, Trans. Soc. Rheol. 4, 233–263 (1960) L.J. Heymans: An Engineering Analysis of Polymer Film Adhesion to Rigid Substrates. Ph.D. Thesis (California Institute of Technology, Pasadena 1983) G.U. Losi, W.G. Knauss: Free volume theory and nonlinear viscoelasticity, Polym. Eng. Sci. 32, 542–557 (1992) I. Emri, T. Prodan: A Measuring system for bulk and shear characterization of polymers, Exp. Mech. 46(6), 429–439 (2006) N.W. Tschoegl, W.G. Knauss, I. Emri: The effect of temperature and pressure on the mechanical properties of thermo- and/or piezorheologically simple polymeric materials in thermodynamic equilibrium – a critical review, Mech. Time-Depend. Mater. 6, 53–99 (2002) R.W. Fillers, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers, Trans. Soc. Rheol. 21, 51–100 (1977) W.K. Moonan, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers. 2. Expansively and compressibility measurements, Macromolecules 16, 55–59 (1983) W.K. Moonan, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers. 3. Substitution of the glassy parameters for those of the occupied volume, Int. Polym. Mater. 10, 199–211 (1984) W.K. Moonan, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers. 4. Measurements in torsion, J. Polym. Sci. Polym. Phys. 23, 623–651 (1985)

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G. Williams, D.C. Watts: Non-symmetrical dielectric behavior arising from a simple empirical decay function, Trans. Faraday Soc. 66, 80–85 (1970) S. Lee, W.G. Knauss: A note on the determination of relaxation and creep data from ramp tests, Mech. Time-Depend. Mater. 4, 1–7 (2000) A. Flory, G.B. McKenna: Finite step rate corrections in stress relaxation experiments: A comparison of two methods, Mech. Time-Depend. Mater. 8, 17–37 (2004) J.F. Tormey, S.C. Britton: Effect of cyclic loading on solid propellant grain structures, AIAA J. 1, 1763–1770 (1963) R.A. Schapery: Thermomechanical behavior of viscoelastic media with variable properties subjected to cyclic loading, J. Appl. Mech. 32, 611–619 (1965) L.R.G. Treloar: Physics of high-molecular materials, Nature 181, 1633–1634 (1958) M.-F. Vallat, D.J. Plazek, B. Bushan: Effects of thermal treatment of biaxially oriented poly(ethylene terephthalate), J. Polym. Sci. Polym. Phys. 24, 1303– 1320 (1986) L.E. Struik: Physical Aging in Amorphous Polymers and Other Materials (Elsevier Scientific, Amsterdam 1978) G.B. McKenna, A.J. Kovacs: Physical aging of poly(methyl methacrylate) in the nonlinear range – torque and normal force measurements, Polym. Eng. Sci. 24, 1138–1141 (1984) A. Lee, G.B. McKenna: Effect of crosslink density on physical aging of epoxy networks, Polymer 29, 1812– 1817 (1988) C. G’Sell, G.B. McKenna: Influence of physical aging on the yield response of model dgeba poly(propylene oxide) epoxy glasses, Polymer 33, 2103–2113 (1992) M.L. Cerrada, G.B. McKenna: Isothermal, isochronal and isostructural results, Macromolecules 33, 3065– 3076 (2000) L.C. Brinson, T.S. Gates: Effects of physical aging on long-term creep of polymers and polymer matrix composites, Int. J. Solids Struct. 32, 827–846 (1995) L.M. Nicholson, K.S. Whitley, T.S. Gates: The combined influence of molecular weight and temperature on the physical aging and creep compliance of a glassy thermoplastic polyimide, Mech. TimeDepend. Mat. 5, 199–227 (2001) J.D. Ferry: Viscoelastic Properties of Polymers, 3rd edn. (Wiley, New York 1980) J.R. Mcloughlin, A.V. Tobolsky: The viscoelastic behavior of polymethyl methacrylate, J. Colloid Sci. 7, 555–568 (1952) H. Leaderman, R.G. Smith, R.W. Jones: Rheology of polyisobutylene. 2. Low molecular weight polymers, J. Polym. Sci. 14, 47–80 (1954)

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W.G. Knauss, V.H. Kenner: On the hygrothermomechanical characterization of polyvinyl acetate, J. Appl. Phys. 51, 5131–5136 (1980) I. Emri, V. Pavˇsek: On the influence of moisture on the mechanical properties of polymers, Mater. Forum 16, 123–131 (1992) D.J. Plazek: Magnetic bearing torsional creep apparatus, J. Polym. Sci. Polym. Chem. 6, 621–633 (1968) D.J. Plazek, M.N. Vrancken, J.W. Berge: A torsion pendulum for dynamic and creep measurements of soft viscoelastic materials, Trans. Soc. Rheol. 2, 39–51 (1958) G.C. Berry, J.O. Park, D.W. Meitz, M.H. Birnboim, D.J. Plazek: A rotational rheometer for rheological studies with prescribed strain or stress history, J. Polym. Sci. Polym. Phys. Ed. 27, 273–296 (1989) R.S. Duran, G.B. McKenna: A torsional dilatometer for volume change measurements on deformed glasses: Instrument description and measurements on equilibrated glasses, J. Rheol. 34, 813–839 (1990) J.E. McKinney, S. Edelman, R.S. Marvin: Apparatus for the direct determination of the dynamic bulk modulus, J. Appl. Phys. 27, 425–430 (1956) J.E. McKinney, H.V. Belcher: Dynamic compressibility of poly(vinyl acetate) and its relation to free volume, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 43–53 (1963) T.H. Deng, W.G. Knauss: The temperature and frequency dependence of the bulk compliance of Poly(vinyl acetate). A re-examination, Mech. TimeDepend. Mater. 1, 33–49 (1997) S. Sane, W.G. Knauss: The time-dependent bulk response of poly (methyl methacrylate), Mech. TimeDepend. Mater. 5, 293–324 (2001) Z. Ma, K. Ravi-Chandar: Confined compression– a stable homogeneous deformation for multiaxial constitutive characterization, Exp. Mech. 40, 38–45 (2000) K. Ravi-Chandar, Z. Ma: Inelastic deformation in polymers under multiaxial compression, Mech. Time-Depend. Mater. 4, 333–357 (2000) D. Qvale, K. Ravi-Chandar: Viscoelastic characterization of polymers under multiaxial compression, Mech. Time-Depend. Mater. 8, 193–214 (2004) S.J. Park, K.M. Liechti, S. Roy: Simplified bulk experiments and hygrothermal nonlinear viscoelasticty, Mech. Time-Depend. Mater. 8, 303–344 (2004) A. Kralj, T. Prodan, I. Emri: An apparatus for measuring the effect of pressure on the timedependent properties of polymers, J. Rheol. 45, 929–943 (2001) J.B. Pethica, R. Hutchings, W.C. Oliver: Hardness measurement at penetration depths as small as 20 nm, Philos. Mag. 48, 593–606 (1983) W.C. Oliver, R. Hutchings, J.B. Pethica: Measurement of hardness at indentation depths as low as 20 nanometers. In: Microindentation Techniques in Materials Science and Engineering ASTM STP 889, ed.

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3.111 R.A. Schapery: A theory of non-linear thermoviscoelasticity based on irreversible thermodynamics, Proc. 5th Natl. Cong. Appl. Mech. (1966) pp. 511–530 3.112 R.A. Schapery: On the characterization of nonlinear viscoelastic materials, Polym. Eng. Sci. 9, 295–310 (1969) 3.113 H. Lu, W.G. Knauss: The role of dilatation in the nonlinearly viscoelastic behavior of pmma under multiaxial stress states, Mech. Time-Depend. Mater. 2, 307–334 (1999) 3.114 H. Lu, G. Vendroux, W.G. Knauss: Surface deformation measurements of cylindrical specimens by digital image correlation, Exp. Mech. 37, 433–439 (1997) 3.115 W.H. Peters, W.F. Ranson: Digital imaging techniques in experimental stress analysis, Opt. Eng. 21, 427–432 (1982) 3.116 M.A. Sutton, W.J. Wolters, W.H. Peters, W.F. Ranson, S.R. McNeil: Determination of displacements using an improved digital image correlation method, Im. Vis. Comput. 1, 133–139 (1983) 3.117 H. Lu: Nonlinear Thermo-Mechanical Behavior of Polymers under Multiaxial Loading. Ph.D. Thesis (California Institute of Technology, Pasadena 1997) 3.118 D.R. Lide: CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton 1995) 3.119 G.B. McKenna: On the physics required for prediction of long term performance of polymers and their composites, J. Res. NIST 99(2), 169–189 (1994) 3.120 P.A. O’Connell, G.B. McKenna: Large deformation response of polycarbonate: Time-temperature, time-aging time, and time-strain superposition, Polym. Eng. Sci. 37, 1485–1495 (1997) 3.121 J.L. Sullivan, E.J. Blais, D. Houston: Physical aging in the creep-behavior of thermosetting and thermoplastic composites, Compos. Sci. Technol. 47, 389–403 (1993) 3.122 I.M. Hodge: Physical aging in polymer glasses, Science 267, 1945–1947 (1995) 3.123 N. Goldenberg, M. Arcan, E. Nicolau: On the most suitable specimen shape for testing sheer strength of plastics, ASTM STP 247, 115–121 (1958) 3.124 M. Arcan, Z. Hashin, A. Voloshin: A method to produce uniform plane-stress states with applications to fiber-reinforced materials, Exp. Mech. 18, 141–146 (1978) 3.125 M. Arcan: Discussion of the iosipescu shear test as applied to composite materials, Exp. Mech. 24, 66– 67 (1984) 3.126 W.G. Knauss, W. Zhu: Nonlinearly viscoelastic behavior of polycarbonate. I. Response under pure shear, Mech. Time-Depend. Mater. 6, 231–269 (2002) 3.127 W.G. Knauss, W. Zhu: Nonlinearly viscoelastic behavior of polycarbonate. II. The role of volumetric strain, Mech. Time-Depend. Mater. 6, 301–322 (2002)

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J.L. Loubet, B.N. Lucas, W.C. Oliver: Some measurements of viscoelastic properties with the help of nanoindentation, International workshop on Instrumental Indentation, ed. by D.T. Smith (NIST Special Publication, San Diego 1995) pp. 31–34 B.N. Lucas, W.C. Oliver, J.E. Swindeman: The dynamic of frequency-specific, depth-sensing indentation testing, Fundamentals of Nanoindentation and Nanotribology, Vol. 522, ed. by N.R. Moody (Mater. Res. Soc., Warrendale 1998) pp. 3–14, MRS Meeting, San Francisco 1998 R.J. Arenz, M.L. Williams (Eds.): A photoelastic technique for ground shock investigation. In: Ballistic and Space Technology (Academic, New York 1960) Y. Miyano, T. Tamura, T. Kunio: The mechanical and optical characterization of Polyurethane with application to photoviscoelastic analysis, Bull. JSME 12, 26–31 (1969) A.B.J. Clark, R.J. Sanford: A comparison of static and dynamic properties of photoelastic materialsm, Exp. Mech. 3, 148–151 (1963) O.A. Hasan, M.C. Boyce: A constitutive model for the nonlinear viscoelastic-viscoplastic behavior of glassy polymers, Polym. Eng. Sci. 35, 331–334 (1995) J.S. Bergström, M.C. Boyce: Constitutive modeling of the large strain time-dependent behavior of elastomers, J. Mech. Phys. Solids 46, 931–954 (1998) P.D. Ewing, S. Turner, J.G. Williams: Combined tension-torsion studies on polymers: apparatus and preliminary results for Polyethylene, J. Strain Anal. Eng. 7, 9–22 (1972) C. Bauwens-Crowet: The compression yield behavior of polymethal methacrylate over a wide range of temperatures and strain rates, J. Mater. Sci. 8, 968– 979 (1973) L.C. Caraprllucci, A.F. Yee: The biaxial deformation and yield behavior of bisphenol-A polycarbonate: Effect of anisotropy, Polym. Eng. Sci. 26, 920–930 (1986) W.G. Knauss, I. Emri: Non-linear viscoelasticity based on free volume consideration, Comput. Struct. 13, 123–128 (1981) W.G. Knauss, I. Emri: Volume change and the nonlinearly thermo-viscoelastic constitution of polymers, Polym. Eng. Sci. 27, 86–100 (1987) G.U. Losi, W.G. Knauss: Free volume theory and nonlinear thermoviscoelasticity, Polym. Eng. Sci. 32, 542–557 (1992) G.U. Losi, W.G. Knauss: Thermal stresses in nonlinearly viscoelastic solids, J. Appl. Mech. 59, S43–S49 (1992) W.G. Knauss, S. Sundaram: Pressure-sensitive dissipation in elastomers and its implications for the detonation of plastic explosives, J. Appl. Phys. 96, 7254–7266 (2004)

References

97

Composite M 4. Composite Materials

Peter G. Ifju

4.2.3 Shear Testing ............................... 105 4.2.4 Single-Geometry Tests .................. 106 4.3

Micromechanics.................................... 4.3.1 In Situ Strain Measurements .......... 4.3.2 Fiber–Matrix Interface Characterization........................... 4.3.3 Nanoscale Testing ........................ 4.3.4 Self-Healing Composites ...............

107 107 108 109 111

4.4 Interlaminar Testing ............................. 4.4.1 Mode I Fracture............................ 4.4.2 Mode II Fracture........................... 4.4.3 Edge Effects .................................

111 111 113 114

4.5 Textile Composite Materials ................... 4.5.1 Documentation of Surface Strain .... 4.5.2 Strain Gage Selection .................... 4.5.3 Edge Effects in Textile Composites ..

114 114 116 117 117 118 118 119 120 120

4.1

Strain Gage Applications ....................... 98 4.1.1 Transverse Sensitivity Corrections ... 98 4.1.2 Error Due to Gage Misalignment ..... 99 4.1.3 Temperature Compensation ........... 100 4.1.4 Self-Heating Effects ...................... 101 4.1.5 Additional Considerations ............. 101

4.6 Residual Stresses in Composites ............. 4.6.1 Composite Sectioning ................... 4.6.2 Hole-Drilling Methods .................. 4.6.3 Strain Gage Methods .................... 4.6.4 Laminate Warpage Methods .......... 4.6.5 The Cure Reference Method ...........

4.2

Material Property Testing ...................... 102 4.2.1 Tension Testing ............................ 103 4.2.2 Compression Testing ..................... 103

4.7

References .................................................. 121

The advancement of civilization has historically been tied to the materials utilized during the era. For example, we document the technological progression of mankind through the stone age, the bronze age, the iron age, etc. [4.1]. If such a description were to be used for the second half of the 20-th century, it is conceivable that the era could be called the composites age. AdR vanced composites such as carbon fiber and Kevlar reinforced polymers, and ceramic matrix materials represent the pinnacle of structural material forms during this era. These are the material systems of choice when high specific strength and stiffness are required [4.2–4].

A composite is defined as a material composed of two or more constituents whose mechanical properties are distinctly different from each other, and phase separated such that at least one of the constituents forms a continuous interconnected region and one of the constituents acts as the reinforcement (and is typically discontinuous) [4.2–4]. The resulting composite has physical properties that differ from the original constituents. By nature, experimental stress analysis on composites can be considerably difficult, since composites can be highly anisotropic and heterogeneous. Many of the well-established experimental techniques

Future Challenges ................................. 121

Part A 4

The application of selected experimental stress analysis techniques for mechanical testing of composite materials is reviewed. Because of the anisotropic and heterogeneous nature of composites, novel methodologies are often adopted. This chapter reviews many of the more applicable experimental methods in specific research areas, including: composite-specific strain gage applications, material property characterization, micromechanics, interlaminar testing, textile composite testing, and residual stress measurements. It would be impossible to review all test methods associated with composites in this short chapter, but many prevalent ones are covered.

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require special procedures, different data analysis, or may not be applicable. Additionally, for anisotropic materials the principal strain direction does not necessarily coincide with the principal stress direction. Failure mechanisms in composites can be also be significantly more complicated than for traditional isotropic materials. There are three main forms of composites: particulate reinforced, lamellar, and fiber reinforced. Some composite materials combine two or more of these forms; for example, a sandwich structure is a lamellar form that can incorporate fiber reinforced face sheets. In this chapter we will focus on advanced fiber and particulate reinforced composites. The objective of this chapter is to help guide the experimentalist by introducing modern experimental stress analysis techniques that are applicable to composites. The chapter is organized by research topics within the composites field. These include: special consideration when testing with electrical resistance strain gages,

coupon and material property testing, micromechanics studies, interlaminar fracture, textile composites, residual stress, and thermal testing. Other chapters of this manual cover time-dependent testing, high-strain-rate experiments, thermoelastic stress analysis, and fiberoptic strain gages, all of which are also important to the study of composites. In 1989 the Composite Materials Technical Division (CMTD) of the Society for Experimental Mechanics, sponsored publication of the Manual on Experimental Methods for Mechanical Testing of Composites [4.5]. The publication was edited by Pendelton and Tuttle. In 1998, Jenkins and the CMTD revised and updated the manual [4.6]. The most recent version is a 264-page book that covers broad aspects of composite testing and analysis. This chapter is significantly more condensed, and highlights applications as well as updates the most recent manual. Often during the course of the chapter, the author will refer the reader to this manual for more details.

4.1 Strain Gage Applications To this day, strain gages are still the most used experimental method to measure mechanical strain on isotropic as well as composite material applications. The previous chapters on strain gages clearly outline the theory and application of strain gages for general testing. For composite applications there are a number of special considerations [4.6] and procedures that should be followed in order to insure accurate and repeatable measurements. This section expands on the use of strain gages for composite material applications by providing a description of these special considerations and showing examples of how severe errors may occur if precautions are not implemented.

4.1.1 Transverse Sensitivity Corrections To obtain accurate strain measurements, transverse sensitivity corrections must be performed in all but two cases [4.7, 8]. The first case is when the transverse sensitivity coefficient K t is equal to zero. This is very rare: typically the transverse sensitivity coefficient is nonzero. The second is when the axial strain is measured on a material with a Poisson’s ratio equal to that of the strain gage calibration device and the state of stress is identical to the calibration device. For the latter case, in practice, this means that the tensile strain is measured for a uniaxial state of stress on a material with a Pois-

son’s ratio ν0 equal to 0.285 (steel). In general, errors due to transverse sensitivity are greatest when the transverse strain with respect to the gage εt is large compared to the axial strain with respect to the gage εa and the transverse sensitivity coefficient is high. Equation (4.1) describes the percentage error E if no corrections are performed due to transverse sensitivity. E=

K t [(εt /εa ) + ν0 ] × 100 . 1 − ν0 K t

(4.1)

For a simple experiment to determine the Poisson’s ratio of metallic material using strain gages, shown in Fig. 4.1, one would apply two strain gages to a tensile specimen, one aligned in the axial direction with respect to the load and one mounted in the transverse direction with respect to the load. If the specimen were aluminum (with a Poisson’s ratio of 0.33) and the transverse sensitivity correction factor of the gage was 2%, then the error in the axial gage would be very small (E = −0.09%) and the error in the transverse gage would be larger (E = −5.46%). If one were to completely ignore these corrections then one would calculate the Poisson’s ratio to be 0.312, rather than 0.33. Although most strain gages have a transverse sensitivity correction factor less than 2%, one can appreciate that corrections should be made.

Composite Materials

σx

4.1 Strain Gage Applications

99

to correct for transverse errors in the other gage. When orthogonal gages are used, as shown in Fig. 4.2, biaxial strain gage rosette corrections may be applied. The correction equations are given in (4.2), which gives the true strains ε1 and ε2 , obtained from the indicated strains ε1 and ε2 , and the transverse sensitivity correction factors K t1 and K t2 for gages 1 and 2.

σx

ε1 (1 − ν0 K t1 ) − K t1 ε2 (1 − ν0 K t2 ) , 1 − K t1 K t2 ε (1 − ν0 K t2 ) − K t2 ε1 (1 − ν0 K t1 ) ε2 = 2 . (4.2) 1 − K t1 K t2 Note that for each gage in the rosette, there may be a different transverse sensitivity coefficient. Similarly, the equations for a rectangular rosette (as shown in Fig. 4.2) are provided in (4.3). Often K t1 and K t3 are equal to each other. ε1 =

Fiber direction

Isotropic material

Unidirectional composite

Even for a seemingly routine test, if the specimen were unidirectional graphite/epoxy, large errors can arise if corrections are not made. For instance, if the objective is to measure ν21 , consider a specimen loaded in the direction transverse to the fibers (Fig. 4.1). For graphite/epoxy a typical ν21 is around 0.015 and therefore the strain gage used to measure the normal strain transverse to the loading direction experiences εt on the order of 66 times that of εa . For K t = 2% the strain gage measuring the strain transverse to the loading direction would yield a value with −132% error if transverse sensitivity corrections are not applied. Therefore the Poisson ratio ν21 would have an error of the order 132% since the error in the axial gage is very small. In order to perform an accurate correction for transverse sensitivity, at least two gages are required. For instance, in the above example, one gage may be used y

y

2

3

2 45°

x 1 Two gage biaxial rosette

1

x

Three gage rectangular rosette

Fig. 4.2 Biaxial and rectangular rosettes

ε1 (1 − ν0 K t1 ) − K t1 ε3 (1 − ν0 K t3 ) , 1 − K t1 K t3 ε (1 − ν0 K t2 ) ε2 = 2 1− K   t2 − K t2 ε1 (1 − ν0 K t1 )(1 − K t3 )  + ε3 (1 − ν0 K t3 )(1 − K t1 )  −1 × (1 − K t1 K t3 )(1 − K t2 ) , ε3 (1 − ν0 K t3 ) − K t3 ε1 (1 − ν0 K t1 ) ε3 = . 1 − K t1 K t3

ε1 =

Part A 4.1

Fig. 4.1 Measurement of the Poisson ratio ν for an isotropic material and ν21 for a unidirectional composite material

(4.3)

4.1.2 Error Due to Gage Misalignment Whenever a practitioner mounts strain gages on a mechanical component or test coupon, gage alignment is an important consideration [4.9]. If the gage is not aligned precisely with respect to the loading direction or geometry of the component, the strain information that is recorded may differ from the value obtained if it were aligned properly. This can lead to errors and/or misinterpretation of the data. For isotropic materials, errors of the order of a couple degrees typically do not lead to significant errors in strain measurement. For instance, for a simple tension test on a steel bar, where both axial and transverse (with respect to the load) gages are utilized, there would be an error in the axial strain of − 0.63% and in the transverse strain of 2.2%, for a misalignment of 4◦ . This may or may not be within the acceptable error for such a measurement. Most able practitioners can routinely align gages to within 2◦ . The errors for such a 2◦ misalignment are − 0.16% and 0.55% for the axial and transverse gages, respectively.

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θ

Percent strain measurement error 80

σx

β = –4°

β

60 β = –2°

40 20 0 β

–20 β = 2°

–40 –60 –80

β = 4°

0

10

20

30

40

50

60

70 80 90 Fiber angle (deg)

Fig. 4.5 Transverse strain errors as a function of fiber an-

σx

gle and gage misalignment error (courtesy M. E. Tuttle)

Part A 4.1

Fig. 4.3 Gage misalignment nomenclature

For unidirectional composites, misalignment can introduce significant errors because of the deleterious interaction between the loading direction and the orthotropy of the material. Take the same unidirectional graphite/epoxy material from the previous section where the moduli are E 1 = 170 GPa, E 2 = 8 GPa, G 12 = 6 GPa, and the Poisson’s ratio is ν12 = 0.32. A case where the gage is intended to line up with the load will be used for an example. For a tension test, where the fiber direction with respect to the loading Percent strain measurement error 20

direction is denoted by θ, and the gage misalignment denoted by β, Figs. 4.3, 4.4, and 4.5 illustrate the relationship between the misalignment angle β and the percentage strain measurement error and the fiber angle θ. Figure 4.4 pertains to errors in the axial direction of the specimen and Fig. 4.5 pertains to the transverse direction. Errors exceeding 15% can occur at fiber angles near 12◦ for the axial gage and greater than 60% near a 12◦ fiber angle for the transverse gage. Because of the exaggerated error when testing composites, special procedures may be warranted in the experimental program to ensure proper gage alignment.

4.1.3 Temperature Compensation

β = –4°

15 β = –2°

10 5 0 –5

β = 2°

–10 –15 –20

β = 4°

0

10

20

30

40

50

60

70 80 90 Fiber angle (deg)

Fig. 4.4 Axial strain errors as a function of fiber angle and

gage misalignment angle (courtesy M. E. Tuttle)

When a strain gage is mounted on a mechanical component or test coupon which is subjected to both mechanical loads and temperature variations, the gage measurements are a combination of strains induced by both effects [4.10, 11]. It is often the objective to separate the effects of the mechanical loads, a process known as temperature compensation. The strain gage output which results from temperature variation can occur for two reasons: the electrical resistance of the gage changes with temperature, and the coefficient of expansion of the gage may be different from that of the underlying material. Temperature compensation is more difficult when dealing with composites than with homogeneous isotropic materials. Some of the options for metallic materials are not applicable to composites. For instance,

Composite Materials

4.1.4 Self-Heating Effects According to Joule’s law, the application of voltage to a strain gage (a resistor) creates a power loss which results in the generation of heat [4.7, 12]. This heat is dissipated in the form of conduction by the substrate material and the surrounding environment. The heat that is generated is primarily related to the excitation voltage of the strain gage circuit, as well as to the resistance of the strain gage. The ability of the substrate material to stabilize or neutralize self-heating is related to its power density, which is a measure of

its ability to act as a heat sink. When self-heating is not neutralized it can result in gradual heating of the substrate, which can lead to creep of the underlying material, as well as signal hysteresis and drift. This is typically not acceptable and therefore steps must be implemented in order to avoid self-heating. In general, one can choose a combination of gage resistance, excitation voltage, and gage size to avoid the problem. Larger gage resistance levels develop less self-heating, larger grid areas allow more efficient dissipation, and lower excitation levels decrease self-heating. However, by decreasing the excitation level, the circuit sensitivity also decreases. Additionally, higher-resistance and larger gages can be more expensive. A balance must be struck in order to optimize the experimental configuration [4.12]. Most composite materials are fabricated with polymer matrices. Since polymers have low power densities (and are thus poor heat sinks), it is very important to select gages with a higher resistance and larger grid area, and/or use lower excitation levels as compared to strain gages used for traditional materials. A range of acceptable power densities for composites is 0.31–1.2 kW/m2 (0.2–0.77 W/in.2 ). As such, a rule-of-thumb recommendation was developed by Slaminko [4.11] for strain gage applications on composites: Size: 3 mm(0.125 in.) or larger; Resistance: 350 Ω or higher; Excitation voltage: 3 V or less. Smaller gages can be used, if necessary (to measure in locations with high strain gradients, for instance), by reducing the excitation levels.

4.1.5 Additional Considerations The process of performing a successful experiment with strain gages starts with the selection of the gage. Vishay [4.13] suggests the following order should be followed when selecting a gage: 1. 2. 3. 4. 5. 6.

gage length gage pattern gage series options resistance self-temperature compensation

Note that the gage size is the first, and arguably the most important, consideration. This is especially

101

Part A 4.1

the self-temperature compensation (S-T-C) numbers provided by the gage manufacturer do not match most composites and therefore the temperature compensation data (polynomial and graphical corrections) cannot be applied. As a result other methods of compensation are required. The so-called dummy compensation method is a robust method for performing compensation on composites. Dummy compensation requires a second gage to be mounted in a half-bridge configuration with the active gage. Three requirements must be fulfilled in order to perform dummy compensation. First, the dummy gage must be from the same gage lot as the gage that measures the combined thermal and mechanical components. Second, the dummy gage must be mounted on an unconstrained piece of the same material, of approximately the same volume and aligned in the same orientation with respect to the orthotropy as the mechanically loaded specimen. Finally, the two specimens (one loaded and the other unconstrained) must be in the same thermal environment and in close proximity. Violation of any of these requirements can result in large errors. The two gages are then connected in a half-bridge configuration where the thermal output from each gage cancel. One can also perform a precalibration test to separate the thermal strain from the mechanical strain. This is performed by mounting a gage on the specimen to be mechanically tested through a range of temperatures. Then, the specimen is subjected to the thermal cycle only and the apparent strain is recorded as a function of temperature. Once this is established, the specimen can be mechanically tested over the same temperature range. The strains due to thermal effects are subtracted from the results of the test with both thermal and mechanical contributions. An accurate temperature measurement is required for this technique as well as postprocessing of the data.

4.1 Strain Gage Applications

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true for composites, since the scale associated with homogeneity is often significantly larger than that for traditional materials. For textile composite forms, which will be discussed later, the optimal strain gage size depends on the textile architecture. In general, smaller gages will produce more experimental scatter than larger gages and therefore should be avoided, unless strains are to be measured in areas of high gradients. Additionally, in some instances the average value for a highly nonuniform strain distribution is desired and thus larger gages, or those that span the entire test section, are more appropriate [4.14]. The selection of the gage pattern is largely dependent on the objectives of the experiment, for instance, whether one two or three axes are of interest, or whether normal or shear strains are required. If multidirectional gages are used there is often a choice of planar or stacked configurations. For experiments where high stress gradients are present and the material is more heterogeneous, stacked configurations are desired. Stacked configurations, however, can be more expensive, and create more reinforcement effects and self-heating problems. The gage series relates to the materials (foil and backing) that are used in the strain gage. The available choices relate to the temperature range, fatigue characteristics, and stability. The most common sensing grid alloys used for composites are constantan and Karma. Both have good sensitivity, stability, and fatigue resistance. Karma is more stable at temperatures over 65 ◦ C (150 ◦ F), but is more difficult to solder to than constantan. Polyimide backing materials are well suited for composites and allow for contourability and maximum

elongation, while fiber glass reinforced phenolics offer better temperature stability. As in noncomposite applications there may be instances where options such as preattached leads are required. As far as the resistance of the gage is concerned, Slaminko recommends that 350 Ω or greater should be used. Finally, since S-T-C is generally not an option for composites, there is no need to consider this number and any S-T-C value is appropriate. There are other considerations that the experimentalist should be aware of when using strain gages on composites. For example, the ideal measurement technique should be nonintrusive. For strain gage applications this means that, by applying the gage to the underlying material, the strain data recorded represents that for the case where there is no gage present. Since the gage and the adhesive used to attach the gage have finite stiffness, there is always a reinforcing effect. For measurement of material systems where the modulus is much higher than the gage material this effect is typically considered insignificant and can be ignored. When taking measurements on relatively compliant materials, however, gage reinforcement can lead to errors. In the case of composites, the stiffness of the material is highly orthotropic in nature, ranging from modulus values similar to that of common strain gages to values that are significantly higher. Additionally, when considering the entire installation, adhesive selection may be an important consideration when taking measurements along compliant axes (such as transverse loading). A detailed description of reinforcement effects is presented by Perry in the manual [4.15, 16] for both isotropic and anisotropic materials.

4.2 Material Property Testing With the rapid development of composite material systems throughout the last three decades, there has been a need to characterize them accurately. The primary loading conditions of tension, compression, and shear applied to more than one loading direction introduced many new mechanical test methods. Because of this rapid progress, standardization has not been entirely achieved for all loading conditions. The American Society for Testing and Materials (ASTM), under the D30 subcommittee, is the primary organization for composite test standards. The organization consists of a voluntary membership and conducts round-robin testing and standards writing. Recent activ-

ity by the US Military Standards (Mil Specs) under the MIL-Handbook-17 (Polymer Matrix Composites) has expanded standards to include fabrication. Five volumes of the handbook, released in 2002, include: Guidelines for Characterization of Structural Materials, Polymer Matrix Composites: Material Properties, Materials Usage, Design, and Analysis, Metal Matrix Composites, and “Ceramic Matrix Composites”. Planning is under way for the release of a volume on Structural Sandwich Composites. Internationally, the British Standards Institute (BSI) in the United Kingdom, the Association Française de National (AFNOR) in France, the Deutsches Institut für Normung (DIN) in Germany,

Composite Materials

4.2.1 Tension Testing Tension testing under ASTM D 3039 [4.19] has become a well-accepted standard for testing of high-modulus and high-strength composites. The standard defines a tabbed, straight, flat specimen of high aspect ratio (Fig. 4.6). The tabs, with a beveled entry into the test section, are added to the specimen to allow for substanASTM D 3039

ASTM D 608

Fig. 4.6 ASTM standard tension test specimen geometries

tial clamping forces at the grip. These tabs are typically made from G-10 glass/epoxy and are adhered using a high-strength epoxy. Untabbed, dog-boned specimens (Fig. 4.6), under ASTM D 608 [4.20] may be use for some composites that have lower tensile strength values (for instance transverse tension testing). However, for composite systems with higher tensile strength, often the enlarged portion shears off parallel to the long dimension, thus reducing the specimen to a flat, straight specimen similar to that used in ASTM D 3039, without tabs. In general, ASTM D 3039 is appropriate for almost any composite form. Both mechanical and hydraulic grips may be used for ASTM D 3039. The clamping pressure on wedgetype mechanical grips is generally proportional to the applied loads. This reduces the potential for crushing the specimen in the tabbed area. Hydraulic grips add more versatility, since the clamping force is independent of the load, although care must be taken to avoid specimen crushing.

4.2.2 Compression Testing Compression testing of composites has received a considerable amount of attention and has proven to be more complicated than tension testing [4.17]. Since the compressive strength (in the fiber direction) is generally lower than the tensile strength, and since there is a desire to use a larger fraction of the ultimate load capacity, there is a need to characterize the compressive strength accurately. Because relatively thin specimens are desired, buckling may occur if the unsupported length is too large. Therefore, in general, most compression tests are designed such that the specimen fails due to true compressive failure of the material and not due to Euler buckling. There are three methods [4.21–23] of introducing compressive loading into the specimen: via shear through end tabs (ASTM D 3410), by direct end loading (ASTM D 695), and by bending of a composite sandwich panel (ASTM D 5476). The latter is rarely used since the specimens are typically large, difficult to manufacture, and expensive. Additionally, there is a concern that prevails among the composites community that the compressive failure strength may be artificially elevated by the presence of the core being bonded along the entire face sheet. Introducing the compressive load via shear through end tabs is similar to the way loads are introduced in tensile tests, except that the loads are of opposite sign and therefore the wedges in the grips are inverted. In ASTM D 3410 two test fixtures are described: the

103

Part A 4.2

Japanese Industrial Standards (JIS) and Ministry of International Trade and Industry (MITI) distributed by the Japanese Standards Association, are the primary standardization organizations. Whether testing composite materials in tension, compression or shear, there are three basic objectives that should be realized. First, since material systems are often expensive (especially when they are initially developed), thin laminates and small specimens are generally desired. Second, the test method should be capable of measuring the elastic properties as well as the failure strength. As such, the state of stress in the test section of the specimen must be pure (without other stresses) and uniform. Third, the test should be devised such that common universal testing machines can be utilized. Often, these considerations rule out the use of complicated geometries and testing in specialized testing machines. The Manual on Experimental Methods of Mechanical Testing of Composites [4.17, 18] has a comprehensive chapter written by Adams on test methods. This section will only briefly describe the most used test methods and summarize the appropriate procedures.

4.2 Material Property Testing

104

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Solid Mechanics Topics

Fig. 4.7 The Celanese compression fixture (Wyoming Test

Fixtures Inc.)

Part A 4.2

Celanese compression fixture [4.24] and the IITRI compression fixture [4.25]; they are shown in Figs. 4.7 and 4.8. Both of these test fixtures have a specified gage length (unsupported length) of 12.8 mm (0.5 in.). Testing of thin specimens may lead to buckling if the slenderness ratio is too high. The original standard was written for the Celanese fixture in 1975. The fixture incorporates cylindrical wedges and has an alignment sleeve. It has been criticized for being potentially unstable if precautions, outlined in the standard, are not followed. These instabilities arise because the wedge grips travel in a conical track and some degrees of freedom are not well constrained. Additionally, the fixture is limited to specimens with a maximum width of 6.4 mm (0.25 in.). For testing of textile composites and materials with heterogeneity on a larger scale, the limited width can be a problem. The IITRI fixture was developed to eliminate the problem of instability by utilizing a flat wedge grip, thus minimizing degrees of freedom. The fixture also allows for testing of wider and thicker specimens. As a result, the fixture is significantly larger than the Celanese fix-

Fig. 4.8 The IITRI compression fixture (Wyoming Test Fixtures Inc.)

ture and weighs in at almost 46 kg (100 lb). This effects ease of handling and increases the cost (typically there is a proportional relationship between the weight and cost of the fixture). By combining the favorable features of the Celanese and IITRI fixtures, Adams et al. developed the Wyoming-modified Celanese compression fixture [4.26], shown in Fig. 4.9. The fixture can accommodate a specimen of 12.8 mm (0.5 in.) wide and 7.6 mm (0.3 in.) thick, twice the values of the original Celanese fixture. Another modification of the Celanese fixture is the German DIN 65 380 compression fixture [4.27]. It uses flat wedge grips and an alignment sleeve. The second common method used to introduce load into the compression specimen is through the ends of the specimen (ASTM D 695) [4.22]. This method can be used with or without end tabs. End tabs increase the bearing surface and thus prevent bearing failure prior to compression failure inside the test section. The clamped portion of the fixture confines the specimen and also suppresses bearing failure, which usually occurs in the form of brooming. Some fixtures can accommodate dog-bone geometries without tabs, again to increase the bearing surface area. Generally speaking, for specimens with compression failure strengths less than 700 MPa (100 ksi) dog-boning may be adequate. For those above that value tabs are usually required because the bog-boned portion shears, and subsequent bearing failure occurs at the ends. Examples of end-loaded fixtures include the Wyoming end-loaded side-supported fixture [4.28], the modified ASTM D 695 compression test fixture, and the NASA short block compression test fixture. Specimens may

Fig. 4.9 The Wyoming modified compression fixture

(Wyoming Test Fixtures Inc.)

Composite Materials

also be loaded through both end loading and shear. An example of such a fixture is the Wyoming combinedloading compression fixture (ASTM D 6641) [4.29,30]. The fixture incorporates the advantages of both load introduction schemes.

4.2.3 Shear Testing

and colleagues at the University of Wyoming along with the development of dedicated loading fixtures. The Iosipescu specimen, illustrated in Fig. 4.10, is a relatively small, rectangular beam with V-shaped notches to concentrate the shear in the test section (the volume between the notches). The notches serve two purposes. They reduce the cross-sectional area in the test section (thus allowing for failure to be confined) and they produce a more uniform state of shear. The latter is highly dependent on the notch angle and the orthotropy of the material being tested. In general, this test violates one of the prerequisites for material strength measurements, since the state of stress in the test section is neither truly pure nor is it perfectly uniform. There exists a free edge in the test section, where the shear stresses are zero (in this case at the notch roots). The shear stresses then rise from the free edge and form a distribution that varies with fiber orientation and notch angle. This large gradient must be accommodated by normal stress gradients, according to equilibrium, and thus the state of shear is neither pure nor uniform. Although, strictly speaking, flat coupons cannot produce the perfect shear field, the Iosipescu geometry provides a practical approach, since the shear stress and strain distribution is near uniform. Shear strain information is recorded with strain gages located in the test section. Two styles have been proposed, a centrally located small ±45◦ rosette and a ±45◦ rosette that spans the entire test section [4.14]. Displacements (μm) –70

0 y x

19 Experimental ux contours –202

–15 Experimental uy contours

Fig. 4.10 The modified Wyoming Iosipescu shear test fix-

ture (Wyoming Test Fixtures Inc.)

Fig. 4.11 Loading condition and displacement fields for a proposed T-shaped specimen (courtesy M. Grediac)

105

Part A 4.2

Shear testing has also received considerable attention in recent years. Unlike tension and compression testing, it is difficult to create a pure and uniform shear field using a specimen made from a flat coupon. Ideally, thin-walled torsion testing provides the most uniform and pure shear state of shear. However, cylindrical composite specimens are inordinately expensive to produce, torsion test stands are relatively rare, and the obtained shear properties extended to flat laminates are questionable. Therefore, numerous specimens have been proposed for shear testing of flat coupons [4.17]. These include the two- and three-rail shear tests, the off-axis tension test, the ±45 tension test, short beam shear, notched compression, picture frame, and notch tests such as the Arcan and Iosipescu specimens. The latter [4.31–33] (ASTM D 5379) has become the most widely used test method for composites. First developed by Nicolai Iosipescu in the early 1960s for isotropic materials, it was extended for use on composites by Adams

4.2 Material Property Testing

106

Part A

Solid Mechanics Topics

Load cell

0°

[(0/90)3]s r = 2

αA

Surface covered with a grating

Fixed

Moveable

265 εxyexp (i, j)

εxyana (i, j)

εxyexp (i, j) – εxyana (i, j)

Part A 4.2 Fig. 4.12 Open-hole specimen and loading configuration with experimental shear strain distribution, analytic shear strain distribution, and differences after the application of difference minimization using the Levenberg–Marquardt algorithm (courtesy J. Molimard)

The latter was developed to correct for the nonuniform shear stress distributions and reduces the experimental scatter when employed on textile composites or materials with significant heterogeneity. It is suggested that gages be placed on both sides of the specimen to cancel the affects of twist [4.34]. By using two Iosipescu strain gages (one on each side of the specimen) the coefficient of variation in modulus can be reduce to less than 2%. A number of fixtures have been proposed for loading the Iosipescu specimen, the most popular being the modified Wyoming shear test fixture (ASTM D 5379) [4.31], which is mounted in a universal testing machine, loads the specimen through wedge clamps, and incorporates a linear bearing for alignment. Loads are imparted to the specimen on the upper and lower edges. For some material systems, tabs may be required if local crushing of the edges is evidenced.

4.2.4 Single-Geometry Tests In order to determine the multiple elastic constants for orthotropic materials, a number of tests must be per-

formed. This can add considerable time and expense. Several researchers have proposed the use of unique geometries and loading conditions in combination with full-field optical methods to determine multiple elastic constants. These tests are generally for elastic constants and are typically not well suited for strength measurements since failure may be from combined effects. Two recent studies in experimental mechanics will be used as examples of such methods [4.35]. The first, proposed in 1999 by Grediac et al., utilizes a unique T-shaped specimen loaded as shown in Fig. 4.11. The investigators used a phased stepped grid method to determine the displacement fields, as shown in the figure, and then determined the strain fields by differentiation. The principal of virtual work, relying on global equilibrium, was employed. Four virtual fields were chosen to satisfy four elastic constants. In practice the four virtual fields act like filters that enable the extraction of the unknown rigidities (constants). From this, the stiffness components can be identified. The axial elastic constants were determined to within 15% of those values obtained using traditional means. It was determined

Composite Materials

that the shear elastic constant could not be determined with reasonable accuracy, since, as the authors point out, the specimen deformation is generally not governed by shear. An open-hole tension specimen was employed by Molimard et al. [4.36] to measure four orthotropic plate constants from a single geometry. They used phase-shifted moiré interferometry to measure full-field surface displacement and strain. Figure 4.12 shows the geometry and loading fixtures used in their tests. Displacement and strain contours were measured in the area around the open hole. The principal employed was

4.3 Micromechanics

107

to minimize the discrepancy between experimental and theoretical strain results using a Levenberg–Marquardt algorithm. Comparisons between the experimental and analytical shearing strain values are presented in Fig. 4.12. The method takes into consideration the optical system, signal processing, and the mechanical aspects. Cost functions were investigated leading to a simple mathematical form. Two models were used: an analytical model based on the Lekhnitskii approach and the finite element method. The researchers were able to identify the four elastic constants to within 6% of those measured using traditional means.

4.3 Micromechanics

a)

b)

be broken down into critical experiments. For instance, the interface can be probed by dedicated tests such as the single-fiber pull-out test. Other critical experiments have been performed on testing of the constituents individually in bulk form. These include polymer studies, fiber strength tests, and nanofiber characterization. Also, by including microencapsulated adhesives with dimensions on the fiber scale, self-healing composites have been developed. In this section full-field fiberscale testing, interface testing, nanocomposites, and self-healing composites will be addressed.

4.3.1 In Situ Strain Measurements As mentioned above, the length scale associated with micromechanics challenges the experimentalist. There have been numerous studies to create scaled-up model materials (to match the capabilities of various experimental techniques such as moiré methods, photoelasticity, etc.) of an ideal geometry (square and hexagonal array) for validation of analytical and numerical modc)

Fig. 4.13 (a) E-beam moiré grating, (b) fringe pattern prior to interlaminar fracture, (c) fringe pattern after fracture (courtesy J. W. Dally)

Part A 4.3

Experimental micromechanical characterization is extremely challenging because of the length scales involved. For example, the diameter of a carbon fiber is of the order 8 μm, boron fibers are of the order 200 μm, and carbon nanotubes are measured in nanometers. As such, there have only been a few notable examples of full-field experimental documentation of the strain field on the fiber scale. Any experimental technique that is suitable for such a task must have extreme spatial resolution, high sensitivity, and be used in conjunction with high magnification capabilities. Traditional experimental techniques, such as moiré interferometry, cannot be used without enhancement. Additionally, most of the interesting micromechanical phenomena occur in the interior of the ply and not on the surface, therefore lack of visual access (whether through traditional optical microscopy or scanning electron microscopy) prohibits the investigation of such problems. As a result, modeling efforts are far ahead of the experimental efforts and precious little experimental evidence supports these models. Nevertheless, aspects of micromechanics can

108

Part A

Solid Mechanics Topics

Fiber pull-out

Fiber push-in

Fiber push-out

Fig. 4.14 Schematic illustrations of the single-fiber pull-out, pushin, and push-out tests

els. This methodology can be useful in understanding the interaction between the fiber and matrix, but often the essence of the micromechanical phenomenon is lost. Here, only implementation of full-field techniques on real composites will be presented.

Part A 4.3

Striker bar

Incident bar

Transmitter bar

Strain gage

Strain gage

3.2 mm Incident bar

Transmitter bar

Punch Specimen

4.3.2 Fiber–Matrix Interface Characterization

Support

Force (N) 0

–1000

5m/s

–2000

7 m/s

–3000 15 m/s

–4000

0

The first example is a novel method developed by Dally and Read called electron beam moiré [4.37– 40]. The method involves writing very high-frequency (10 000 lines/mm) gratings on sectioned composites using e-beam polymethyl methacrylate (PMMA) resists and performing a moiré test in the scanning electron microscope (SEM) using the scanning lines as the reference grating. While details of the technique are given in the reference, the example presented on glass fiber reinforced epoxy will be illustrated here. The interface between the 0◦ and 90◦ plies on the edge surface of a [02 /902 ]2s laminate was interrogated for a specimen loaded in tension using a small testing machine located within the SEM specimen chamber. The crossed-line grating on the surface of a small region of the specimen is shown in Fig. 4.13, along with moiré fringe patterns at two load levels, one prior to and one after a delamination crack. Xing et al. [4.41] also extended the used of e-beam moiré, by measuring the interfacial residual stresses in SiC/Ti-15-3 silicon fiber reinforced ceramic composites. Additional in situ high-spatial-resolution studies using novel experimental techniques include the use of micro-Raman spectroscopy [4.42–44] for the study of fiber breakage in compression specimens and the interaction between neighboring fibers.

10

20

30

40

50

60 70 Time (μs)

Fig. 4.15 Effect of sliding speed on the push-out force for a rough

aluminum fiber (12 mm long) in an epoxy matrix (courtesy J. Lambrose)

Fiber–matrix interface properties can dominate the mechanical behavior of a composite. The properties of the interface can influence delamination, microcracking, kink band formation, residual stresses, and numerous other failure modes. Therefore, the characterization of the interfacial strength is of importance when developing micromechanical models to predict the strength of composites. The interface, defined as the boundary between the fiber and matrix, actually has a finite volume. This is because the fiber typically has a sizing to control adhesion of the matrix and the matrix morphology varies near the fiber. Hence, the term interphase has been adopted to describe this finite volume. The interphase volume is on a length scale even smaller than the fiber scale, and therefore introduces serious challenges to the experimentalist. However, the importance of this volume has spurred numerous researcher efforts. A number of tests that have been proposed to measure the adhesive strength of the fiber–matrix interphase. These tests include the single-fiber pull-out, push-in, and push-out tests shown schematically in Fig. 4.14.

Composite Materials

4.3 Micromechanics

1 μm

Stress σ (×106 Pa) 30 25 20 15 10 Dow DER732+DER331 Nanocomposite 5wt%

5 0

0

1

2

3

4 Strain ε (%)

Fig. 4.16 TEM micrograph of clay platelet reinforced polymer and the tensile stress–strain response of unreinforced and 5% clay reinforced polymer (courtesy I. M. Daniel)

They are common in that a traction is placed on a single fiber to initiate and propagate bond failure between the fiber and matrix either as an adhesive or cohesive failure. An example of such a test is presented in this section. In 2002 Zhouhua et al. [4.45] conducted experiments to determine the dynamic fiber debonding and frictional push-out in a model composite system. The investigators utilized a modified split Hopkinson pressure bar with a tapered punch to apply a compressive load on a single fiber (modeled using aluminum and steel) embedded in an epoxy matrix. In the push-out experiment, the fiber model is pushed from the surface

4.3.3 Nanoscale Testing The quest to achieve improved mechanical properties has led to the development of composites that incorporate nanoparticle, nanofiber, or nanotube reinforcement. These nanocomposites are a class of composite material where one of the constituents has dimensions in the range of 1–100 nm. Over the past decade, the development of nanoreinforcements, processing them into composites, and assessing their mechanics has rapidly evolved. In this section we will briefly describe two studies: the incorporation of nanoclay particles in polymers and a study of the mechanical characterization of multiwalled carbon nanotubes. Some polymer–inorganic nanocomposites have demonstrated pronounced improvements in stiffness, strength, and thermal properties over unreinforced polymers without sacrificing density, toughness, and processibility. The advantages of nanocomposites over traditional composites come from the high surface-areato-volume ratio of the reinforcement, and thus have a direct effect on the interfacial mechanics. Measuring the stress and strain distribution on the nanoscale

Part A 4.3

and punches through a hole in a support. The test allows real-time measurement of the relative fiber–matrix displacement and push-out force. Figure 4.15 shows a schematic of the setup and an example force–time plot to demonstrate the effect of sliding speed (directly related to the loading rate). The technique documented the effects of loading rate, material mismatch, fiber length, and surface roughness on the push-out event. It was observed that surface roughness played an important role in the dynamic interfacial strength, and that the maximum push-out force increased with loading rate. The plot in the figure shows a markedly different behavior between slow and fast sliding speeds, with a transition occurring around 10 m/s. At slow speeds the transmitted force reaches a maximum in two steps, indicating that the load levels required for initiating and propagating the debonding crack differ. A finite element model was constructed to extract the interface strength and toughness values. On the scale of real composites (versus model materials) the single-fiber push-in test is more applicable since the only specimen preparation required is cross-cutting and polishing the specimen. Micro- and nanoindenters have been utilized for applying loads over the small area of the fiber cross-section. These devices have load and position readout and are well suited for static loading cases.

109

110

Part A

Solid Mechanics Topics

a)

increased the tensile modulus and strength by as much 50%, as can be seen in the figure. When adding additional reinforcement, the modulus increased, but the strength significantly degraded. These findings were thought to be attributed to increased voids, inclusions, and agglomeration of the nanoparticles during processing. Recent research by Zhu and Espinosa [4.47] has lead to the development of very small tensile testing machines that are capable of testing single multiwalled carbon nanotubes. The system utilizes a micro-

b)

5 μm

c)

d)

Microcapsule Catalyst Crack

Part A 4.3

Fig. 4.17 The procedure to attach a nanowire to the cross-heads of the MEMS-based testing machine involves welding the wire using electron-beam-induced deposition (courtesy Y. Zhu)

is a current challenge to the experimentalist. Many of the experimental mechanics contributions to this field pertain to using traditional test methods applied to the macroscopic behavior of nanocomposites. Examples include using tensile tests, fracture tests, and shear tests to validate the inclusion of nanoparticles in matrix materials. Daniels et al. [4.46] characterized clay nanoparticle platelet reinforced epoxy via tension tests to determine how process parameters (such as the weight percentage of the reinforcement) affect the stress–strain response. Figure 4.16 shows a tunneling electron microscope micrograph of the exfoliated Cloisite 30B clay reinforced Dow DER331/DER732 epoxy system. It was found that moderate nanoclay concentrations on the order of 5% a)

Healing agent

Polymerized healing agent

20 μm

b)

500 nm

500 nm

Fig. 4.18 Tension tests of a multiwalled carbon nanotube with detailed insets to shown the wall thickness and morphology (courtesy Y. Zhu)

Fig. 4.19 Self-healing concept for thermosetting matrix with SEM image of broken microcapsule (courtesy E. N. Brown)

Composite Materials

4.3.4 Self-Healing Composites Researchers at the University of Illinois have developed a self-healing polymer that they have incorporated into the fabrication of fiber reinforced composite materials [4.48, 49]. The concept is to heal a crack in polymers using a microencapsulated healing agent that

is released upon the intrusion of a crack. The healing agent is then hardened by an embedded catalyst. The dicyclopentadiene (DCPD) healing agent, stabilized by 100–200 ppm p-tert-butylcatechol encapsulated by a poly-ureaformaldehyde shell, catalyzes when exposed to bis(tricyclohexylphosphine)benzylidine ruthenium (IV) dichloride (otherwise called Grubbs’ catalyst), which is imbedded in the composite matrix. Mean capsule diameters are of the order 166 μm. Figure 4.19 shows a schematic of the fracture/healing process, where a crack penetrates through the bulk polymer and the shells of the microcapsules. The healing agent then wicks into the crack and comes into contact with the embedded catalyst, leading to polymerization of the healing agent and subsequent bridging of the crack. The researchers also conducted extensive fracture studies on unreinforced (no fibers) EPON 282 epoxy tapered double-cantilever beam specimens, and found that, once healed, the polymer containing the healing agent exhibited 90% of its virgin fracture toughness [4.50,51]. Studies were then performed by combining the self-healing matrix material with a plain-weave graphite fiber fabric. Fracture tests were performed using a width-tapered double-cantilever beam specimen. It was found that as much as 45% of the virgin fracture toughness was recovered by autonomic healing, and as much as 80% was recovered by increasing the temperature to 80 ◦ C during recovery.

4.4 Interlaminar Testing Since most composite materials are layered with only the matrix material binding each layer to its neighbor, weak planes exist. As a result, delamination between layers is a common failure mode. Delamination may initiate from a number of different loading conditions, including impact (high and low velocity), tension, compression, and flexure. The resulting interlaminar normal and shear stresses can then lead to interlaminar fracture and possible subsequent catastrophic failure of a structural component. Often, interlaminar fracture initiates at a free edge where the interlaminar stresses may be much higher than the nominal applied stresses. Figure 4.20 provides dye penetrant enhanced radiography images showing delamination (dark areas) initiated at the free edge, at a stress concentration and at an impact site. This section will cover the experimental characterization of

mode I (opening mode), mode II (shearing mode), and mixed-mode interlaminar fracture, as well as a study of stresses due to free edge effects.

4.4.1 Mode I Fracture Mode I interlaminar fracture toughness is traditionally measured using a double-cantilever beam (DCB), as shown in Fig. 4.21. The specimen is fabricated such that a starter or initial crack is produced using a Teflon film sandwiched between the layers where fracture is intended to propagate (typically the midplane). When loaded, the crack propagates in the plane of delamination. Mode I fracture toughness can be determined in two ways from the critical energy release rate. First, the load at which the delamination starts to propagate can

111

Part A 4.4

electromechanical system (MEMS) for in situ electron microscopy. Loads in the nanonewton and displacements in the subnanometer range can be resolved with the device. The system was used to measure the stress–strain response of free-standing polysilicon films, metallic nanowires, and carbon nanotubes. Figure 4.17 shows electron microscopy images of a nanowire bridging the two cross-heads of the loading device. Small welds secure the fiber in place. Figure 4.18 shows a tunneling electron microscope (TEM) image of a multiwalled carbon nanotube (with an outer diameter of 130 nm and inner diameter of 99 nm) being loaded. Upon rupture, crystallization is clearly seen in the inset of the lower plot in Fig. 4.18. The failure stress was measured at 15.84 GPa with a failure strain of 1.56%. These correspond to a modulus of more than 1000 GPa. These strength and stiffness values are of the order a fivefold improvement over the best traditional carbon fiber (10 μm diameter), and confirm the extraordinary potential that exists from the processing and mechanics of these materials.

4.4 Interlaminar Testing

112

Part A

Solid Mechanics Topics

be used to compute G IC , the critical energy release rate: F 2 a2 (4.4) G IC = C , bE I where FC is the load at which the crack propagates, a is the current crack length, b is the beams width, and E I is the equivalent flexural rigidity of the upper and lower arms of the specimen. In the second method, the specimen is unloaded after the crack propagates a distance Δa. The area under the load–deflection diagram, bounded by the loading portion, load drop due to crack extension, and unloading portion, represents the work done, ΔU, for crack propagation and the critical energy release rate, G IC , is computed as ΔU (4.5) . bΔa One of the challenges associated with this methodology is the accurate determination of the crack position. Optical microscopes, digital image correlation, and moiré interferometry have been utilized for this purpose. For visual inspection, the edge of the specimen is painted white so that the crack is more pronounced. Perry et al. [4.52] used full-field moiré interferometry to measure the displacement fields for the calculation of the J-integral, in order to determine the strain energy release rate on the DCB specimen. They also developed a method to determine the energy release rate for arbitrary geometries (not DCB) and used the DCB test to validate the methodology. An example

Initial crack

F

Δa a

F Load F

G IC =

Load point displacement

Part A 4.4

Fig. 4.21 The double-cantilever beam specimen, loading, and typical load–displacement diagram

of the phase wrapped moiré displacement pattern for the DCB test is shown in Fig. 4.22. Numerous other researchers have implemented a modified version of the DCB method. Some of these modifications include tapering the specimen in either the vertical direction by adding material to the top and bottom surfaces or tapering the specimen in the widthwise direction [4.53,54]. These modifications transition the crack growth from unstable to stable, and crack propagation occurs at a near-constant load level. For through-thickness stitched composites, it was found that the force required to open the crack breaks the arms of the specimen. The failure was governed by compres-

2 mm

Cracks emanating from the free-edge during loading

Cracks emanating from a stress riser

Damage from impact

Fig. 4.20 Delamination and cracking due to cyclic loading and im-

pact

Fig. 4.22 Moiré interferometry, wrapped, fringe pattern, at the crack tip, for the vertical displacements on a DCB specimen (courtesy K. E. Perry)

Composite Materials

113

in a sliding mode horizontally as the beam bends. The critical energy release rate can be determined using the same analyses as those used for the DCB test. For the instantaneous mode II critical energy release rate

F F/2

F/2

4.4 Interlaminar Testing

Load F

3 FC2 a2 (4.6) . 64 bE I Additionally, by determining the area in the load– displacement diagram (Fig. 4.23), bounded by the loading portion, load drop due to crack extension, and unloading portion, the critical energy release rate is determined in the following equation. G IIC =

ΔU (4.7) . bΔa Typically, the location of the crack front is difficult to determine since the crack does not open, but rather slides. Full-field techniques can be of value for locating the crack tip. Perry et al. used moiré interferometry for both crack front location as well as displacement mapG IIC =

Load point displacement

Fig. 4.23 The end-notched flexure specimen, loading, and typical load–displacement diagram

Ny –5 y

x

5

4.4.2 Mode II Fracture Mode II fracture toughness can be determined using the end-notched flexure (ENF) specimen [4.52]. The specimen is fabricated in the same manner as that used in the double-cantilever beam test, with a Teflon starter crack. It is loaded, however, in a three-point bend configuration, as shown in Fig. 4.23. The crack propagates

V field Ny fringes

0

10

Graphite/ epoxy

15

(O2 /±452 /902)n

20

f = 2400 lines/mm

γxy /|εyav|

γxy x 106

4 8000 2 0

0 –2

–8000 –4

2 mm

–16000

–6

0

8

16

24

32

40

Ply thickness = 0.19 mm (= 0.0075 in.) 48

Ply no.

Fig. 4.24 Horizontal displacement field, at the crack tip,

for an ENF specimen (courtesy K. E. Perry)

Fig. 4.25 Moiré interferometry fringe pattern on a portion of the edge of a quasi-isotropic laminate. Strong interlaminar shear strain are present between +45◦ and −45◦ plies (courtesy D. Post)

Part A 4.4

sive instability. Additional horizontal forces were added by Chen et al. [4.55, 56] to suppress this instability. Under the basic assumptions of superposition of axial load and bending moments in beams, the compressive stresses were reduced and more bending moment could be applied. Crack propagation was realized for stitched materials and traditional DCB analysis showed that the fracture toughness increased by more than 60-fold.

114

Part A

Solid Mechanics Topics

ping to calculate the energy release rate. Figure 4.24 shows the wrapped fringe pattern used in the analysis.

4.4.3 Edge Effects

Part A 4.5

Often, delamination initiates at the free edge of a composite laminate because of highly localized stresses between layers of dissimilar orientation. This free edge effect forms a stress gradient that dissipates to the nominal stress level of the interior within a distance equivalent to a few ply thicknesses. These stress distributions can be highly complex and three dimensional, involving all three normal and all three shear stresses, and are dependent on the loading condition, the stacking sequence, and material properties. Some are restricted to the interior and cannot be measured on any of the free surfaces, making a comprehensive experimental characterization difficult. Additionally, if residual stresses are ignored in the overall analysis, the absolute stresses cannot be determined. In order to study the free edge affect, very high spatial resolution is required, since the strain gradients are high and the geometric features are small. Post et al. [4.57,58] conducted a series of tests using moiré interferometry to study the interlaminar strains

on both cross-ply and quasi-isotropic thick laminates. They performed both in-plane and interlaminar compression tests and documented the strain distributions on the ply scale. As an example of the experiments performed, a (02 / ± 452 /902 )n , graphite–epoxy laminate subjected to in-plane compressive loading will be presented. Figure 4.25 shows the specimen geometry, the moiré interferometry fringe pattern of the displacement in the vertical direction (loading direction), and the shear strain distribution along a line in the thickness direction on the free surface. The fringe pattern shows significant waviness, where there would only be horizontal, straight, evenly spaced, parallel fringes for a homogeneous isotropic specimen (for instance, a monolithic metallic specimen). By analyzing the horizontal fringe gradient ∂v/∂x the shear strains were determined. Strong gradients were present between the +45◦ and −45◦ plies. The shear strains were of the order of five or five times the applied normal strain. Additionally, from the u-field patterns, it was found that tensile strains in the transverse direction (Poisson effect) varied significantly with peak tensile strains of the order of the applied axial compressive strain.

4.5 Textile Composite Materials Textile composites are a form of composite that utilizes reinforcement in the form of a fabric. The fabric may be woven, braided, knit or stitched and then combined with matrix materials, via resin transfer techniques, hand lay-up, or prepreg methods. Textiles have distinct advantages over traditional laminates made from unidirectional layers, including better contourability over complex three-dimensional (3-D) tools, enhanced interlaminar strength, cost savings through near-net-shape production, and better damage tolerance. Disadvantages include lower fiber volume fraction, the existence of resin-rich volumes between yarns, and degraded inplane properties (as a result of the nonstraight path that the yarns must accommodate). Even so, they are extensively used in applications from the sporting goods industry to advanced aerospace vehicles. Textile composites pose additional challenges to the experimentalist since they have an additional level of heterogeneity. Not only is there heterogeneity on the fiber scale, but also on the scale of the textile architecture and the laminate scale (if multiple layers of cloth are used). For many textile forms a yarn may contain

12 000 fibers, or more, and the repeating unit in the textile architecture may have linear dimensions of the order 1 cm or more. This can pose significant challenges to the experimentalist, even for seemingly routine tests to determine the elastic properties.

4.5.1 Documentation of Surface Strain It has been assumed that the textile architecture induces repeating spatial variation of strain on the surface coincident with the architecture itself. In order to document the strain distribution associated with the architecture, high spatial and measurement resolution are required. Recent studies using moiré interferometry, digital phase-shifting shearography, and Michelson interferometry have documented the strain field on the surface of composites. These results can be used to guide instrumentation practices for strain gages as well as to validate modeling efforts. Lee et al. [4.59] used digital phase-shifting grating shearography to characterize experimentally plainweave carbon–epoxy composites under tensile load.

Composite Materials

Measuring area

8 N/mm 14.8 N/mm 22.7 N/mm 28 N/mm

4500 8 N/mm

4000

x1

Two peaks on the fill arm

4

6

115

Fig. 4.26 Tensile strain contour maps for four load levels. Plots along two lines show the variation of strain and correspondence to the textile architecture (courtesy J. Molimard)

Tensile strain (με) 5000

4.5 Textile Composite Materials

3500 3000 2500 2000 1500 1000 500

14.8 N/mm

0

0

2

8

10

12

14

16

Distance along x1-line (mm) Tensile strain (με) 5000

Measuring area

4000 22.7 N/mm

3500

Part A 4.5

8 N/mm 14.8 N/mm 22.7 N/mm 28 N/mm

x2

4500

Two volleys on the warp yarn

3000 2500 2000 1500 1000 500

28 N/mm

0

0

2

4

6

8

10

12

14

16

Distance along x2-line (mm) –353

174

701

1228

1756

2283

Tensile strain (με)

The experimental technique is similar to moiré interferometry and can be used in conjunction with Michelson interferometry to determine the three in-plane strain components (εx , ε y , γxy ) on a full-field basis, as well as the out-of-plane displacements and surface slopes. They found that the strain varied in a cyclic manner and followed the weave architecture. Figure 4.26 shows contour plots as well as line plots (at two different locations with respect to the architecture) of the tensile strain for four load levels. The maximum strains were of the order of three times greater than the minimum strain values. In the transverse direction they found that the normal strain is predominantly compressive (Poisson effect), but locally there are regions of tensile strain. These effects are attributed to yarn crimp and bending as a function of axial load.

2810

3338

Δ = 264

Shrotriya et al. [4.60] used moiré interferometry to measure the local time–temperature-dependent deformation of a woven composite used for multilayer circuit boards. They studied both the deformation fields in the plane as well as over the cross-section through a temperature range of 27–70 ◦ C. Their measurements revealed the influence of the fabric architecture in the deformation field. They noted that the variation in strain was greater when the composite was loaded in the fill direction (versus the warp direction) due to higher crimp angles (the angle that defines the undulation of a yarn as is passes over and under the transverse yarn). They also found that the total deformation increased with temperature and time (reflecting what was previously measured using strain gages) but the shape and distribution remained almost identical for all the loading cases and

116

Part A

Solid Mechanics Topics

A

Stacked nesting case B 1.2 mm

Split – span nesting case

C

y (a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

Diagonal nesting case

x Prepreg tows

Fig. 4.27a–i Moiré fringe patterns in the fill direction at 27 ◦ C for: (a) initial u-field, (b) u-field at 0.1 min, (c) u-field at 1 min, (d) ufield at 10 min, (e) composite microstructure, (f) initial v-field, (g) vfield at 0.1 min, (h) v-field at 1 min, (i) v-field at 10 min (courtesy

Prepreg tape

Part A 4.5

P. Shrotriya).

sample configurations. Figure 4.27 represents a sample of the fringe patterns taken on the edge of the composite for various times after the load was applied. In the figure, both the horizontal and vertical displacement fields are presented, as well as a schematic of the fabric architecture.

Fig. 4.29 Idealized textile composites can be used to val-

4.5.2 Strain Gage Selection

idate models. Three nesting configurations were tested by the investigators (courtesy D. O. Adams)

Because of the pronounced strain variation on the surface of textile composites, strain gage size (length and Coefficient of variation of strain readings (%) 7 6 Gage data Moire data

5 4 3 2 1 0

0

2

4

6

8 10 12 14 Gage length (mm)

Fig. 4.28 Coefficient of variation (CV) results using differently sized strain gages and CV results from the moiré pattern analysis (courtesy S. Burr)

Tensioned

width) selection is critical in order to avoid significant measurement variation. Studies performed by Burr and Ifju [4.61, 62] showed that, if the strain gage length is sufficiently smaller than the repeating unit cell size of the textile architecture, larger coefficients of variation in material property measurements are observed for simple tension testing. After testing with a variety of gage sizes, it was also observed that, as the gage length increased, the coefficient of variation decreased. Moiré interferometry fringe patterns on the surface of graphite–epoxy triaxial braided specimens were used to determine the appropriate gage length for a variety of braided materials (reinforcement in three directions) with unit cell sizes ranging from 2.5 mm to 22.4 mm. A representative gage area was superimposed over the fringe pattern and a procedure to measure the average strain in the gage area was developed. The representative area was then moved throughout the pattern and the variation of average values was used to determine the relationship between gage area and unit cell size. Figure 4.28 shows a fringe pattern with a represen-

Composite Materials

tative gage area superimposed along with a plot of the gage length versus coefficient of variation for both strain gage experiments and moiré interferometry patterns. Good correlation was found between the moiré prediction of the gage variation and the actual gage variation itself. It was suggested that, prior to performing multiple strain gage experiments, where specimen fabrication and instrumentation costs are prohibitive, a simple moiré experiment can be used to guide gage selection, with the reduction in measurement variability as the payoff.

4.5.3 Edge Effects in Textile Composites

woven composites, where the free edge is typically oriented with the yarn directions, these shearing stresses are less pronounced. It has been shown, however, that small differences in the in-plane fiber angle can lead to significant interlaminar shear on the free edge, where shear would not be expected to exist. For instance, in a weave with fibers running 0/90, there should not be strong shear stresses on the free edge. However, if the fiber angle deviates even a small amount from 90◦ or 0◦ then shear strains will be present, as evidenced by Hale et al. [4.63]. Modeling of the stress and strain fields in textile composites assumes idealized geometries. For multilayered textile composites, the nesting of the layers can influence these fields. Adams and colleagues [4.64] have developed a methodology to produce idealized textile composites with near-perfect nesting, so that experiments performed using full-field optical methods can be directly related to the model. The technique utilizes a method of curing the composite on the actual loom and controlling the tension and precise positions of individual warp and fill yarns. The result is almost perfect alignment of each layer such that layer-to-layer nesting can be controlled. Figure 4.29 shows some of the fabric nesting configurations that were studied.

4.6 Residual Stresses in Composites The act of combining two constituents in the fabrication process of composites typically induces residual stresses in the material.These stresses can arise from two separate sources, the chemical shrinkage of the matrix material (if the matrix undergoes a chemical reaction such as polymerization) as well as stresses due to differences in the coefficient of thermal expansion (CTE) between the fiber and matrix. The stresses can exist on both the fiber scale as well as the ply scale of the laminate. Additionally, for textile composites, these stresses can exist on the fiber architecture scale. The residual stresses due to chemical shrinkage typically occur during cure, but may also evolve with time as additional curing and physical aging transpires. The stresses that arise from CTE mismatch occur when the material is subjected to temperatures that differ from that of the cure temperature, and as such are strongly dependent on temperature. Many composite material systems, especially the high-tech graphite–epoxy systems used for aerospace applications, are cured in a high-pressure high-temperature apparatus called an

autoclave. This allows the materials to be operated at higher temperatures and typically leads to higher fiber volume fraction and lower void fraction. However, when the composite is utilized at a temperature much lower than the cure temperature, large residual stresses can arise. These stresses superimpose with the applied loads and may cause premature failure of the component. In some cases residual stresses may be high enough to lead to failure, even before loads are applied. For laminated forms, residual stresses on the ply scale exist because of the orthotropic nature of the thermal expansion coefficients in the principal material directions. A typical value for the CTE in the fiber direction α1 is very near zero if not slightly negative, while that in the transverse direction α2 is quite high and dominated by the CTE of the matrix material. For instance, a typical value for α1 is − 0.018 × 10−6 /◦ C for graphite–polymer and 6.3 × 10−6 /◦ C for glass– polymer unidirectional material. Typically, α2 is around 24 × 10−6 /◦ C for both graphite–polymer and glass– polymer materials. This value is nearly the same as

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As seen in laminate forms, edge effects can produce significant shear and normal stress components that may be many times greater than in the interior of the laminate. In textile composites a similar phenomenon transpires, except it occurs on the yarn scale, and can be highly localized. If a free edge cuts through a yarn that is neither parallel nor perpendicular to the free surface, large shearing strains may be present in the interface between the yarn and its nearest neighbor. In the case of braided materials, the yarns are typically inclined to the free edge, leading to large shear stresses and strains. For

4.6 Residual Stresses in Composites

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Part A 4.6

that of a typical aluminum. A unidirectional composite is rarely used on mechanical components since the transverse properties are so poor. By laminating the composite, desired mechanical properties can be achieved, however the lamination process leads to residual stresses as neighboring plies constrain each other. A recent example of the seriousness of these stresses is documented by the failure of the NASA X-33 liquid hydrogen fuel tanks [4.65]. A full-size prototype tank was fabricated with honeycomb sandwich core and quasi-isotropic, graphite–epoxy face sheets. Upon filling the tank with liquid hydrogen at cryogenic temperatures, the face sheets failed (cracked) because of residual stress induced by a CTE mismatch on the ply scale superimposed with chemical shrinkage effects on the ply scale. Hydrogen leaked into the core and expanded upon emptying the tank, leading to face sheet separation. The resulting failure led to the cancellation of the entire X-33 program. Given the seriousness of these stresses, there has been a concerted effort to develop experimental techniques to quantify them. This section will document five such methods: the composite sectioning method, the hole-drilling method, the strain gage method, the nonsymmetric lay-up method, and the cure reference method. Generally speaking, the methods can be diHorizontal displacement field (O2/904)13s AS/3501-6 Sectioning here

Mid-plane

vided into two groups, destructive (require damaging the specimen to extract the stress information) and nondestructive. The first two methods are considered destructive, while the following three are nondestructive.

4.6.1 Composite Sectioning Whenever a multidirectional laminate is cut, a free edge is created and stresses are liberated. Stresses at the free edge must satisfy equilibrium and thus the normal stress perpendicular to, and the shear stress components parallel to, the newly developed surface must be zero. Stresses still exist in the interior of the composite, and thus there is a gradient established by forming the free edge. One can measure the strain relief as a result of the creation of the free edge and then relate this to the stresses that existed in the laminate before the cut. There is an underlying assumption that the cutting method itself does not create a deformation field (i. e., no plastic deformation). For most fiber reinforced polymer materials, a well-lubricated slow-speed diamond-impregnated cut-off wheel has been shown to produce negligible local deformation. This can be verified by cutting a unidirectional composite, where-ply level residual stresses do not exist. A number of researchers have utilized this method to back-out the residual stress in laminates [4.66–69]. Joh et al. [4.66] used a combination of moiré interferometry and layer separation to determine residual stresses in a thick (02 /904 )13s , AS/3501-6, cross-ply composite. A grating was attached to the composites before it was sectioned. Cutting was performed to separate plies, and subsequently, moiré experiments were conducted to reveal the displacement field. Figure 4.30 shows the moiré interferometry fringe pattern after ply separation. The displacements were used to determine the strain and subsequent stresses through laminate theory. The method was shown to be capable of determining the residual stresses as a function of ply position and thickness. Tensile stresses of the order the transverse strength of a single ply were reported. Gascoigne [4.67] used a similar procedure to investigate a (020 /9020 /020 )T cross-ply that was composed of thick layers. Similar findings were reported by other investigators [4.69].

4.6.2 Hole-Drilling Methods Fig. 4.30 Moiré interferometry fringe pattern of the hori-

zontal displacements after ply separation (courtesy D. Joh)

The hole-drilling method, originally developed for anisotropic materials by Mathar [4.70], has been ex-

Composite Materials

y x

drilled to two depths (h in the figure). The investigators were able to determine the residual stresses in each ply of a (02 /902 )2s laminate. Later, the method was simplified by Sicot et al. [4.74] by the use of the residual stress rosette in combination with incremental hole drilling. Instead of the displacement information used in Wu’s work, Sicot utilized strain information at the location of the residual stress gages.

4.6.3 Strain Gage Methods Strain gages may be mounted on a cured composite in order to evaluate the thermal coefficient of expansion. This method requires extensive calibration of the strain gage and typically dummy gages mounted on a material that has coefficients of thermal expansion with well-known values. To measure the strain induced during cure however, strain gages must be mounted on the composite and co-cured with it. These gages may be mounted either on top of or embedded in the interior, between plies. The gages must be calibrated through the temperature range and they must be of a gage series that is appropriate for the temperature range. In order to determine the coefficient of expansion through a large temperature range a detailed proce-

υ-field h = 375 μm

u-field h = 112 μm

Finite element model utilizing symmetry about the mid-plane and hole υ-field h = 112 μm

υ-field h = 375 μm

Residual stress (MPa) 80 60 40 20 0 –20 –40 –60

In y-direction

In x-direction

Fig. 4.31 Moiré fringe patterns for

0

112.5

225

337.5

450

562.5

675

119

787.5

900

Thickness direction (μm)

two drill depths, the finite element model used for calibrating coefficients, and the residual stresses as a function of thickness (courtesy Z. Wu)

Part A 4.6

tended for use on orthotropic materials by Schajer et al. [4.71]. This method is similar to the composite sectioning techniques, except instead of a straight cut the free edge is created by a drill-type end mill. The method involves drilling a flat-bottom hole progressively through a laminate (layer by layer), and analyzing the deformation relieved by the process. A special strain gage rosette [4.72] is used to measure the strain field near the hole created by the stress relief. For complicated multidirectional laminates, where the stresses in each individual ply can be different, the simple orthotropic solution cannot be applied. Wu et al. [4.73] introduced the experimental technique of incremental hole drilling, with a 2 mm-diameter drill bit, in combination with the optical method of full-field moiré interferometry. The drilling increments were computer controlled and were coincident with the interfaces between plies. An expression describing the relationship between displacements on the surface, stresses in each layer, in-plane direction cosines, and a set of coefficients was employed. A rigorous calibration of the constants was performed using the full-field moiré displacement fields, and a 3-D finite element model of the 16-ply composite. Figure 4.31 shows two sets of moiré displacement patterns around the hole after the mill had

4.6 Residual Stresses in Composites

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dure was reported by Scalea in [4.75]. He derived analytical expressions for the CTEs of orthotropic lamina, accounting for transverse sensitivity and gage misalignment effects. Experiments were performed on glass–epoxy specimens and Invar as a reference material. He developed rules for temperature dwell time in order to ensure thermal equilibrium.

4.6.4 Laminate Warpage Methods

Part A 4.6

In 1989 [4.76] a method was introduced for the measurement of residual stresses in laminated material systems by fabrication of a nonsymmetric lay-up, more specifically a (0n /90n )T configuration. Numerous researchers have since utilized the method. After cure, the curvature induced from residual stress can be measured for a narrow strip of the laminate and the residual stresses calculated using laminate theory. The curvature can be measured with a variety of methods, including projection moiré, shadow moiré, digital image correlation or simply using a dial gage and assumptions of the shape. The resulting residual stresses are a combination of those induced both thermally and chemically. By heating the specimen to the cure temperature, the thermal contribution can be separated from the contribution due to chemical shrinkage. Figure 4.32 illustrates

the specimen geometry and the deformed shape. In the lower image, a shadow moiré fringe pattern of the curved specimen is presented.

4.6.5 The Cure Reference Method The cure reference method [4.77, 78] introduced in 1999 utilizes moiré interferometry and the application of a diffraction grating on the surface of a composite during the autoclave curing process, as shown in Fig. 4.33. This grating forms a datum from which subsequent thermal stresses can be referenced. Additionally, the method is capable of measuring the combination of the thermally induced and chemically induced components of strain. Unlike many of the former methods, it is capable of determining the residual stresses on any laminate stacking sequence (not just cross-plies), alPorous release film Breather ply Bleeder ply Vacuum bag Vacuum line

Autoclave tool

Non-porous release film Grating Pre-preg

During cure at elevated temperature

Pre-preg 3501-6 epoxy layer Evaporated aluminum 3501-6 epoxy layer

0° 90°

Astrosital autoclave tool (O2/452)2s AS4/3501-6

After cure at room temperature

u displacement field y, υ Shadow moire contour plot documenting the curvature

Fig. 4.32 Warping of a nonsymmetric laminate can be

used to determine residual stresses

x, u

υ displacement field

Fringe patterns were taken over a 1 in. diameter area contered on the face of the composite panel

Fig. 4.33 CRM grating replication and fringe patterns

Composite Materials

the coefficient of thermal expansion α2 , the modulus E 2 , and the shear modulus G 12 on temperature (these three properties are highly dependent on temperature while the others are nearly independent of temperature). These temperature-dependent material properties were then utilized in the analysis. It was shown that the laminate configuration that was presumed to be within a safe operating condition at cryogenic condition, with a predicted factor of safety of 1.3, actually had a safety factor of 0.8, once the temperature-dependent properties and chemical shrinkage residual stresses were incorporated into the analysis. Unfortunately, this analysis was performed after the failure of the fuel tank and the cancellation of the entire X-33 program. In a parallel optimization study, an alternative laminate sequence was developed in order to carry the applied loads, and at the same time, resist residual stress failure. The resulting angle-ply laminate, (±25)n , was tested using the cure reference method in parallel to the X-33 laminate. It was found that it retained a safety factor of 1.8 even when the chemical shrinkage term and the temperature-dependent material properties were used in the analysis.

4.7 Future Challenges Even though many experimental stress analysis techniques have been presented in this chapter, the fact is that modeling efforts are significantly more prevalent. This is especially true on the nanoscale, where molecular and multiscale modeling is proceeding at a strong rate, and the experimental tools to validate these models lag. As such, many of the models remain experimentally unsubstantiated, with little or no validation. Currently, there is a dearth of experimental methods that probe the nanoscale to investigate the complex interaction between composite phases. The traditional continuum breaks down and interatomic mechanics rules the behavior. On the nanoscale, full-field stress analysis techniques are nearly absent and thus this area of research is primed for major contributions. Experimental methods to measure strain must be developed in the atomic force microscope, scanning electron microscope, and tunneling electron microscope.

As new material systems are invented there will be a continuous need to characterize them. Nondestructive evaluation methods discussed in other chapters will play a pivotal role in the adoption of composite materials and their widespread utilization. On the macroscale, with the confidence bought by experimental and modeling efforts and the ability to produce and inspect composite structures, there will be an increase in the number of structures that will be built out of composites. This is especially true in our civil infrastructure, such as bridges and buildings. Continued advances in flight vehicles, lightweight ground vehicles, and sporting goods will require composites because of their advantageous properties. The future of composites is bright and the need to experimentally characterize them will only increase. Tomorrow’s composites may look very different than today’s but the challenges will be similar.

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though standard Kirchoff assumptions are made in the analysis (therefore variations due to through-thickness cure gradients cannot be measured). If the specimen is brought to the cure temperature and the strain is monitored, the thermal and chemical contributions can be separated. By applying the cure reference method to a unidirectional material, the free residual strain, defined by the thermal expansion and chemical shrinkage terms combined, can be measured. Then, by applying the cure reference method to a laminate (in the same autoclave cycle) the stresses can be calculated on the ply scale from the laminate strain information and the free residual strains, within the context of laminate theory. Experiments on the X-33 laminate were conducted at the University of Florida using a combination of the cure reference method and strain gages. The strain on the surface of multidirectional and unidirectional composites was measured through the temperature range from cure to liquid nitrogen. It was determined that approximately 20% of the strain at cryogenic temperatures originates from chemical shrinkage and the remainder from a thermal expansion mismatch. A series of experiments were performed to determine the dependence of

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4.77

4.78

– Carbon/Epoxy [02 /902 ]2s, Proc. SEM Spring Conference (1998) O. Sicot, X.L. Gong, A. Cherout, J. Lu: Determination of residual stress in composite laminates using the incremental hole-drilling method, J. Compos. Mat. 37(9), 831–844 (2003) F.L. Scalea: Measurement of thermal expansion coefficients of composites using strain gages, Exp. Mech. 38(4), 233–241 (1998) I.M. Daniel, T. Wang: Determination of chemical cure shrinkage in composite laminates, J. Comp. Technol. Res. 12(3), 172–176 (1989) P. G. Ifju, B. C. Kilday, X. Niu, S. C. Liu: A novel means to determine residual stress in laminated composites, J. Compos. Mat. 33(16), 1511–1524 (1999) P.G. Ifju, X. Niu, B.C. Kilday, S.-C. Liu, S.M. Ettinger: Residual strain measurement in composites using the cure-referencing method, J. Exp. Mech. 40(1), 22–30 (2000)

Part A 4

125

Fracture Mec 5. Fracture Mechanics

Krishnaswamy Ravi-Chandar

In this chapter, the basic principles of linearly elastic fracture mechanics, elastic plastic fracture mechanics, and dynamic fracture mechanics are first summarized at a level where meaningful applications can be considered. Experimental methods that facilitate characterization of material properties with respect to fracture and analysis of crack tip stress and deformation fields are also summarized. Full-field optical methods and pointwise measurement methods are discussed; many other experimental methods are applicable, but the selection presented here should provide sufficient background to enable implementation of other experimental methods to fracture problems.

5.2

Fracture Mechanics Based on Energy Balance ................................ Linearly Elastic Fracture Mechanics......... 5.2.1 Asymptotic Analysis of the Elastic Crack Tip Stress Field.. 5.2.2 Irwin’s Plastic Correction ............... 5.2.3 Relationship Between Stress Analysis and Energy Balance – The J-Integral .............................. 5.2.4 Fracture Criterion in LEFM ..............

126 128 128 129

Elastic–Plastic Fracture Mechanics.......... 5.3.1 Dugdale–Barenblatt Model............ 5.3.2 Elastic–Plastic Crack Tip Fields ....... 5.3.3 Fracture Criterion in EPFM ............. 5.3.4 General Cohesive Zone Models ....... 5.3.5 Damage Models ...........................

132 132 134 135 136 136

5.4 Dynamic Fracture Mechanics .................. 5.4.1 Dynamic Crack Initiation Toughness ................................... 5.4.2 Dynamic Crack Growth Toughness... 5.4.3 Dynamic Crack Arrest Toughness .....

137 138 139 139

5.5 Subcritical Crack Growth........................ 140 5.6 Experimental Methods .......................... 5.6.1 Photoelasticity ............................. 5.6.2 Interferometry ............................. 5.6.3 Lateral Shearing Interferometry ..... 5.6.4 Strain Gages ................................ 5.6.5 Method of Caustics ....................... 5.6.6 Measurement of Crack Opening Displacement ...... 5.6.7 Measurement of Crack Position and Speed...................................

140 141 143 147 151 152 153 155

130 131

References .................................................. 156

Assessment of the integrity and reliability of structures requires a detailed analysis of the stresses and deformation that they experience under various loading conditions. Many early designs were based on the limits placed by the strength of the materials of construction; in this approach the structure never attains loading that will cause strength-based failure during its entire lifetime. However, many spectacular failures of engineered structures occurred, not by exceeding strength limitations, but due to inherent flaws in the material and/or the structure, or due to flaws that grew to critical dimensions during operation. For example, during World

War II, Liberty ships simply broke into two pieces while sitting still in port as a result of cracking. The first commercial jet aircraft, the de Havilland Comet, failed in flight due to fractures emanating from window corners that grew slowly under repeated loading, eventually becoming critical. Such events provided the impetus for analysis that takes into account material and structural defects and spawned the development of the philosophy of flaw-tolerant design – such that the structure will operate safely, even in the presence of certain anticipated modes of failure. Such a damage- or flawtolerant approach to structural integrity and reliability

Part A 5

5.1

5.3

126

Part A

Solid Mechanics Topics

Part A 5.1

has become possible through a combination of material characterization, flaw detection and fracture mechanics analysis. The success of the flaw-tolerant design approach may be seen in the mid-flight tearing of fuselage panels of the Aloha Airlines Boeing 737 in 1988. While a significant portion of the fuselage was blown off due to linking of multiple cracks, the deflection of crack growth through bulkhead tear-straps limited crack growth, retaining the structural integrity of the airplane, which was able to land safely. The theory of fracture mechanics and the experimental methods used in fracture characterization are described in this chapter. The fundamental ideas of the theory of rupture were formulated elegantly by Griffith [5.1]; the crux of this theory is captured in Sect. 5.1. Griffith’s analysis brought a key element to the theory: that the maximum load-carrying capacity of a structural element depends not only on the strength of the material, but also on the size of flaws that may exist in the structural element – the larger the flaw, the smaller the load capacity or strength. Formalization of this into the practical theory of fracture mechanics – now called linear elastic fracture mechanics (LEFM) – was spearheaded through the efforts of Irwin [5.2, 3] who introduced the idea of the stress intensity factor that combined the stress and flaw issues into one parameter, and by many others;

LEFM is discussed in Sect. 5.2. The limitations of elastic analysis were recognized by many investigators (beginning with Griffith) and a theory of fracture that accounts for elastic–plastic material behavior, the theory of elastic–plastic fracture mechanics, described in Sect. 5.3, was developed. For fast-running cracks, the effect of inertia becomes important; the fundamentals of dynamic fracture theory and practical implementation aspects are discussed in Sect. 5.4. The experience from the Comet airplane incident indicated that, even if the structure was not fracture-critical under monotonic loading, crack extension could occur under repeated loading through a process of fatigue. Other processes such as stress corrosion and creep also engender timedependent crack growth under subcritical conditions. Subcritical crack growth is handled phenomenologically by correlating a loading parameter to crack speed (crack extension per cycle or per unit time). A short description of subcritical crack growth is provided in Sect. 5.5. Experimental investigations and specific experimental methods of analysis have played a crucial role in the development of fracture mechanics. Optical methods that provide full-field information over a region near the crack tip and pointwise measurement of some component of stress or deformation are both in common use; these methods are discussed in detail in Sect. 5.6.

5.1 Fracture Mechanics Based on Energy Balance Consider the equilibrium of a linearly elastic medium containing a crack; a special geometry, called the double-cantilever beam (DCB) is illustrated as an example in Fig. 5.1. This specimen contains a crack of length a and the other relevant geometrical quantities are shown in the figure; the specimen width is taken to be unity, indicating that all loads are considered to be per unit thickness. The total energy of the system E can be partitioned into two components: the potential energy Π and the energy associated with the crack Us . Thus, E = Π + Us .

to be the fracture energy per unit area of crack surface and not specifically attributed to the surface energy. We consider a quasistatic problem and hence the kinetic energy has been neglected. The energy of the system P, ΔT CM P, Δ

(5.1)

The potential energy is the sum of the work done by the applied external tractions WE and the stored energy in the system U. Us was defined originally by Griffith [5.1] as the surface energy of the crack. Orowan [5.4] generalized this to include plastic dissipation during creation of the surface and thus Us can be considered generally

2h

a

Fig. 5.1 Geometry of the double-cantilever beam fracture

specimen

Fracture Mechanics

must be an extremum when the system is in equilibrium; thus dE/ dA = 0, where A represents the crack surface area, which for the unit thickness considered is equal to the crack length a. We define the potential energy release rate per unit crack surface area G and the fracture resistance per unit surface area R as dUs dΠ (5.2) , R≡ . da da The equilibrium condition can be written as an expression that the potential energy released by the body in extending the crack is equal to the fracture resistance: G(a) ≡ −

G(ac ) = R ,

(5.3)

Δ (5.4) , P where P is the load per unit thickness and Δ is the displacement of the load point. Let the specimen be attached to a pinned support at the bottom and attached to a load through a spring at the top. The total displacement ΔT (considered to be fixed) is   ΔT = C(a) + CM P , (5.5) C(a) =

where CM is the compliance of the loading machine, with CM = 0 indicating a fixed-grip loading and CM → ∞ implying a dead-weight loading on the specimen. The potential energy in the system – the specimen and the loading machine – at fixed total displacement is Π=

Δ2 (ΔT − Δ)2 + . 2CM 2C(a)

(5.6)

127

The potential energy release rate can then be calculated either at fixed load or fixed grip conditions: 1 2 d C(a) (5.7) P . 2 da Thus, calculation of the compliance of a cracked body will yield the potential energy release rate directly. For the specific example of the double-cantilever beam, elementary beam theory can be used to estimate the 3 compliance of the beam segments: C(a) = E8ah 3 . Substituting in (5.7) and equating to the fracture resistance, we find that the equilibrium crack length can be written either in terms of the beam deflection or the applied load:   3 1/2  3Eh 3 Δ2 1/4 Eh R = . (5.8) ac = 16R 12P 2 G=

Obreimoff [5.5] used this idea and successfully measured the fracture resistance of mica subjected to cleavage between its layers. It is to be noted that the compliance of the loading machine does not enter into the determination of the equilibrium crack length, but will influence the stability of crack growth. The most remarkable aspect of the formulation of the fracture problem in terms of the energy approach is that the issue of analysis of the stresses and deformations in the vicinity of the crack is completely circumvented. In the case of the DCB specimen, as illustrated above, and in a few other simple cases, the compliance may be estimated rather easily and hence the energy approach is easily implemented. In other cases, the compliance can be obtained in the process of experimentation or through a numerical simulation of the structural problem and hence used in the analysis of fracture. The main obstacle to applying this approach in general is the determination of the fracture resistance. If the fracture energy per unit area is to be regarded as a material property, determined in analog laboratory tests and used in applications, the conditions under which crack growth occurs must be similar in the laboratory tests and field applications. This requires consideration of how the deformation and stresses evolve in the vicinity of the crack tip region. Analysis of the stress and deformation in a solid containing a crack is considered next, within the context of the theory of linear elasticity.

Part A 5.1

where ac is the equilibrium crack length. If the fracture resistance is taken to be a material parameter, then the problem of determining the equilibrium crack length is reduced to the problem of determining the potential energy release rate. This is the essence of fracture mechanics as outlined by Griffith [5.1] and augmented by Orowan [5.4]. Calculation of the potential energy release rate can be accomplished experimentally, analytically or numerically by solving the appropriate boundary value problems. A simple analytical method is described here, with application to the double-cantilever beam as an illustrative example. In general, the compliance of any specimen can be written as

5.1 Fracture Mechanics Based on Energy Balance

128

Part A

Solid Mechanics Topics

5.2 Linearly Elastic Fracture Mechanics The Griffith theory of fracture discussed above was developed further through analysis of the details of the stress and deformations in an elastic body containing a crack. We summarize these analyses under the assumption that the material may be modeled as a linearly elastic medium up to the point of failure. Criteria for failure under these conditions are also discussed.

5.2.1 Asymptotic Analysis of the Elastic Crack Tip Stress Field

Part A 5.2

The nature of the stress field near the crack tip in a linearly elastic solid was established through the efforts of Inglis [5.6], Mushkhelisvili [5.7], Williams [5.8], Irwin [5.2], and many other investigators. In general, consideration of three different loading symmetries is sufficient to decompose any arbitrary loading with respect to a crack; as illustrated in Fig. 5.2; these are typically called mode I or opening mode, mode II or inplane shearing mode, and mode III or antiplane shearing mode. For a homogeneous, isotropic, and linearly elastic solid, the structure of the solutions to the equations of equilibrium subjected to traction-free crack surfaces can easily be determined for each of the three modes of loading; this stress field in the vicinity of the crack for modes I and II may be written in the following form: KI I (θ, 1) + σ0x δ1α δ1β Fαβ σαβ (r, θ) = √ 2πr ∞  n I + (θ, n) An r n/2 Fαβ 2 n=3

K II II +√ Fαβ (θ, 1) 2πr ∞  n II + (θ, n) Bnr n/2 Fαβ 2 n=3 ⎧ ⎨0 plane stress , σ33 (r, θ) = ⎩νσ (r, θ) plane strain

(5.9)

where the θ dependence is given by Kobayashi [5.9]   I (θ, 1) = cos 12 θ 1 − sin 12 θ sin 32 θ , F11

II F22 (θ, 1) = sin 12 θ cos 12 θ cos 32 θ ,

The corresponding displacement field can be expressed as  σ0x 2   KI r I  G θ, 1 + rG I θ, 2 u α (r, θ) = μ 2π α μ 1+ν α ∞   An n/2 I  r G α θ, n + 2μ n=3  K II r II  + G α θ, 1 μ 2π ∞   Bn n/2 II  + r G α θ, n , 2μ ⎧ n=3     ⎨− νxE3 σαα r, θ plane stress , (5.10) u 3 r, θ = ⎩0 plane strain where

αα

I F12 (θ, 1) = sin 12 θ cos 12 θ cos 32 θ ,   I F22 (θ, 1) = cos 12 θ 1 + sin 12 θ sin 32 θ ,   II F11 (θ, 1) = − sin 12 θ 1 + cos 12 θ cos 32 θ ,   II F12 (θ, 1) = cos 12 θ 1 − sin 12 θ sin 32 θ ,

   I (θ, n) = 2 + (−1)n + 12 n cos 12 n − 1 θ F11     − 12 n − 1 cos 12 n − 3 θ ,

   I F12 (θ, n) = − (−1)n + 12 n sin 12 n − 1 θ     + 12 n − 1 sin 12 n − 3 θ ,

 n    I F22 (θ, n) = 2 − −1 − 12 n cos 12 n − 1 θ     + 12 n − 1 cos 12 n − 3 θ , for n ≥ 2 ,

 n    II F11 (θ, n) = − 2 + −1 + 12 n sin 12 n − 1 θ     + 12 n − 1 sin 12 n − 3 θ ,

     n II F12 (θ, n) = −1 + 12 n cos 12 n − 1 θ     + 12 n − 1 cos 12 n − 3 θ ,

 n    II F22 (θ, n) = − 2 − −1 − 12 n sin 12 n − 1 θ     − 12 n − 1 cos 12 n − 3 θ . for n ≥ 2 .

 G I1 (θ, 1) = cos 12 θ 1 − 2ν + sin2  G I2 (θ, 1) = sin 12 θ 2 − 3ν − cos2  2 1 G II 1 (θ, 1) = sin 2 θ 2 − 2ν + cos  2 1 G II 2 (θ, 1) = cos 2 θ 1 − 2ν + sin



1 2θ ,  1 2θ ,  1 2θ ,  1 2θ ,

G I1 (θ, 2) = cos θ , G I2 (θ, 2) = −ν sin θ ,

  n n n G I1 (θ, n) = (3 − 4ν) cos θ − cos −1 θ 2 2 2   n n n + (−1) cos θ , + 2 2

Fracture Mechanics

  n n n G I2 (θ, n) = (3 − 4ν) sin θ + sin −1 θ 2 2 2   n n + (−1)n sin θ , − 2 2 for n ≥ 3 ,   n n n G II (θ, n) = −(3 − 4ν) sin θ + sin − 1 θ 1 2 2 2   n n + (−1)n sin θ , − 2 2   n n n II −1 θ G 2 (θ, n) = −(3 − 4ν) cos θ + cos 2 2 2   n n + (−1)n cos θ , − 2 2 for n ≥ 3 .

σ22 σ12 σ11

y r θ x

Fig. 5.3 Crack tip coordinate system and notation for

stress components

The stress intensity factors are a function of the applied load, the geometry of the specimen or structure, and the length of the crack, and must be determined by solving the complete boundary value problem in linear elasticity that includes the full geometric description of the cracked structure and the applied load; the stress intensity factor has dimensions √ of (FL3/2 ) and in SI units is typically indicated as MPa m. The displacement field in (5.10), on the other hand, is bounded and u α → 0 as r → 0. While it may be objectionable to consider extending linear elastic calculations to infinite stresses and strains, there are two simple arguments to reconcile this difficulty. First, even though the stresses are unbounded, the energy in a small volume near the crack tip is always bounded because the displacements tend to zero. Secondly, inelastic deformations occur in the regions of high stress and hence, these expressions are taken to be applicable for small values of r relative to structural dimensions, but outside of a distance rp where inelastic, nonlinear, and fracture processes dominate the deformation of the material; an estimate of rp is obtained in the next section. For distances that are very large in comparison to rp , terms involving higher powers of r may become important. The nonsingular term is labeled σ0x in keeping with the literature in experimental mechanics, but it is also called the T-stress in the general fracture literature. This term in (5.9) is independent of r and arises only in the σ11 component of stress. These terms may not necessarily be important in characterizing fracture itself – although there is some debate in recent literature that suggests otherwise – but must be taken into account while evaluating experimental data collected from fullfield optical techniques discussed later in this chapter.

5.2.2 Irwin’s Plastic Correction Fig. 5.2 Three loading symmetries in a cracked specimen

129

Irwin [5.3] suggested that the effect of the plastic zone near the crack tip can be approximated by a very simple

Part A 5.2

The corresponding expressions of the stress and deformation fields for mode III can also be written; we do not address mode III problems in this chapter and hence these fields are not provided here. The corresponding expressions for plane strain may be obtained by replacing ν by ν/(1 + ν). The standard notation of linear elastic theory is used; the Cartesian components of stress are written in terms of the polar coordinates centered at the crack tip (see Fig. 5.3 for the coordinate system and notation). δαβ is the Kronecker delta and E is the modulus of elasticity. Standard index notation is used; Greek subscripts take the range 1, 2 and summation with respect to repeated subscripts is implied. These fields are seen to be separable in r and θ; the θ dependence for the two different modes are given above for completeness. From the first term in (5.9), the stress components are seen to be square-root singular as r → 0 and hence (5.9) cannot be valid very close to the crack tip. The strength of the singularity is determined by the two scalar quantities K I and K II , called the stress intensity factors in modes I and II, respectively.

5.2 Linearly Elastic Fracture Mechanics

130

Part A

Solid Mechanics Topics

such as B, the specimen thickness a, the crack length, and any other length defining the geometry of the specimen. This gives a qualitative meaning to the concept of small-scale yielding, and a more precise restriction must be obtained from experimental characterization of materials.

σ22

5.2.3 Relationship Between Stress Analysis and Energy Balance – The J-Integral

ry

rp

x

Fig. 5.4 Irwin model of the crack tip plastic zone

model; assume that the stress in the plastic zone near the crack tip must be limited to some multiple of the yield stress: σ22 (r < rp , 0) = βσY , where rp is the extent of the plastic zone. However, in order to maintain equilibrium, the above assumption requires that the stress estimated from (5.9) for some distance r < ry must be redistributed to regions r > ry , as indicated in Fig. 5.4. This is expressed by the following equation

Part A 5.2



ry σ22 (r, 0) dr = rp βσY , 0

1 KI ry = 2π βσY

2 . (5.11)

Evaluating this equation yields rp = 2ry . In this estimate, the square-root singular field is considered to have its origin shifted to the point r = ry , effectively considering the crack to be of length aeff = a + ry . Hence the stress intensity factors need to be evaluated using this effective crack length. If we assume a Tresca yield criterion, for thin plates with σ33 = 0, it is easy to show that β = 1. However, for conditions of plane strain expected expects to prevail near the crack tip, with σ33 > 0, one √ β to be larger; Irwin suggested a value of β = 3, resulting in an estimate of plastic zone size   1 KI 2 . (5.12) rp = 3π σY This simple estimate of the zone of yielding does remarkably well in capturing the extent of yielding near the crack tip. More importantly, this analysis indicates clearly that the characteristic scale in the frac2  length ture problem is obtained as σKYI . In applying the ideas of LEFM, it is now clear that one must ensure that (K I /σY )2 is much smaller than other relevant lengths

The energy release rate described in Sect. 5.1 and the stress field discussed above must, of course, be related in some manner. This relationship can be obtained in a formal way by considering the change in potential energy, using ideas of conservation integrals. Eshelby [5.10] introduced these in a general sense for defects; Cherepanov [5.11] and Rice [5.12] defined a special case – the J-integral – for cracks. Knowles and Sternberg [5.13] developed these as general conservation laws for elastostatic problems, but Rice’s paper was the key to recognizing the importance to fracture mechanics. The J-integral is defined as 

 ∂u α (5.13) Un 1 − σαβ n β ds , J= ∂x1 Γ

where U =

ε

σαβ dεαβ is the mechanical work in de-

0

forming to the current strain level, and n α are the components of the unit outward normal to the contour of integration Γ shown in Fig. 5.5; this integral is independent of the path of integration. By considering a contour along the boundaries of the specimen, it can be shown that dΠ (5.14) =G. J =− da Furthermore, by considering a contour to lie in the region close to the crack tip such that the asymptotic field in (5.9) and (5.10) is valid, it can be established that 1 − ν2 2 1 − ν2 2 1 + ν 2 (5.15) KI + K II + K III . J= E E E x2

x1 s Γ

Fig. 5.5 Contour path for the J-integral

n

Fracture Mechanics

For plane stress, the factor (1 − ν2 ) in the first two terms of (5.15) is replaced with unity. This equivalence between the potential energy release rate and the stress intensity factor forms the basis of the local approach to fracture.

5.2.4 Fracture Criterion in LEFM Equations (5.3) and (5.14) provide the basis for formulating the fracture criterion in terms of the energy release rate or the J-integral; for the case of small-scale yielding (5.15) indicates that the fracture criterion may be posed in terms of the stress intensity factor as well. The crack is now considered to be fracture critical under mode I loading when KI = KC ,

(5.16)

KC

the specimen thickness B obey the following inequality   K IC 2 a, B ≥ 2.5 . (5.17) σY The characteristic length scale can be seen to arise from Irwin’s scaling analysis in Sect. 5.2.2; the numerical factor was determined through numerous tests in different materials and loading geometries. Standard techniques for characterization of the fracture toughness are provided by the American Society for Testing and Materials, the British Standards Institution, the Japanese Society of Mechanical Engineers, and the International Organization for Standardization (ISO). In particular, conditions for preparation and testing of specimens that produce a valid measurement of the plane-strain fracture toughness that obeys the restriction in (5.17) are discussed in these standards. The typical range of values of the fracture toughness for different materials is provided in Table 5.1. This methodology of assessing fracture with very limited inelastic deformation is called linearly elastic fracture mechanics (LEFM). As an illustration of LEFM, consider a simple example: a large panel with a crack of length 2a subjected to uniform stress σ; the stress√intensity factor for this configuration is simply K I = σ πa. Applying the fracture criterion in√(5.16) results in the critical condition, expressed as σ πa = K IC . This condition can be used in one of three ways in fracture-critical structures. First, at the design stage, one sets the crack length to be at the limit that is detectable by nondestructive inspection techniques. Then, for a desired design load, a material with the appropriate fracture toughness can be selected or alternatively, for a given material, the maximum permissible stress can be determined. Second, for a given structural application with fixed fracture toughness and stress, the critical crack length can be calculated and Table 5.1 Typical values of the fracture toughness for various materials

KIC

2.5 Normalized specimen thickness B

σY KIC

2

Fig. 5.6 Dependence of the fracture toughness on specimen thickness

Material

K IC , MPa

Ductile metals: Cu, Ni, Ag A533 Steel Mild steels High-strength steels Al alloys Ceramics: Al2 O3 , Si3 N4 Polyethylene Polycarbonate Silica glass

100– 350 200 140 50– 150 20– 45 3–6 2 1 – 2.5 0.7

√ m

131

Part A 5.2

where K C is the critical stress intensity factor, considered to be a material property; note that it must be evaluated with special care to ensure that conditions of small-scale yielding are assured. One expects that the plastic work in a thinner specimen is larger than in that in a thicker specimen; this is reflected in the typical variation of K C with specimen thickness as shown in Fig. 5.6. The measured value of K C is high for small specimen thicknesses and reaches a nearly constant lower plateau at large specimen thicknesses; it is this value of the critical stress intensity factor, labeled K IC , that a new material property called the plane strain fracture toughness that is taken to be a new material. The main condition that arises here ensures that the radius rp of the inelastic region near the crack tip is small enough; thus, in tests performed to evaluate the fracture toughness, one must have the crack length a and

5.2 Linearly Elastic Fracture Mechanics

132

Part A

Solid Mechanics Topics

used in inspections to determine how close the structure is to fracture criticality. Considering that cracks grow during subcritical loading (Sect. 5.5), one may also determine how long the crack will grow in the time interval between inspections. Finally, for a given crack length and material, one can impose limits on loading such that critical conditions are not reached during operation. The fracture criterion described above is valid only for the mode I loading symmetry. Under a combination of modes I and II, the crack typically kinks from its initial direction and finds a new direction; hence the fracture criterion should not only determine the load at which the crack will initiate under mixed-mode loading, but also determine the direction of the new crack. Applying the energy balance idea to this problem, the fracture criterion may be stated as follows: The crack will follow the path along which the potential energy release rate is maximized. While this is the most appropriate form of the fracture criterion, it is difficult to apply since the energy release rate has to be calculated for different paths and then maximized. A number of alternative criteria exist: the maximum tangential

stress criterion, the principle of local symmetry, the strain energy density criterion, etc., but their predictions are not clearly distinguishable in experimental observations. The maximum tangential stress criterion, which provides the same estimate for the crack direction as the energy-based criterion, can be stated as follows: The crack will extend in the direction γ such that ∂σθθ /∂θ is maximized. Using the stress field in (5.9),  ⎫ ⎧ ⎨ K I − K I2 + 8K II2 ⎬ . γ = 2 arctan (5.18) ⎭ ⎩ 4K II The energy release rate for extension in that direction G(γ ) must be equal to the fracture resistance R at the onset of crack initiation. The fracture criterion appropriate for mode III loading is significantly more complicated. A straightforward extension of the maximum tangential stress criterion discussed in this paragraph for use in combined mode I and mode II can be shown to breakdown simply from symmetry considerations. In fact, the crack front breaks up into a fragmented structure and appropriate criteria for characterizing this are still not well developed.

Part A 5.3

5.3 Elastic–Plastic Fracture Mechanics In cases where the extent of the plastic zone becomes large compared to the specimen thickness and crack length, the LEFM methodology of Sect. 5.2 breaks down and one is forced to contend with the inelastic processes that occur within r < rp . This is the subject of elastic–plastic fracture mechanics (EPFM). First, an elegant and simple way of incorporating the inelastic effects – through the Dugdale–Barenblatt model – is introduced; this is followed by an evaluation of the stress and deformation field using the deformation theory of plasticity. Then we return to an extension of the Dugdale–Barenblatt model in the form of generalized cohesive zone models for inelastic fracture. Finally, a generalized damage model based on void nucleation and growth is described.

the crack tip, interatomic (cohesive) forces that attract the two separating crack surfaces cannot be ignored in the analysis. Dugdale considered a larger-scale picture where a linear region in the prolongation of the crack tip would limit the development of stress to some multiple of the yield stress (similar to the observation of Irwin discussed in Sect. 5.2.2). Both these suggestions amount

δt

α

5.3.1 Dugdale–Barenblatt Model The idea of eliminating the source of the singularities in the elastic analyses was proposed independently by Barenblatt [5.14] and Dugdale [5.15] and has been generalized into what are now commonly called cohesive zone models. Barenblatt suggested that, close to

δt

Fig. 5.7 Schematic diagram of the Dugdale–Barenblatt model of fracture

Fracture Mechanics

Problem a

factors:

 K Ia = σ ∞ π(a + rp ) ,   a + rp a b −1 (5.19) . cos K I = −2βσY π a + rp Note that the negative sign in problem b is a mere formality arising from the fact that the loads pinch the crack tip closed. The condition that the resulting problem does not contain singular stresses is enforced by requiring the sum of the stress intensity factor from the two component problems to be zero; this results in the following estimate for the extent of the inelastic zone:  ∞ rp πσ (5.20) −1 . = sec a 2βσY The crack-tip opening displacement (CTOD) is defined as the separation between the top and bottom surfaces of the physical crack tip located at x1 = a; this can also be calculated as the superposition of the elastic solutions to the two problems; for plane stress, this results in δt = u 2 (x1 = a, x2 = 0+ ) − u 2 (a, 0− )   ∞  8βσY a πσ = (5.21) . ln sec πE 2βσY A key feature of the DB condition is that the crack opening profile is not parabolic, as indicated by the singular solution, but a smooth cusp-like closing from x1 = a to x1 = a + rp ; this is indicated in Fig. 5.9. Crack opening displacement – πE δ 2βσY a 10

5

σ∞ Problem b

133

0

βσ Y –5 2(a+rp) 2(a+rp) –10 –2

–1

0

1 Position –

2 x1 a

Fig. 5.8 Dugdale–Barenblatt model of fracture for a center- Fig. 5.9 Crack opening profile in the Dugdale–Barenblatt cracked specimen model of fracture

Part A 5.3

to a model for the fracture process and provide a way to regularize the stress near the crack tip. Since then generalizations of the cohesive zone ideas to craze failure in polymers (Knauss [5.16] and Schapery [5.17]), fracture in weakening solids such as concrete by Hillerborg et al. [5.18], and for ductile failure in metallic materials [5.19] have been developed. A schematic diagram of the Dugdale–Barenblatt (DB) model is shown in Fig. 5.7. Assuming that the extent of the inelastic zone ahead of the crack tip rp is large in comparison to the specimen thickness B, it can be modeled as a line segment on x2 = 0, along which the normal stress σ22 can be taken to be limited by the yield stress as in the case of the Irwin model: σ22 (r < rp , 0) = βσY . However, the extent of the inelastic zone is obtained by ensuring that the singularity indicated by the elastic analysis is removed, rather than from the local equilibrium analysis of Irwin; this is called the DB condition. This requires the solution of the appropriate boundary value problem for the cracked elastic solid; the procedure is illustrated here for the case of a large plate with a central crack of length 2a loaded in uniform remote tension as shown in Fig. 5.8. Consider that the material in the DB zone is removed and replaced with its equivalent effect, i. e., a constant stress of magnitude βσY over the length rp . (Indeed, there is no reason to limit this to constant stress; the generalization of nonlinear cohesive zone models simply relies on introducing a nonlinear traction separation law over the line of the inelastic zone.) This is represented as the superposition of two elastic problems, problem a, with uniform far-field stress σ ∞ of a crack of length 2(a + rp ), and problem b, with uniform closing stress of magnitude βσY only over the extended crack surfaces rp near either tip. The solutions to these problems result in the following stress intensity

5.3 Elastic–Plastic Fracture Mechanics

134

Part A

Solid Mechanics Topics

While the introduction of the DB model has removed the singularity and provided a way to estimate the extent of the inelastic zone, it has also eliminated the possibility of imposing a fracture criterion based on the stress intensity factor. One has to resort to the energy or J-based criterion for fracture; the latter can be accomplished by evaluating the J-integral for a contour indicated in Fig. 5.7 that closely follows the DB zone; for a < x1 < a + rp , we have n 1 = 0, σ11 = 0, σ12 = 0, σ22 = βσY ; thus, a+α   

∂u 2  ∂u 2  J =− − (−βσY ) dx1 ∂x1 (x2 ,0+ ) ∂x1 (x1 ,0− ) a

= βσY δt .

(5.22)

Part A 5.3

The equivalence indicated in (5.22) between the Jintegral and the CTOD holds in general for all elastic–plastic crack tip models, but the particular form above is valid for all crack geometries based on the DB model. Now, the fracture criterion may be applied by setting J = Jc at the onset of crack initiation, or equivalently in terms of the CTOD. For the particular case of the center-cracked geometry, substituting from (5.21), setting J = Jc at the failure stress σ ∞ = σf , and rearranging, we get an estimate for the failure stress under the DB model as:    2 πE σf J = sec−1 exp (5.23) . c βσY π 8(βσY )2 a A plot of the failure stress as a function of the crack length is shown in Fig. 5.10. For comparison, the estiNormalized failure stress – 2

mate based on small-scale yielding is also shown in the figure. It is clear that, for short cracks, the failure stress is large enough that large-scale yielding becomes important and LEFM does not provide a correct estimate of the fracture strength.

5.3.2 Elastic–Plastic Crack Tip Fields In order to account in greater detail for the development of plastic deformation in the vicinity of the crack tip, Hutchinson [5.20], and Rice and Rosengren [5.21] performed an asymptotic analysis of the fields in a nonlinear elastic solid characterized by a power-law constitutive model. The stress–strain relationship is given as  n−1 sαβ εαβ 3 σe = α , (5.24) εY 2 σY σY where α is a material constant, n ≤ 1 is the strainhardening exponent, σY is the yield stress in uniaxial tension and εY = σY /E is the corresponding yield strain, sαβ = σαβ − σγγ δαβ /3  are the components of the stress deviatoric, and σe = 3sαβ sαβ /2 is the effective stress. The field near the crack tip was determined to be   J 1 1/(n+1) σ˜ αβ (θ, n) , σαβ (r, θ) = σY ασY εY In r  εαβ (r, θ) = αεY

ε˜ αβ (θ, n) , (5.26)



σf βσY

J 1 ασY εY In r

(5.25)

n/(n+1)

1 J u α (r, θ) = αεYr ασY εY In r

n/(n+1) u˜ α (θ, n) , (5.27)

1.5

1

0.5

0

0

1

2

3

Normalized crack length –

4 8(βσY)2 a πEJc

Fig. 5.10 Comparison of the failure stress calculated from

LEFM (dashed line) and the DB model (solid line)

where J is the value of the J-integral, In is a dimensionless quantity that depends only on the strain hardening exponent (see, for example, Kanninen and Popelar [5.22] for calculations of I (n)), and σ˜ αβ (θ, n), ε˜ αβ (θ, n), and u˜ αβ (θ, n) express the angular variation of the stress, strain, and displacement variation near the crack tip, determined numerically; the description of the field given above is called the HRR field, in honor of the original authors, Hutchinson, Rice and Rosengren. It is clear that the stress and strain components given in (5.25) and (5.26) are singular as r → 0, but with the order of singularity different from the corresponding linear elastic problem and different for the stress and strain components. The strength of the singularity

Fracture Mechanics

is determined by the value of the J-integral. Equations (5.25)–(5.27) cannot be valid very close to the crack tip and a small region in the vicinity of the crack tip – called the fracture process zone, rfpz – must be excluded from consideration. The fracture process zone is taken to be the region in which the actual process of fracture – void nucleation, growth, and coalescence – occurs and results in the final failure of the material. Furthermore, it is clear from (5.27) that the CTOD, defined arbitrarily as the opening at the intercept of two symmetric 45◦ lines from the crack tip and the crack profile, should be proportional to the J-integral: δt = d(n, εY )J/σY , where d(n, εY ) can be calculated in terms of the HRR field; see Kanninen and Popelar [5.22] for a plot of this variation. Thus, in EPFM, J plays multiple roles – determining the amplitude of the plastic stress and deformation field, determining the CTOD, and determining the potential energy release rate. It is possible to incorporate terms that are higher order in r but evaluation of such terms is much more difficult in the elastic–plastic problem in comparison to the elastic problem. Attempts have been made to incorporate the next-order term (the nonsingular term) in fracture analysis [5.23, 24], but these are not addressed here.

Unlike the case of LEFM, there is no simple way in which to estimate the size of the process zone rfpz and hence of the region of dominance of the HRR field expressions given above. Some attempts using moiré interferometry are discussed in Sect. 5.6.2. However, it is clear from discussions of the plastic zone in Sect. 5.2.2 and the CTOD that the only characteristic length scale relevant to this problem can be written as J/σY . Thus, the remaining uncracked segment b and the thickness of specimen B must obey the following inequality in order to justify the use of the HRR field b, B > κ

J . σY

(5.28)

Estimates of how large this should be vary significantly depending on the geometry of the specimen; estimates based on finite element analysis show that κ varies over a range from about 20–300 depending on the specimen geometry. Under such conditions, the fracture criterion may be posed as the equivalent of the energy criterion discussed above. Hence, we have J(σ , a) = JC .

(5.29)

135

Here it is necessary to calculate J(σ , a), a somewhat more challenging task than the corresponding calculation of the stress intensity factor for a problem in LEFM. Under laboratory testing conditions, since the J-integral is the change in potential energy of the system, one can use (5.14) and the experimentally measured load versus load-point displacement data to evaluate J. Typically, this results in a simple expression of the form J=

ηA , bB

(5.30)

where A is the area under the load–displacement curve, b is the remaining uncracked ligament, B is the specimen width, and η is a factor that depends on the specimen geometry. Therefore, monitoring the load–displacement curve is adequate in the evaluation of the J-integral. The value of the J-integral at onset of crack extension should result in an experimental measurement of the fracture toughness under elastic–plastic conditions, not restricted to small-scale yielding. In applying this approach of EPFM to other structural configurations, one encounters two difficulties. First, whereas many different handbooks tabulate solutions of stress intensity factors for different geometric and loading conditions, few such solutions are available for elastic plastic problems. The Ductile Fracture Handbook [5.25] provides a tabulation of estimates of the J-integral for a few cases associated with circular cylindrical pipes and pressure vessels. However, with modern computational tools, this is hardly a limitation and one should, in principle, be able to calculate the J-integral for arbitrary loading and geometric conditions. The second – and more challenging – difficulty arises from the fact that the measured crack resistance in many common structural materials increases significantly with crack extension and that such an increase depends critically on the constraint to plastic deformation provided in the particular geometric condition of the test. This brings into question the transferability of laboratory test results to full-scale structural integrity assessment. Therefore the methodology presented in this section remains limited in its use to a few configurations of cylindrical pipes and pressure vessels where there is significant experience in its use. Recent advances in elastic–plastic fracture methodology have focused more on generalized cohesive zone models and damage models as discussed in the next two sections. Such generalized cohesive zone models are

Part A 5.3

5.3.3 Fracture Criterion in EPFM

5.3 Elastic–Plastic Fracture Mechanics

136

Part A

Solid Mechanics Topics

now embedded into commercial finite element software packages.

5.3.4 General Cohesive Zone Models

Part A 5.3

Cohesive zone models for nonlinear fracture problems are simply a straightforward extension of the Dugdale– Barenblatt model. In these models, the inelastic region (including the plastic zone and the fracture process zone) is assumed to be in the form of a line ahead of the crack tip, as in the DB model. However a relationship between the traction across this line T and the separation δ between the two surfaces T = F(δ) is postulated. Such a relationship may be obtained directly from experimental measurements (as in the case of some polymeric materials) or more typically from a micromechanical model of the void nucleation, growth, and coalescence that are the underlying fracture processes. A typical traction-separation law for cohesive zone models is shown in Fig. 5.11. Two crucial features can be identified: the peak stress that corresponds to the maximum stress that can be sustained by the material in the fracture process zone and the maximum displacement that corresponds to the critical CTOD. The area under the traction separation curve is the fracture energy JC . Details of such generalized cohesive models may be found in a number of papers by Knauss [5.16], Schapery [5.17], Hilleborg et al. [5.18], Kramer [5.26], Normalized traction T 1.5

1

0.5

0

0

0.2

0.4

0.6 0.8 1 Normalized separation δ

Fig. 5.11 A typical traction–separation law used in cohesive zone models is shown. Traction is normalized by the peak cohesive traction and the separation is normalized by the critical CTOD; the area under the curve is the fracture energy

Carpinteri [5.27], Needleman [5.28], Xu and Needleman [5.29], Yang and Ravi-Chandar [5.30], Ortiz and Pandolfi [5.31], and many other references. Incorporation of such a generalized cohesive zone model into finite element computational strategies allows for estimation of crack growth to be performed automatically, without imposing additional criteria for fracture. Irreversible models of the cohesive zone have also been proposed based on the maximum attained crack surface separation [5.30–32]. In particular, persistence of previous damage in the cohesive zone, and contact and friction conditions upon unloading, are important aspects to include in a proper description of the cohesive zone model. Numerous investigators have shown that crack growth simulations based on cohesive zone models are somewhat insensitive to the detailed form of the traction-separation curve and that it is much more important to capture the correct fracture energy. However, care should be exercised in using this idea since there is an inherent dependence on the mesh size and geometry dependence of the results obtained (Falk et al. [5.33]).

5.3.5 Damage Models In cases where the cohesive zone model may not be appropriate because of the diffuse nature of the inelastic zone, one may resort to continuum damage models for modeling fracture. The Gurson–Tvergaard–Needleman model is perhaps the most commonly used model for simulations of fracture [5.34]. In this model, the yield function describing the plastic constitutive model is represented as   σ2 3q2 σH −1 g(σe , σH , f ) = e2 + 2 f ∗ q1 cosh 2σ σ 2  (5.31) − q1 f ∗ , where σe is the effective stress, σH = σii/3 is the mean stress, and σ is the flow stress of the material at the current state, q1 and q2 are fitting parameters used to calibrate to model predictions of periodic arrays of spherical and cylindrical voids, f is the void volume fraction, and f ∗ is a bilinear function of the void volume fractions that accounts for rapid void coalescence at failure and is given as ⎧ ⎨f f < fc , (5.32) f∗ = ⎩ f c + (1/q1 − f c ) ( f − f c ) f ≥ f c ( ff − fc )

where f f represents the void volume fraction corresponding to failure; the value of f ∗ at zero stress f c

Fracture Mechanics

is the critical void volume fraction where rapid coalescence occurs. The above description of the yield function needs to be augmented with evolution equations for both the flow stress σ and the void volume fraction f . The instantaneous rate of growth of the void fraction depends both on nucleation of new voids and growth of preexisting voids. Thus, f˙ = f˙nucleation + f˙growth .

(5.33)

Chu and Needleman [5.35] assumed that void nucleation will obey a Gaussian distribution; then straincontrolled nucleation can be expressed as    1 ε¯ p − εn ˙ p fn ε¯ (5.34) f˙nucleation = √ exp − 2 sn sn 2π where sn is the standard deviation, εn is the nucleation threshold strain, f n is a constant, and ε¯ p is the average plastic strain. The latter is related to the plastic strain through a work equivalence expressed as (1 − f )¯εp σ = σ : ∂Φ ∂σ . The growth rate is obtained by enforcing that the plastic volumetric strain rate in the matrix (outside of the voids) is zero; thus, p f˙growth = (1 − f )˙εkk .

(5.35)

where m is the strain rate hardening exponent, and g(¯ε, T ) is the temperature-dependent effective stress versus effective strain response calibrated through a uniaxial tensile test at a strain rate ε˙¯ 0 and temperature T .

Setting f = 0 results in the above formulation reducing to a standard von Mises yield surface for an isotropic material. Equations (5.31)–(5.35) augment the standard plasticity equations in the sense that void nucleation and growth are now incorporated into the model. If the above model is evaluated under monotonically increasing uniaxial strain on a cell of unit height, initially a linear elastic behavior is obtained; this is followed by nonlinearity associated with plastic deformation and eventually a maximum stress associated with the onset of decohesion or failure of the unit cell. Beyond this level of displacement, the cell is unstable to load control, but deforms in a stable manner under displacement control. Thus, the damage zone model mimics the response that one assumes in the generalized cohesive zone models discussed above. Two main issues need to be addressed with respect to this type of damage model. First, there are numerous constants embedded into the theory. In addition to the usual elastic constants E and ν, one has the plastic constitutive description g(¯ε, T ), the strain-rate hardening parameter m, the four parameters that define the yield function q1 , q2 , f c , f f , and the three parameters that define the nucleation criterion f n , sn , εn . These need to be determined through calibration experiments for each particular material. Second, the introduction of a cell is a micromechanical construct: it introduces an artificial length scale and geometric effect that must be considered carefully. For example, does calibrating the model parameters under uniaxial straining provide sufficient characterization to handle biaxial or triaxial loading conditions? Active research continues in this area to explore these questions further and to validate the approach of the damagemodel-based fracture analysis.

5.4 Dynamic Fracture Mechanics In many applications, it is necessary to consider cracks that grow rapidly at speeds that are a significant fraction of the stress wave speeds in the material. In such cases, material inertia effects cannot be ignored in considerations of crack growth. Within the setting of small-scale yielding, analysis of the stress and displacement field near the tip of a crack moving at a speed v can be accomplished easily within linear elastodynamic theory [5.36, 37]; the crack tip stress and displacement fields for in-plane loading conditions may be written in

terms of polar coordinates moving with the crack tip as KI K II II I f αβ f αβ (θ, v) (θ, v) + √ σαβ (r, θ) = √ 2πr 2πr   + σox αd2 − αs2 δα1 δβ1 + · · · , ⎧ ⎨0 plane stress σ33 (r, θ) = , (5.37) ⎩νσ (r, θ) plane strain αα

137

Part A 5.4

Power-law hardening of the plastic material is represented as  m σ ˙ε¯ = ε˙¯ 0 , (5.36) g(¯ε, T )

5.4 Dynamic Fracture Mechanics

138

Part A

Solid Mechanics Topics

√ KI r I u α (r, θ) = √ gα,β (θ, v) 2π √ K II r II + √ gαβ (θ, v) + · · · , ⎧ 2π ⎨− νx3 σ (r, θ) plane stress E αα . (5.38) u 3 (r, θ) = ⎩0 plane strain I (θ, v) and f II (θ, v) are The functions f αβ αβ I f 11 (θ, v) =

1 R(v)



G(v) =

  cos 12 θd  1 + αs2 1 + 2αd2 − αs2 1/2 γd

− 4αd αs

1



1 2 θs

cos , 1/2 γs  2 cos 12 θd  1 I (θ, v) = − 1 + αs2 f 22 1/2 R(v) γd  1 + 4αd αs 1/2 cos 12 θs , γs    1 + αs2 sin 12 θd sin 12 θs 2α d I − , f 12 (θ, v) = 1/2 1/2 R(v) γ γs d

Part A 5.4

II f 11 (θ, v) = −

2αs R(v)



(5.39)

 sin 12 θd  1 + 2αd2 − αs2 1/2 γd

  1  − 1 + αs2 1/2 sin 12 θs , γs    2 2αs 1 + αs sin 12 θd 1 II 1 (θ, v) = − sin θ f 22 2 s , 1/2 1/2 R(v) γd γs  cos 12 θd 1 II f 12 (θ, v) = 4αd αs 1/2 R(v) γd   cos 12 θs  − 1 + αs2 (5.40) , 1/2 γs where Cd and Cs are the dilatational and distortional wave speeds, respectively, and  γd = 1 − (v sin θ/Cd )2 and  (5.41) γs = 1 − (v sin θ/Cs )2 , tan θd = αd tan θ , tan θs = αs tan θ , 2  R(v) = 4αd αs − 1 + αs2 .

gularity dictated by the dynamic stress intensity factor for the appropriate mode of loading. In analogy with the quasistatic crack problems, it is possible to define a dynamic energy release rate; it is the energy released into the crack tip process zone per unit crack extension. Introducing the elastodynamic singular stress field, G can be related to the dynamic stress intensity factor

(5.42) (5.43)

As in the case of quasistatic LEFM, the crack tip stress field is square-root singular, with the strength of the sin-

 1 − ν2

AI (v)K I2 + AII (v)K II2 , E

v2 αd and (1 − ν)Cs2 R(v) v2 αs . AII (v) = (1 − ν)Cs2 R(v)

(5.44)

AI (v) =

(5.45)

E is the modulus of elasticity and ν is the Poisson’s ratio. The fracture criterion in (5.3) can be imposed in this problem as well – the dynamic energy release rate must be equal to the dissipation; thus G(v) = R provides the equation of motion for the crack velocity. The functions AI (v) and AII (v) in (5.45) are singular as v → CR . To satisfy the energy balance equation, the dynamic stress intensity factors K I (t, v) and K II (t, v) must tend to zero as v → CR . This implies that the limiting crack speed in modes I and II is the Rayleigh wave speed. Experimental measurements by many investigators, beginning with the pioneering work of Schardin [5.38], have found that in practice cracks never reach that speed, branching into two or more cracks at speeds of only about one half of the Rayleigh wave speed. For use in applications, the deviations from the energy criterion discussed above has been circumvented by the use of distinct criteria for dynamic crack initiation, dynamic crack growth, and dynamic crack arrest as described in the following sections.

5.4.1 Dynamic Crack Initiation Toughness Since the state of stress near the crack tip is described in terms of the dynamic stress intensity factor K I (t), crack initiation can be identified with the stress intensity factor reaching a critical value; the dynamic crack initiation criterion is postulated as   (5.46) K I (tf ) = K Id T, K˙ I . The right-hand side represents the dynamic initiation toughness, with the subscript ‘d’ replacing the subscript ‘C’ used for the critical stress intensity factor for the quasistatic fracture toughness. The dependence

Fracture Mechanics

of the dynamic crack initiation toughness on the temperature and rate of loading (represented by the rate of increase of the dynamic stress intensity factor, K˙ I ) is indicated through the arguments; this dependence must be determined through experiments covering the range of temperatures and rates of loading of interest. The temperature dependence of the dynamic crack initiation toughness arises from the increase in ductility with increasing temperature or from heating associated with inelastic deformation in the near-tip zone or a combination of both. The left-hand side of (5.46) represents the applied stress intensity factor at time tf when crack propagation commences. Standard procedures do not exist for characterization of the dynamic crack initiation toughness at rates on the order of K˙ I ≈ 1 × 104 MPa m3/2 s−1 or larger, but there exist a large database on specific materials; a survey is presented by Ravi-Chandar [5.37].

5.4.2 Dynamic Crack Growth Toughness

5.4 Dynamic Fracture Mechanics

139

An interesting consequence of this difference between crack initiation and growth toughness is that the crack will jump to a large finite speed immediately upon initiation. Second, experimental efforts to   determine K ID v, K˙ I , T have met with mixed success, particularly for nominally brittle materials; variations in the measurements with different specimen geometries [5.39, 40] and loading rates [5.41] have not been resolved completely. Also, indications that the measurements are hysteretic – meaning that accelerating and decelerating cracks exhibit different behavior – have been reported [5.42]. For ductile materials, (5.47) appears to be a more reasonable characterization of dynamic fracture; see Rosakis et al. [5.43]. As in the case of dynamic crack initiation toughness, standard procedures have not been developed for determination of the dynamic crack growth toughness, but there is a large literature on methods of characterization and data on specific materials; Ravi-Chandar [5.37] presents a survey of available results on this topic.

5.4.3 Dynamic Crack Arrest Toughness

The upper-case subscript ‘D’ is used to indicate the dynamic crack growth toughness instead of the lowercase ‘d’ that was used to indicate the dynamic initiation toughness. Once again, the right-hand side represents the material property to be characterized through experiments and the left-hand side represents the dynamic stress intensity factor calculated from the solution of the boundary initial value problem in elastodynamics. The dynamic crack growth toughness is commonly referred to as the K I –v relation. Some important limitations must be recognized in using (5.47) for dynamic crack growth problems. First, the crack initiation point is not on the curve characterizing the dynamic crack growth criterion. Thus,     K ID v → 0, K˙ I , T = K Id T, K˙ I . This could be possibly due to bluntness of the initial crack, the intrinsic rate dependence of the material, or inertial effects.

In applications, the most conservative design approach would utilize (5.48), thus assuring that the dynamic stress intensity factor for all possible loading conditions never exceeds the crack arrest toughness. Thus a dynamically growing crack is never encountered in the lifetime of the structure and one avoids the complications in computing the dynamic stress intensity factors for growing cracks and in determining the dynamic crack growth criterion. Based on round-robin tests and an accumulation of data a standard test procedure, the ASTM E-1221 standard has been established that describes the determination of crack arrest toughness in ferritic steels. A rapid crack growth-arrest sequence is generated in a compact crack arrest specimen by using a displacement-controlled wedge loading; as the crack grows away from the loading wedge, the stress intensity factor drops quickly and hence results in arrest of the

Crack arrest is not the reversal of initiation and hence the initiation toughness discussed above is not relevant for crack arrest; also, the characterization of the crack growth criterion at very slow speeds is quite difficult. This has led to the postulation of a separate criterion for crack arrest: the dynamic crack arrest toughness is defined as the smallest value of the dynamic stress intensity factor for which a growing crack cannot be maintained; thus the crack arrests when K I (t) < K Ia (T ) .

(5.48)

Part A 5.4

Once dynamic crack growth has been initiated as per the conditions of the dynamic initiation criterion, subsequent growth must be determined though a separate criterion that characterizes the energy rate balance during growth. The dynamic stress field near a growing crack is still characterized by the dynamic stress intensity factor, but now this is a function of loading, time, crack position, and speed and is represented as K I (t, v). The energy rate balance in (5.44) can be expressed as a relation between the instantaneous dynamic stress intensity factor and the toughness:   (5.47) K I (t, v) = K ID v, K˙ I , T .

140

Part A

Solid Mechanics Topics

crack. The dynamic run–arrest sequence that this specimen experiences under the wedge load clearly indicates the need for a dynamic analysis of the problem. However, according to the ASTM standard, a static analysis is considered to be appropriate assuming that the values

measured at 2 ms after crack arrest do not differ significantly from the values measured at 100 ms after crack arrest. The stress intensity factor at arrest is taken to be the crack arrest toughness, K Ia . Further details can be found in the ASTM E-1221 standard.

5.5 Subcritical Crack Growth

Part A 5.6

The discussion above has been focused on fracture under monotonically increasing loading that approaches the critical conditions resulting in quasistatic or dynamic crack growth. The primary design philosophy of flaw-tolerant design is to ensure that the structure should never reach the critical condition during its useful life. However, this does not imply that crack growth does not occur during the lifetime of the structure; crack extension occurs under subcritical conditions; if such subcritical extension is due to repeated loading– unloading cycles, the resulting crack extension is called fatigue crack growth. Such extension could also be due to operational conditions under steady loading; temperature-dependent accumulation of plastic deformation, resulting in creep crack growth in metals above 0.5 TM , and stress-assisted corrosion in glass and some metallic materials are some other examples of subcritical crack growth. In all these cases, crack extension occurs very slowly over a number of cycles or over a long time; eventually such cracks reach a critical length, resulting in catastrophic failure at some point. Subcritical crack growth is typically handled in a phenomenological sense; regardless of the underlying cause, material characterization of crack extension can be performed under specified condition and represented as da (5.49) = f (K I , . . .) , dξ where ξ is the number of cycles for fatigue, and time for time-dependent crack growth; thus the crack growth

rate per unit cycle or unit time is some function of the stress intensity factor (or an equivalent parameter), the environment, and perhaps other variables. The functional form of the dependence on the stress intensity factor (or its range in the case of cyclic loading) is obtained from experimental characterization. For fatigue, the dependence is typically written as ΔK I da = C(ΔK I )b , dN

(5.50)

where C and b are empirical constants, determined through calibration experiments for each material; it is well known that these are not true material constants, but influenced significantly by material microstructure, crack length, loading range, etc. If the critical crack length, ac , at final fracture is known, then the number of cycles to failure can be obtained by integrating (5.50)

ac Nc = a0

da . C(ΔK I )b

(5.51)

Details of fatigue crack growth analysis may be found in specialty books devoted to fatigue of materials [5.44, 45]. Standard methods for fatigue crack growth characterization, such as E 647, have also been established by the ASTM and other organizations. A good summary of progress in creep fracture is provided by Riedel [5.46]. Stress-corrosion and chemical effects are discussed in detail by Lawn [5.47].

5.6 Experimental Methods Many different experimental methods have been developed and used in investigations of fracture. They may be broadly classified into two categories: those based on optical methods that provide full-field information over a region near the crack tip and those based on pointwise measurement of some component of stress or

deformation. While full-field techniques are preferred in research investigations where the sophistication of apparatus and analysis are required in order to elucidate fundamental phenomena, in applications to industrial practice, simpler methods such as those based on strain gages or compliance methods are to be preferred. The

Fracture Mechanics

methods of photoelasticity, moiré and classical interferometry, and shearing interferometry, fall into the first category and are discussed first. The digital image correlation (DIC) method is rapidly gaining popularity in light of the availability of commercial hardware and software. This method is described in detail in Chap. 25 and hence is not discussed in the present chapter. The optical method of caustics and strain-gage-based methods provide much more limited data, but are quite useful in the determination of stress intensity factors; these are discussed next. Finally, methods based on measurement and analysis of the compliance of the entire specimen and loading system are also of significant use in fracture mechanics; this is a straightforward application of (5.7) and is not discussed further. Three different volumes devoted to experimental methods in fracture characterization have been published in the last 25 years [5.48–50]; while the present chapter provides a survey of the methods, these original references remain invaluable resources for details on the different methods discussed in this chapter.

5.6 Experimental Methods

141

5.6.1 Photoelasticity The method of photoelasticity has been applied to fracture problems under both static and dynamic loading conditions. In a circular polariscope, illustrated in Fig. 5.12, the intensity of the light beam is:   Δs(x1 , x2 ) (5.52) , I (x1 , x2 ) ∝ k2 sin2 2 where k is the amplitude of the electric vector and Δs(x1 , x2 ) is the absolute phase retardation (see Chap. 24 for details of the analysis). Thus, in this optical arrangement, the spatial variation of the light intensity is governed only by the phase retardation introduced by the stressed specimen. Introducing the phase difference from (25.2) of Chap. 25, bright fringes corresponding to maximum light intensity are lines in the x1 –x2 plane along which (σ1 − σ2 ) =

N fσ , with N = 0, ±1, ±2, . . . , (5.53) h Axis of polarization Light source

Fast axis Polarizer π/4 σ1 λ/4

σ2 α

Fast axis

Specimen π/4

λ/4

Polarizer

Axis of polarization

Fig. 5.12 Optical arrangement of a circular polariscope in a dark-field configuration

Part A 5.6

*

142

Part A

Solid Mechanics Topics

Table 5.2 Material stress fringe value f σ for selected polymers (λ = 514 nm) Material

fσ (kN/m)

Source

Homalite 100

22.2

Polycarbonate

6.6

Polymethyl methacrylate

129

Dally and Riley [5.51] Dally and Riley [5.51] Kalthoff [5.52]

Part A 5.6

where N is called the fringe order, f σ is the stress-fringe value, which depends on the material, h is the specimen thickness, and σ1 and σ2 are the principal stress components. Typical values of f σ for materials commonly used in fracture studies are given in Table 5.2. Therefore, placing the specimen between crossed circular polarizers reveals lines of constant intensity that are contours of constant in-plane shear stress; these lines are called isochromatic fringes. Such isochromatic fringe patterns are captured with a camera at sufficient spatial and temporal resolution for quasistatic and dynamic problems and analyzed to determine the stress field. In applications to fracture mechanics, it is assumed that the appropriate asymptotic field is applicable in the vicinity of the crack tip and the parameters of the asymptotic field are then extracted by fitting the observed isochromatic fringes to theoretically predicted patterns in a least-squared error minimization process. Assuming that the fringe pattern formation is governed by the asymptotic stress field near the crack tip and substituting for σ1 and σ2 , the geometry of the a)

b)

fringe pattern can be expressed as N fσ (5.54) = g(r, θ, K I , K II , σox , . . .) . h For given values of the stress intensity factors, fringes can be simulated using (5.54). Examples of such simulated isochromatic fringe patterns corresponding to assumed values of K I , K II , and σox are shown in Fig. 5.13; in this representation, only the singular term and the first higher-order term are indicated, whereas in actual applications higher-order terms in the crack tip stress field are also taken into account in interpreting the experimental fringe pattern. An example of the time evolution of the fringe patterns obtained with a high-speed camera is shown in Fig. 5.14 [5.53]. The stress intensity factor K I can be obtained at each instant in time by using a leastsquares matching of the experimentally measured fringe pattern with simulations based on (5.54). We describe the general method of extracting parameters of the stress field from experimentally observed isochromatics. First, the experimental isochromatic fringe pattern is quantified by a collection of (Ni , ri , θi ), measured at M points. The distance r at which these measurements are taken should be appropriate for application of the two-dimensional asymptotic crack tip stress field; this is typically interpreted to hold for r/h > 0.5, based on the experimental results of Rosakis and RaviChandar [5.54], but Mahajan and Ravi-Chandar [5.55] showed that for fringe data based on photoelasticity, one may be as close as r/h > 0.1 and still obtain a good estimate of the stress intensity factors. The sum of the squared error in (5.54) at all measured points is then given by M   Ni f σ e(a1 , a2 , . . . ak ) = h i=1 2 − g(ri , θi , a1 , a2 , . . . ak ) . (5.55)

Fig. 5.13a,b Simulated isochromatic fringe patterns corresponding to a dark-field circular polariscope arrangement (a) K I = 1 MPa m1/2 , K II = 0, and σox = 0 (b) K I = 1 MPa m1/2 , K II = 0, and σox = 50 MPa. f σ = 7 kN/m corresponding to polycarbonate has been assumed. The field of view shown in these figures is 40 mm along one side and v/Cd = 0.1

In (5.55), the stress field parameters (K I , K II , σox , . . .) are represented by the vector a = (a1 , a2 , . . . ak ). The stress field parameters must be obtained by minimizing e with respect to the parameters. Near the minimum, the function e can be expanded as a quadratic form e(a1 , a2 , a2 , . . .) ∼ γ − 2

M  k=1

βk a k +

M M  

αkl ak al ,

k=1 l=1

(5.56)

Fracture Mechanics

5.6 Experimental Methods

143

Fig. 5.14 High-speed sequence showing isochromatic fringe patterns obtained in an experiment with a quasistatically loaded single-edge-notched specimen. Polycarbonate specimen observed in a dark-field circular polariscope arrangement. Frames are 10 μs apart. The height of the field of view presented in the image is about 75 mm (after Taudou et al. [5.53])

where 1 ∂e = 2 ∂ak

i=1



Ni f σ ∂gi − gi h ∂ak

and

M   1 ∂2e ∂gi ∂gi = 2 ∂ak ∂al ∂ak ∂al i=1   2  Ni f σ ∂ gi − − gi h ∂ak ∂al   and gi = g(ri , θi , a1 , a2 , a2 . . .). Since Nihf σ − gi is expected to be small, the second term in αkl is usually neglected and only the first derivative of g needs to be evaluated. The estimate for the increment in the parameters is obtained by setting ∇e = 0. This results in the following equation for the increments in the parameters δak :

αkl =

βl =

M 

αkl δak .

(5.57)

k=1

With the update of the parameters a = (a1 , a2 , . . . ak ) obtained from the above, the procedure is repeated until the parameters converge. For a thorough discussion of the procedures for such least-squared error fitting see Press et al. [5.56]. Curve fitting routines based on the above are also implemented in Mathematica, Matlab,

and other commercial software packages. There appear to be multiple minima for the function in (5.56) and hence it is usually good practice to simulate the isochromatic fringe patterns with the converged parameters and perform a visual comparison of the recreated fringes to the experimentally observed pattern. Numerous examples of the application of the least-squares fitting method for extraction of the stress intensity factors in quasistatic and dynamic fracture problems can be found in the literature [5.53, 57–61].

5.6.2 Interferometry Classical optical interferometry has been applied by numerous researchers to investigations of fracture problems; two implementations of classical interferometry are commonly used, one to measure the out-of-plane displacement on the free surface of a cracked planar specimen and the other to measure the crack opening displacement in the vicinity of the crack. Complementing this, moiré interferometry has also been used in investigations of the in-plane deformations near a crack tip. The basic principles and interpretation of results are discussed in this section. Measurement of Out-of-Plane Displacements The basic principle of the method is the interference of coherent light beams in a classical two-beam

Part A 5.6

βk = −

M  

144

Part A

Solid Mechanics Topics

Object plane

Specimen

Reference mirror

5 mm

Image plane Imaging optics Beam splitter

Beam-forming optics

Laser

Fig. 5.15 Twyman–Green interferometer for the measurement of

out-of-plane displacements (after Schultheisz et al. [5.62, 63])

Part A 5.6

interferometer; the arrangement of a Twyman–Green interferometer, a variant of the Michelson interferometer, shown in Fig. 5.15 was used by Schultheisz et al. [5.62] to accomplish this interference scheme. Light rays reflected from the deformed surface of the specimen and the reference mirror are brought together to interfere; as a result of path differences introduced by the deformation of the surface, fringe patterns are observed in this arrangement as contours of constant out-of-plane displacement component u 3 . This component, corresponding to the singular term of the asymptotic stress field, is given by νK I I Fαα (θ, 1) , u 3 (r, θ) = − √ E 2πr

Fig. 5.16 Interference fringes obtained in a Twyman– Green interferometer corresponding to contours of constant out-of-plane displacements (after Schultheisz et al. [5.62, 63]). 4340 steel specimen

( f -number) limits the acceptance angle of incoming rays; a high f -number is desirable for maximum acceptance angle. Furthermore, as the fringe density becomes high near the crack tip, the resolution of the imaging and reproduction systems limits the visibility of the fringes. The Twyman–Green interferometer has also (mm) 4

(5.58)

where are known functions of θ given in (5.10). Proper alignment of the optical system is essential in order to record these fringe patterns; typically the experimental arrangement is housed on a floating optical table in order to eliminate random vibrations. Also, the specimen surface must be polished to a high degree of flatness and reflectivity; while this may be easily accomplished in inorganic and organic glasses, it is much more difficult to achieve in metallic specimens. An excellent application of this method to the measurement of the out-of-plane deformation of a 4340 steel specimen is demonstrated by Schultheisz et al. [5.62]; an image of the fringe pattern obtained in a Twyman– Green interferometer is shown in Fig. 5.16. Note that, as one approaches the crack tip, the visibility of the fringes becomes poor. This is due to the fact that the slope of the deformed surface increases as the crack tip is approached and the numerical aperture

2

FαI β (θ)

0

–2

–4 –4

–2

0

2

4 (mm)

Fig. 5.17 Out-of-plane displacement field near a rapidly growing crack in a PMMA specimen. Stress waves radiating from the crack tip damage zone can be seen as perturbations in the fringe pattern, especially behind the crack tip. The crack speed was 0.52 mm/μs (about 0.522 Cs ) (after Pfaff et al. [5.64])

Fracture Mechanics

Crack Opening Interferometry The measurement of crack opening displacements with optical interferometry relies on the transparency of the specimen and generation of specularly reflecting fracture surfaces. The basic principle of the method, called crack opening interferometry (COI), is illustrated in Fig. 5.19. When the light rays reflected from the top

145

u3 (μm) 20 0 –20 –40 –60 –80 –100 Expt 35.0 kN FEM 35.0 kN Expt 52.3 kN FEM 52.3 kN Expt 73.5 kN FEM *73.5 kN

–120 –140 –160 –180

0

0.5

1

1.5

2 r/t

Fig. 5.18 Comparison of measured out-of-plane displacement vari-

ation along θ = 0 with a fine-mesh finite element simulation. Good comparison is obtained at the lower load levels, but the discrepancy increases significantly at higher loads (and hence larger plastic zone size) (after Schultheisz et al. [5.63])

and bottom surfaces of the crack are brought together (rays 1 and 2 in the figure), fringes of equal separation between the surfaces can be observed on the image plane. Along the crack surface, the separation distance is the crack opening displacement (COD) denoted as δ(r); the relationship between the fringe number and the Ray 1 Ray 2

y δ (r)

x

Fig. 5.19 Optical arrangement for crack opening interferometry. Rays 1 and 2 reflect from the top and bottom surfaces of the crack and interfere to form fringes of constant COD

Part A 5.6

been used successfully in dynamic fracture problems where the challenges of alignment are significant; fringe patterns captured in a dynamic experiment is shown in Fig. 5.17 [5.64]. Postprocessing was used to subtract out the initial fringe pattern revealing only the surface deformation due to the loading of the crack. Also, the region near the crack tip that is obscured due to aperture limitations has been removed during image processing. Stress waves radiating from the crack tip damage zone can be seen as perturbations in the fringe pattern, especially behind the crack tip. These are caused by the intermittency in the crack propagation process. The displacements calculated from such fringes are pure measurements in the sense that no assumptions about the deformation or stress state are imposed in obtaining the displacements from the measurements. Fitting the observed fringe patterns ahead of the crack to a prediction based on (5.58) using a least-square error method, the stress intensity factor can be determined, in the same way as discussed in Sect. 5.6.1, for the method of photoelasticity. A major difficulty in using this method arises from the fact that, while the measurements of the displacement components are themselves very accurate, the application of the plane asymptotic fields in the interpretation of the crack tip field parameters introduces significant errors, particularly when one approaches the crack tip. Therefore, such experiments may serve as verification on the limitations of models or the accuracy of more precise elastoplastic numerical simulations. Schultheisz et al. [5.63] compared the measured out-of-plane displacement ahead of the crack to a fine-mesh finite element simulation of the same problem (Fig. 5.18) and suggested that good agreement could be obtained with a proper model of the constitutive response of the material as long as the applied loads were low and three-dimensional deformation of the specimen was modeled appropriately. The last aspect is especially important in using classical interferometry since the technique provides the out-ofplane displacements only on the outer surface of the specimen.

5.6 Experimental Methods

146

Part A

Solid Mechanics Topics

COD can be written as nλ δ(r) = u 2 (r, π) − u 2 (r, −π) = , 2 n = 0, 1, 2, . . . ,

(5.59)

where λ is the wavelength of light used and n is the fringe order; it is assumed that the gap between the crack surfaces is filled with air and that the incidence angle of the light beam is normal to the crack surface. Thus, monitoring the fringe patterns results in a direct measurement of the COD; once again, as in the out-ofplane measurements, this is a pure measurement devoid of assumptions on the nature of the deformation and needs to be interpreted in terms of a theoretical model. The measured COD can be compared either with the elastic stress field in (5.10) or the elastic–plastic field corresponding to one of the plasticity models for extraction of the stress intensity factor or the J-integral. For example, if the crack tip deformation is governed by (5.10), then ⎧  ⎨ 8K I r plane stress E 2π  . (5.60) δ(r) = 2 ⎩ 8(1−ν )K I r plane strain E 2π

Part A 5.6 Thickness (μm) In unbroken craze stress ≈ 0 for this configuration (note different scales)

2 1 0

Crack

–1 –2

Craze Fracture surface layer thickness ≈ 0.58 μm –80 –70 –60 –50 –40 –30 –20 –10

0

10

20

30

40

Length (μm)

Fig. 5.20 COI fringes for a crack-craze system in PMMA

are shown at the top. The graph indicates the variation of crack and craze opening displacement calculated from the fringe pattern (after Kambour [5.65])

Comparing the measured COD to the calculation based on the elastic field above, the stress intensity factor can be extracted. An example of the COI fringe pattern in a Dugdale–Barenblatt-type craze zone near a crack tip in a polymethyl methacrylate (PMMA) specimen is shown in Fig. 5.20; the displacement profile extracted from this measurement is also shown in the figure. In this example, one can clearly see from the fringe spacing that the COD changes from concave to convex as the crack tip is approached. This corresponds to a physical crack with a craze that develops in the fracture process zone of the PMMA; note the similarity between the COD measured and the one shown in Fig. 5.9. The DB model is more appropriate for this situation; numerous investigators [5.26, 65, 66] have demonstrated that such a model can be applied to crazes. It is also possible to apply nonlinear cohesive zone models to interpret such measurements. One major advantage of COI is that variations in the crack front (three dimensionality) can be rendered visible directly; however, this method comes with big penalties: the optical arrangement is rather complicated since one has to approach the interior of the specimen and more importantly, since only normal opening is measured, effects of mixedmode loading cannot be assessed with this measurement alone. Numerous investigators have used COI to explore problems of brittle fracture [5.67–69], crazing in polymers [5.26, 65, 66], interfacial cracks [5.70], and elastic–plastic problems [5.71]. Moiré Interferometry The method of moiré interferometry is discussed in detail in Chap. 22; here a brief description of the method and its applications to fracture problems is presented. The basic principle relies on diffraction and interference. Consider a diffraction grating adhered to the specimen surface; if two incident beams strike the specimen symmetrically with respect to the normal to the surface such that the first-order diffraction from both beams is normal to the surface, then the two diffracted beams will interfere and provide uniform illumination. If the specimen deforms nonuniformly, then the two diffracted beam will, in general, contain phase differences and hence provide fringe patterns that reflect the deformation of the surface. The fringes are to be interpreted as contours of equal in-plane displacement components with the direction given by the grating used. Details of different systems for implementation of moiré interferometry are discussed by Post et al. [5.72] and are not repeated here. This method is insensitive to out-of-plane displacements near the crack tip

Fracture Mechanics

18 500 N 10 mm

5.6 Experimental Methods

147

in Sect. 5.3.2. It is striking that, while the component of strain perpendicular to the crack line matches the HRR singularity quite well, the strain parallel to the crack exhibits significant deviation. Such detailed strain measurements, obtained without a priori assumptions regarding the material or the deformation, are quite useful in outlining the limits of validity of analytical estimates. Schultheisz et al. [5.62, 63] also performed moiré interferometric experiments on a 4340 steel specimen; however, they compared the measured displacements to a fully three-dimensional finite element simulation (Fig. 5.23) and demonstrated excellent agreement with the experiments, but only for low loading levels. At higher load levels, they found that there was a significant discrepancy; this was attributed to tunneling of the crack front in the interior of the specimen and the resulting three dimensionality of the displacement field not captured in the numerical simulations.

5.6.3 Lateral Shearing Interferometry

Fig. 5.21 Moiré interferometric fringes corresponding to

displacements u 1 (upper image) and u 2 (lower image) (after Schultheisz et al. [5.62])

and hence provides a nice complement to the out-ofplane measurements through classical interferometry. Applications to quasistatic as well as dynamic loading problems in elastic, elastic–plastic, and damaging materials have been considered [5.62, 63, 72–74]. In the case of elastic problems, the interest in application of moiré interferometry is in identifying the crack tip displacement field, outlining the regions of dominance of the elastic singularity, and eventually in the extraction of the stress intensity factor. The moiré interference fringes from a fracture experiment indicating the u 1 and u 2 components of displacement are shown in Fig. 5.21. Dadkhah and Kobayashi [5.73] examined the displacement fields near a crack in a biaxially loaded Al 2024-T3 alloy specimen. Their result for a biaxiality of two is reproduced in Fig. 5.22; here the strain components evaluated from the moiré interferometric fringes are compared with an estimate based on the HRR singularity discussed

where Δs is the absolute phase angle of the components, which depends on the state of stress of the specimen at the point (x1 , x2 ); we assume that the specimen is optically isotropic. Shearing of images is

Part A 5.6

18 500 N 10 mm

The principle of shearing interferometry is very similar to other two-beam interferometric techniques; the main difference is that, rather than using a reference wavefront to interfere with the wavefront of interest, the wavefront is made to interfere with a copy of itself after introducing a predetermined, selectable, (lateral) shear displacement between the two components. Tippur et al. [5.75] developed a variant based on a diffraction grating that enabled them to examine the surface deformations near stationary and dynamically propagating cracks; they termed their implementation the coherent gradient sensor (CGS). Lee and Krishnaswamy [5.76] later introduced a simpler implementation of the experiment based on a calcite crystal; a variant of this scheme is described here. The analysis of the light propagation through Jones calculus makes the manipulations simple [5.77, 78]; see the chapter on photoelasticity for details of Jones’s calculus. The optical arrangement is shown in Fig. 5.24; a light beam polarized at an angle of π/4 with respect to the global x1 -axis passes through the stressed specimen that acts as a phase retarder. The electric vector of the light emerging from the specimen is given by   −iΔs(x 1 ,x 2 ) iωt e , (5.61) E = ke e−iΔs(x1 ,x2 )

148

Part A

Solid Mechanics Topics

Strain εx 0.01

Strain εy 0.02 θ = 0° B=2

0.008 Moire

0.006

Moire HRR – 0.93

0.01

θ = 0° B=2

– 0.5

0.004 HRR 0.002

0

0

10 20 Distance from crack tip r (mm)

0

– 0.923 0

10 20 Distance from crack tip r (mm)

Fig. 5.22 Comparison of the strain determined from moiré interferometry with calculations based on the HRR field; note that, while the normal strain ε22 compares favorably, the crack parallel strain ε11 shows significant deviation (after Dadkhah and Kobayashi [5.73])

Part A 5.6

accomplished by introducing a uniaxial optical crystal (for example, calcite) into the path of the light beam. Uniaxial crystals produce two refracted beams – the ordinary ray with a refractive index n  and the extraordinary ray with a refractive index n  – for each incident beam, each polarized in mutually orthogonal planes (see Born and Wolf [5.79] for a discussion of uniaxial crystals). Let the uniaxial crystal be aligned such that the ordinary ray is polarized along x1 , the extraordinary ray is polarized along the x2 -axis, and that these rays are sheared by an amount Δx1 along the x1 -axis. At every point in the field, there are now two beams: the ordinary ray that entered the specimen at the point (x1 , x2 ) and the extraordinary ray that entered the specimen at the point (x1 + Δx1 , x2 ). Thus the light beam leaving the crystal may be written as   e−iΔs(x1 ,x2 ) iωt . (5.62) E = ke e−iΔs(x1 +Δx1 ,x2 ) These two components are brought together by a polarizer oriented at an angle of π/4 with respect to the global x1 -axis. The output electric vector is given by    1 −iΔs(x 1 ,x 2 ) + e−iΔs(x1 +Δx1 ,x2 ) iωt 2 e E = ke   . 1 −iΔs(x 1 ,x 2 ) + e−iΔs(x1 +Δx1 ,x2 ) 2 e (5.63)

The intensity of this light beam is the time average of the electric vector E2 , averaged over a time significantly longer than the period; if the beam displacement Δx1 ,

is small, the intensity may be written as   k2 Δx1 ∂Δs . cos2 I (x1 , x2 ) = 2 2 ∂x1

(5.64)

Thus, the light intensity variation observed in this optical arrangement depends on the gradients of the phase difference in the x1 -direction. In applying this method to problems in mechanics, it remains to evaluate the angular phase difference Δs; for an optically opaque specimen, if the surfaces are finished to be specularly reflecting, the phase difference is given by hν Δs = − (σ11 + σ22 ) ≡ hcr (σ11 + σ22 ) , (5.65) 2π/λ E where cr = −hν/E represents the sensitivity of the method. Therefore, bright fringes corresponding to maximum light intensity are lines in the x1 –x2 plane along which mλ ∂(σ11 + σ22 ) = , ∂x1 Δx1 with m = 0, ±1, ±2, . . . ,

hcr

(5.66)

where m is the fringe order. Therefore, fringes observed in the shearing interferometer or the coherent gradient sensor are lines of constant gradient ∂(σ11 + σ22 )/∂x1 . Obviously, by reorienting the ordinary axis of the calcite fringes representing lines crystal with the   x2 -direction, of constant ∂ σ11 + σ22 /∂x2 can be obtained. A similar analysis can be performed for transparent materials, with only a slight modification in the constant cr .

Fracture Mechanics

Expt 35.0 kN FEM 35.0 kN Expt 52.3 kN FEM 52.3 kN Expt 73.5 kN FEM *73.5 kN

0

0.5

1

1.5

2

2.5

3

r/t

u2 (μm) 400 Expt 35.0 kN FEM 35.0 kN Expt 52.3 kN FEM 52.3 kN Expt 73.5 kN FEM *73.5 kN

350 300 250 200 150

50 0

0.25

0.5

0.75

1

1.25

1.5

r/t

Fig. 5.23 Comparison of the u 1 and u 2 displacement components determined from moiré interferometry with a fine-mesh finite element analysis. While good agreement is observed at lower load levels, a large departure is observed at the highest load (after Schultheisz et al. [5.62])

The above description of the shearing interferometer is quite general and can be applied to many problems in mechanics. Also,from (5.64), it is clear that the gradient  of u 3 x1 , x2 , − h2 may be determined directly, without relating it to the plane-stress calculation of the stress component; therefore, the method can also be used to determine surface profiles of objects. Explicit equations for the light intensity can be obtained through the introduction of the asymptotic crack tip stress field in (5.9). The singular term, the higher-order terms in the steadystate asymptotic expansion, and the dynamic asymptotic field have all been used in interpreting the fringe pattern observed near stationary and dynamically growing cracks. We will describe only the quasistatic applica-

The entire analysis presented above carries over if the image shearing is introduced in the x2 -direction; in this case, the x2 gradient of the field is obtained. Simulated interference fringe patterns corresponding to the x1 gradient for assumed values of the stress intensity factors are shown in Fig. 5.25 for pure mode I and mixed-mode loading. An example of the shearing interference fringe patterns obtained in a shearing interferometer is shown in Fig. 5.26. In comparing the images in these figures, it should be noted that only the singular term in the crack tip asymptotic field has been used in the simulations, whereas the complete field will influence the patterns observed in the experiments. In the discussion above, only the singular term was introduced in the evaluation of the fringe patterns. However, since the asymptotic field is not expected to establish dominance at large distances from the crack tip where the fringe patterns are typically analyzed, higher-order nonsingular terms in the expansion in (5.9) must be introduced as we discussed in the case of photoelasticity. The stress intensity factor, K I , can be obtained at each instant in time by using a leastsquares matching of the experimentally measured fringe pattern with simulations based on (5.67). First, the experimental fringe pattern is quantified by a collection of (m i , ri , θi ), measured at M points. The distance r at which these measurements are taken should be appropriate for the application of the two-dimensional asymptotic crack tip stress field; based on the experiments of Rosakis and Ravi-Chandar [5.54], it should be recognized that the distance r must be larger than 0.5h to be away from the zone of three-dimensional deformations. Then, the sum of the squared error in (5.67) at all measured points is then given by e(A0 , A1 , . . . Ak−1 ) 2 M   m i πλ − hcr g(ri , θi , A0 , A1 , . . . Ak−1 ) . = Δx2 i=1

(5.68)

In (5.68), the stress field parameters (K I , K II , . . .) are represented by the vector A = (A0 , A1 , . . . Ak−1 ) and g(r, θ, A0 , A1 , . . . Ak−1 ) is used to represent ∂(σ11 + σ22 )/∂x1 determined from the k-term description of the asymptotic crack tip stress field. Also, since

Part A 5.6

100

0

149

tion. Introducing the singular term from (5.9) into (5.66) results in the equation for bright fringes   3θ 3θ mλ hcr . (5.67) K I cos + K II sin = √ 3/2 2 2 Δx 2πr 1

u1 (μm) 65 60 55 50 45 40 35 30 25 20 15 10 5 0 –5

5.6 Experimental Methods

150

Part A

Solid Mechanics Topics

Axis of polarization

*Light source

π/4 σ1 σ2 α

Polarizer

Specimen

Fast axis Image shearing crystal

π/4

x2 Polarizer

Part A 5.6

x3

x1

Fig. 5.24 Optical arrangement for lateral shearing interferometry

shearing interferometry or the coherent gradient sensing method depends on the gradient of the stress field, the constant nonsingular term σox in the asymptotic expansion does not contribute to fringe formation and cannot be determined in the analysis. The remaining a)

b)

stress field parameters must be obtained by minimizing e with respect to the parameters. The least-squared error method described in connection with photoelastic data analysis in Sect. 5.6.1 can be used here as well to a)

b)

10 mm

Fig. 5.25a,b Simulated shearing interferometric fringe patterns corresponding to the optical arrangement in Fig. 5.24 (a) K I =1 MPa m1/2 , K II =0 (b) K I =1 MPa m1/2 , K II =0.5 The field of view shown in these figures is 40 mm along one side and v/Cd = 0.1. The lack of clarity in the region near the crack tip is a numerical artifact

Fig. 5.26a,b Fringe patterns from the coherent gradient sensing (CGS) arrangement of the lateral shearing interferometer: (a) image shearing parallel to the crack line, (b) image shearing perpendicular to the crack line; the dashed lines are reconstructions based on a threeparameter crack tip field

Fracture Mechanics

evaluate the best fit coefficients A = (A0 , A1 , . . . Ak−1 ). The entire procedure carries over to the dynamic problem if the appropriate asymptotic field in (5.37) is used in (5.66). Tippur et al. [5.80] examined the applicability of the asymptotic stress field by comparing the experimentally observed fringe patterns with the analytically estimated patterns; dashed lines in Fig. 5.26 correspond to theoretical estimates based on a threeparameter fit. They concluded that measurements need to be made at a distance of about one-half of the plate thickness, reinforcing earlier results of Rosakis and Ravi-Chandar [5.54]. A complete discussion of the application of this method to quasistatic and dynamic problems is provided by Rosakis [5.81].

5.6.4 Strain Gages

x2

x'2

x'1 α

θ x1

Fig. 5.27 Location and orientation of strain gage relative to the crack tip

151

type, dimensions, and sensitivity of the strain gage are extremely important and are addressed in the chapters that deal with strain gages and are not addressed here. Dally and Berger [5.84] introduced a very simple idea for the use of strain gages in the evaluation of stress intensity factors. Consider a strain gage mounted at a point (r, θ), with the strain gage itself oriented at an angle α with respect to the global x1 -axis (Fig. 5.27); the strain gage will measure the extensional strain ε11 in the direction x1 . For the case of a stationary crack√under mode I loading, retaining up to terms of order r the strain can be evaluated to be E ε (r, θ) (1 + ν) 11  θ 1 K I (t) 3θ k cos − sin θ sin =√ cos 2α 2 2 2 2πr  3θ 1 sin 2α + A1 (k + cos 2α) + sin θ cos 2 2  √ θ + A2 r cos 12 θ k + sin2 cos 2α 2  1 (5.69) − sin θ sin 2α + . . . , 2 where k = (1 − ν)/(1 + ν). Through a proper choice of θ and α the second and third terms in (5.69) can be made to vanish; this is assured by the conditions cos 2α = −k

and

tan 12 θ = − cot 2α .

(5.70)

For a material with ν = 1/3, we get k = 1/3 and α = θ = π/3. Introducing these values in (5.69), we obtain   π 3 Eε11 r, θ = (5.71) = KI . 3 8πr Thus, by placing a strain gage aligned along a line oriented at an angle θ = π/3, the measured strain can be directly related to the stress intensity factor. It should be noted that the distance r at which the strain gage is placed is still open, but this can be farther from the crack tip than the K -dominant region, since (5.71) is based on a three-term representation of the strain field. This arrangement can be used in the evaluation of dynamic initiation toughness as well. Examples of application of this method are demonstrated by Dally and Barker [5.85], who evaluated the dynamic initiation toughness of a Homalite 100 specimen at high loading rates by imposing an explosively driven stress wave on a crack. Owen et al. [5.86] determined both the dynamic initiation toughness and dynamic propa-

Part A 5.6

While optical methods yield measurements of the stress field components over the complete field of observation, they also require elaborate instrumentation. Full-field optical methods dominated fracture research in the early years of the discipline, primarily because issues related to the dominance of the various asymptotic fields could be evaluated with these schemes. On the other hand, multiple strain gages can be used more readily with simpler instrumentation requirements; this method was used very successfully by Kinra and Bowers [5.82], Shukla et al. [5.83], and many other investigators for dynamic problems. Recent progress in this area was summarized by Dally and Berger [5.84], who describe the application of strain gages to quasistatic as well as dynamic fracture problems. Here we provide a brief description as applied to dynamic problems. In general, strain gages may be placed at different positions (r, θ) to measure one or more components of the strain tensor. These measurements can be used to determine the stress intensity factors K I and K II as well as the higher-order terms by fitting the experiment in a leastsquare sense to the asymptotic fields of quasistatic or dynamic problems. Practical considerations on the

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Part A 5.6

gation toughness for an aluminum 2024-T3 alloy; they found the initiation toughness to be independent of the dyn rate of loading up to K˙ I = 105 MPa m1/2 s−1 , but then to increase threefold as the rate of loading indyn creased to K˙ I = 106 Pa m1/2 s−1 . Owen et al. [5.86] also determined the dynamic propagation toughness, but using (5.71), justifying its use by the fact that the crack speeds were quite low, about 4% of the shear wave speed. For dynamically running cracks, two major errors are encountered in using (5.71); the first is due to the inertial distortion of the crack tip strain field, which is ignored in developing (5.71). While this error is likely to be small at low crack speeds, it cannot be ignored when the crack speed is high; this error can be removed completely by using the appropriate asymptotic strain field. The second, and perhaps more important, contribution to the error arises from the fact that the distance r and the orientation θ of the position of the strain gage relative to the moving crack tip are continuously changing as a result of crack growth. This must be taken into account in the analysis. Therefore, the simplification introduced in (5.71) is not appropriate and one must incorporate the additional feature that both r and θ are now functions of time, given by  (5.72) r(t) = r02 + v2 t 2 − 2r0 vt cos θ0 ,   r0 sin θ0 θ(t) = arcsin (5.73) , r(t) a)

Specimen

Real screen

where r0 and θ0 are the distance and orientation at time t = 0. Of course, it has been assumed that the crack extension is straight along the x1 -direction. RaviChandar [5.87] used the above analysis to evaluate the measurements of Kinra and Bowers [5.82]. More recently, Berger et al. [5.88] have used measurements from multiple strain gages to set up an overdetermined system of equations and obtain improved estimates of the dynamic propagating toughness. In summary, strain-gage-based methods are just as powerful as full-field optical techniques for crack tip field characterization while at the same time requiring only minimal investment in measuring equipment. This is particularly useful in the development of standard methods of measuring the initiation, propagation, and arrest toughness.

5.6.5 Method of Caustics The method of caustics discovered by Schardin [5.38] was popular in early dynamic fracture investigations due to its simplicity, although significant inherent limitations in the method have made it a little used technique. It provides a direct measurement of the stress intensity factor and is applicable to static and dynamic problems. Here we describe the essential ingredients of the method for completeness. The principle of formation is illustrated in Fig. 5.28 for transparent and opaque specimens. Consider a parallel beam of light incident along the x3 -direction, normal b)

x2

x2 r0 (z0)

r0 (z0) x3

D

Crack front

S1

Virtual screen

Specimen

x3

D

Crack front

S2 z0

z0

Fig. 5.28a,b Schematic illustration of the principle of formation of the caustic curve: (a) transmission arrangement for transparent specimens, (b) reflection arrangement for opaque specimens

Fracture Mechanics

a)

153

b)

D

Dmin Dmax

Fig. 5.29a,b Simulated bitmap image of caustics. The field of view represents a square, 20 mm along one side. (a) K I = 1 MPa m1/2 , K II = 0; (b) K I = 1 MPa m1/2 , K II = 1 MPa m1/2

determined. For mixed-mode loading conditions, the variation of the stress intensity factor with time may be obtained by measuring two length dimensions: Dmax and Dmin . The ratio (Dmax − Dmin )/Dmax depends only on μ; this dependence is shown in Fig. 5.30a. Therefore, μ can be determined first from the measurements of Dmax and Dmin and then K I and K II are determined from the following: √  3 2 2π Dmax 2 , (5.75) KI = 2ct hz 0 F(v) g(μ) K II = μK I .

(5.76)

The function g(μ) is shown in Fig. 5.30b. While the method has been used by a number of investigators in evaluating mixed-mode stress intensity factors in quasistatic problems, dynamically growing cracks typically follow a locally mode I path and hence there are very few examples of the evaluation of the mixed-mode stress intensity factors. Examples of the application of the method of caustics to dynamic problems may be found in the papers by Kalthoff [5.89], Ravi-Chandar and Knauss [5.41, 90, 91], and Rosakis et al. [5.43].

5.6.6 Measurement of Crack Opening Displacement While the COI discussed before provides an accurate measurement of the crack opening displacement, it has two major limitations: first, it requires a transparent specimen with a complicated optical arrangement, and second, significant postprocessing of fringe data is needed in order to extract the crack opening. In many practical applications, specimens are opaque and hence one needs other means of measuring the COD. In industrial practice, methods based on replica techniques

Part A 5.6

to the specimen free surface. In a transparent material a light ray passing through a stressed plate is deviated from its path partly due to thickness variation generated by the deformation and partly due to the change in refractive index caused by stress-induced birefringence. In an opaque material, the light ray reflects with a deviation from parallelism dictated by the local slope or equivalently the thickness change. If the plate contains a crack, the rays are deviated from the region around the crack tip and these form a singular curve called a caustic on a reference plane at some distance away from the specimen. The size of the caustic curve can be related to the stress intensity factor by introducing an analysis based on geometrical optics and fracture mechanics. This analysis is described in detail in many references [5.81]; we provide just the final results. As can be seen from the illustration in Fig. 5.28, far away from the crack tip, the light rays pass through the transparent specimen and maintain their parallel propagation; the influence of the stress field on the wavefront is small enough to be neglected. On the other hand, in the region near the crack tip, where the specimen exhibits a concave surface due to the Poisson contraction, the light rays deviate significantly from parallelism. As a result, a dark region called the shadowspot forms on the screen at z 0 , where there are no light rays at all. This shadow region is surrounded by a bright curve, called the caustic curve. The line on the specimen plane whose image is the caustic curve on the specimen is called the initial curve. Light rays from outside the initial curve fall outside the caustic; rays from inside the initial curve fall on or outside the caustic curve and rays from the initial curve fall on the caustic curve. Hence the caustic curve is a bright curve that surrounds the dark region. Figure 5.29 shows simulated caustics corresponding to mode I and mixed-mode loading. For mode I loading, the transverse diameter is related to the stress intensity factor through the following relation: √  3 2 2π D 2 , (5.74) KI = 2cp hz 0 F(v) 3.17 where h is the plate thickness, z 0 is the distance between the midplane and the screen, and    specimen F(v) = 2 1 + αs2 αd2 − αs2 /R(v) is a correction factor for a crack propagating dynamically at a speed v, cp = −ν/E for opaque specimens and cp = c − (n − 1)ν/E for transparent specimens, where c is the direct stress-optic coefficient of the material. Thus, from measurements of the transverse diameter of the caustic curve, the dynamic stress intensity factor can be

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a) (Dmax -Dmin )/Dmax

b) Numerical factor g

0.5

4

0.4

3

0.3 2 0.2 1

0.1 0

0

1

2

3

4 5 6 7 8 ∞ Stress intensity factor ratio µ = KII /KI

0

0

1

2 3 4 5 Stress intensity factor ratio µ = KII /KI

Fig. 5.30 (a) Relationship between the caustic dimensions and the mixed-mode μ (b) g(u) versus μ (after Kalthoff [5.52])

Part A 5.6

or clip gages are commonly used. Replica techniques are quite clumsy; they require that a soft polymer be inserted into the crack under load, cured in-place, and then extracted by breaking the specimen. This replica is then characterized to determine the crack opening displacement. A crucial factor in determining the reliability of this method is the uncertainty associated with the flow of the viscous polymer into the crack opening before curing begins; while the replica technique remains a powerful tool in determining metallographic aspects of fracture, is not a very good quantitative tool for precise COD measurements. Clip gages are more reliably used in such applications; these rely on two flexible beam elements, inserted into the crack along the load line, with strain gages attached on them. Opening of the crack results in variations in the bending strains that are then calibrated in terms of the opening of the crack. Many commercial devices suitable a)

b)

A

c)

α d

Fig. 5.31 (a) Vickers indents on the specimen surface 100 μm apart. (b) Optical arrangement for the interferometric strain/displacement gage. (c) Pattern of light observed, indicating the triangular outline

and the interference fringes

for different environmental conditions (high temperature, etc.) are available and therefore clip gage methods are not reviewed here. An optical technique developed by Sharpe [5.92, 93] called the interferometric strain/displacement gage (ISDG) is quite versatile and can be applied in static and dynamic applications and in different environments. We describe the basic principles and capabilities of this technique. The ISDG takes advantage of two-beam interference just as other classical interferometric techniques discussed in Sect. 5.6.2, with the exception that, instead of attempting full-field measurement, reflections from two points on the specimen are considered. In the simplest implementation of the method, two small microindentations are made on the specimen with a pyramidal Vickers indentor; typically the indents are about 10–20 μm long and are spaced about 100 μm apart (Fig. 5.31a), and the reflections from these two points are made to interfere. The two indents could be placed across the crack line if COD is to be measured at that location; they can also be placed anywhere where the local strain is to be measured. If a light beam impinges on these two indents at normal incidence to the nominal surface, the reflected beams from each face of the pyramidal indent will overlap and generate interference fringes in space; the beam diffraction from the indents ensures that overlap of the two beams occurs. This is a variant of the Fresnel mirror arrangement and can be viewed as equivalent to Young’s two-slit interference scheme [5.79]. The scheme is illustrated in Fig. 5.31b; to an observer at point A, the light beams reflected from the two pyramidal faces of the indent ap-

Fracture Mechanics

pear as two slits separated by a distance d sin α. The fringe formation condition for the Young’s two-slit interference scheme is sd sin α = m Lλ ,

m = 0, ±1, ±2, . . . ,

(5.77)

5.6.7 Measurement of Crack Position and Speed Measurements of crack tip position and speed are often required in order to analyze the fracture problem. Different electrical resistance methods have been used to determine the crack position and speed. The two most popular techniques are the resistive grid and potential drop methods. In the grid method a number of electrical wires are laid across the path of the crack. As the crack propagates, it severs the wires sequentially and provides an electrical signal which can then be used to determine the crack position and speed with time (see, for example, Dulaney and Brace [5.94] and Paxson and Lucas [5.95]). Commercial suppliers of strain gages now provide such grids for crack speed measurements; these grids can be incorporated into standard strain gage bridge circuits to provide the history of wire breakage and hence the crack position as a function of time. While very good estimates of the crack speed can be obtained from such grid techniques, the discrete nature

of the grids dictate that the sampling rate of the crack speed will typically be much lower than that obtained using other methods. In contrast to this discrete measurement, if the change in resistance of a conducting specimen is measured as a function of crack length, then the crack length can be inferred at very high spatial and temporal resolution; this is the basis of the potential drop technique. The method can be used even in nonconducting materials, provided that a thin conducting film is deposited on the surface of the specimen. Carlsson et al. [5.96] demonstrated the application of this method to the measurement of crack speed in PMMA; they used a voltage divider circuit to measure the resistance change. Many other investigators have used this method to determine the speed of running cracks [5.97, 98]. Commercial versions of this technique – such as the KrakGageTM – are now available; it is also quite easily accomplished in the laboratory with thin-film coating methods. ASTM guidelines have also been established for measurement of crack position during fatigue crack growth characterization (ASTM E 647). A review of the method can be found in Wilkowski and Maxey [5.99]. A simple arrangement of the method is shown in Fig. 5.32. The specimen or the conducting coating on the specimen is connected to one arm of a Wheatstone bridge as indicated and balanced initially to give a null output; as the crack length increases, the change in resistance of this segment of the bridge circuit results in an output voltage from the bridge. The relationship between this voltage output and the crack length can be obtained through direct calibration or by calculating the electric field numerically for the particular specimen geometry (see Bonamy and Ravi-Chandar [5.100] for a recent application). Analytical and numerical solutions are available for a number of specimen geometries (see the ASTM E 647 standard for references). Both direct- (DC) and alternating-current (AC) excitation and V0 R1

Ra

Vbat Rb

Fig. 5.32 Potential drop technique for measurement of crack tip position

155

Part A 5.6

where s is the distance between fringes, m is the fringe order, λ is the wavelength of light, and L is the distance between the plane of the indents and the plane of observation. The pyramidal shape of the surface of the indent dictates that the light beam emerging from the surface is triangular as well (Fig. 5.31c). Diffraction effects dictate that the reflected beams will acquire a small divergence angle. It is clear from (5.77) that the distance between the fringes will increase as one moves away from the specimen plane (as L increases). However, in the implementation of ISDG, L is fixed and changes in the spacing between the indents Δd cause changes in the fringe spacing. If these changes are monitored with a photosensor, the movement of the fringes over the sensor yields sinusoidal variations in the intensity at a fixed point. High-resolution data can be captured by measuring the intensity variations with a photosensor array and interpreted in terms of fractional fringe orders. Displacement resolutions on the order of 5 nm can be achieved with this kind of displacement gage. The method is suitable for static and dynamic applications, as well as high-temperature applications. Details of the implementation of the method are described in Sharpe [5.92, 93].

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constant-voltage or constant-current excitation can be used; however, for high-speed cracks, it is preferable to use constant-voltage DC excitation. In fatigue applications, this method is used to determine crack position

to within a resolution of a few micrometers. In dynamic applications, where the crack speed may be in the range of 500–1500 m/s, the speed can be determined to within a few tens of m/s.

References 5.1

5.2

5.3 5.4 5.5 5.6

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5.15 5.16

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5.72 5.73

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5.84

5.85

determine crack profiles, Exp. Mech. 22, 383–391 (1982) K.M. Liechti, Y.-S. Chai: Biaxial loading experiments for determining interfacial fracture toughness, J. Appl. Mech. 58, 680–688 (1991) D. Post, B. Han, P. Ifju: High Sensitivity Moiré (Springer, Berlin, Heidelberg 1994) M.S. Dadkhah, A.S. Kobayashi: HRR field of a moving crack: An experimental analysis, Eng. Fract. Mech. 34, 253–262 (1989) J.S. Epstein, M.S. Dadkhah: Moiré interferometry in fracture research. In: Experimental Techniques in Fracture,, ed. by J.S. Epstein (Wiley VCH, Weinheim 1993) pp. 427–508 H.V. Tippur, S. Krishnaswamy, A.J. Rosakis: A coherent gradient sensor for crack tip deformation measurements: Analysis and experimental results, Int. J. Fract. 48, 193–204 (1990) H. Lee, S. Krishnaswamy: A compact polariscope/shearing interferometer for mapping stress fields in bimaterial systems, Exp. Mech. 36, 404–411 (1996) R.C. Jones: A new calculus for the treatment of optical systems Part I, J. Opt. Soc. Am. 31, 488–493 (1941) R.C. Jones: A new calculus for the treatment of optical systems, Part II, J. Opt. Soc. Am. 31, 493–499 (1941) M. Born, E. Wolf: Principles of Optics, 7th Ed (Cambridge Univ. Press, Cambridge 1999) H.V. Tippur, S. Krishnaswamy, A.J. Rosakis: Optical mapping of crack tip deformations using the method of transmission and reflection coherent gradient sensing: a study of the crack tip K-dominance, Int. J. Fract. 52, 91–117 (1991) A.J. Rosakis: Two optical techniques sensitive to the gradients of optical path difference: The method of caustics and the coherent gradient sensor. In: Experimental Techniques in Fracture, Vol. III, ed. by J.S. Epstein (VCH, Weinheim 1993) pp. 327–425 V.K. Kinra, C.L. Bowers: Brittle fracture of plates in tension. Stress field near the crack, Int. J. Solids Struct. 17, 175 (1981) A. Shukla, R.K. Agarwal, H. Nigam: Dynamic fracture studies on 7075-T6 aluminum and 4340 steel using strain gages and photoelastic coatings, Eng. Fract. Mech. 31, 501–515 (1989) J.W. Dally, J.R. Berger: The role of the electrical resistance strain gage in fracture research. In: Experimental Techniques in Fracture, ed. by J.S. Epstein (Wiley VCH, Weinheim 1993) pp. 1–39 J.W. Dally, D.B. Barker: Dynamic measurements of initiation toughness at high loading rates, Exp. Mech. 28, 298–303 (1988)

5.86

D.M. Owen, S. Zhuang, A.J. Rosakis, G. Ravichandran: Experimental determination of dynamic initiation and propagation fracture toughness in thin aluminum sheets, Int. J. Fract. 90, 153–174 (1998) 5.87 K. Ravi-Chandar: A note on the dynamic stress field near a propagating crack, Int. J. Solids Struct. 19, 839–841 (1983) 5.88 J.R. Berger, J.W. Dally, R.J. Sanford: Determining the dynamic stress intensity factor with strain gages using a crack tip locating algorithm, Eng. Fract. Mech. 36, 145–156 (1990) 5.89 J.F. Kalthoff: On the measurement of dynamic fracture toughness – a review of recent work, Int. J. Fract. 27, 277–298 (1985) 5.90 K. Ravi-Chandar, W.G. Knauss: An experimental investigation into dynamic fracture - I. Crack initiation and crack arrest, Int. J. Fract. 25, 247–262 (1984) 5.91 K. Ravi-Chandar, W.G. Knauss: An experimental investigation into dynamic fracture – III. On steady state crack propagation and branching, Int. J. Fract. 26, 141–154 (1984) 5.92 W.N. Sharpe Jr.: An interferometric strain/displacement measuring system, NASA Tech. Memo. 101638 (1989) 5.93 W.N. Sharpe Jr.: Crack-tip opening displacement measurement techniques. In: Experimental Techniques in Fracture, Vol. III, ed. by J.S. Epstein (VCH, Weinheim 1993) pp. 219–252 5.94 E.N. Dulaney, W.F. Brace: Velocity behavior of a growing crack, J. Appl. Phys. 31, 2233–2236 (1960) 5.95 T.L. Paxson, R.A. Lucas: An investigation of the velocity characteristics of a fixed boundary fracture model. In: Dynamic Crack Propagation, ed. by G.C. Sih (Noordhoff, Leiden 1973) pp. 415–426 5.96 J. Carlsson, L. Dahlberg, F. Nilsson: Experimental studies of the unstable phase of crack propagation in metals and polymers. In: Dynamic Crack Propagation, ed. by G.C. Sih (Noordhoff International, Leyden 1973) pp. 165–181 5.97 B. Stalder, P. Beguelin, H.H. Kausch: A simple velocity gauge for measuring crack growth, Int. J. Fract. 22, R47–R54 (1983) 5.98 J. Fineberg, S.P. Gross, M. Marder, H.L. Swinney: Instability in dynamic fracture, Phys. Rev. Lett. 67, 457–460 (1991) 5.99 G.M. Wilkowski, W.A. Maxey: Review and applications of the electric potential method for measuring crack growth in specimens, flawed pipes and pressure vessels, ASTM STP 791, II266–II294 (1983) 5.100 D. Bonamy, K. Ravi-Chandar: Dynamic crack response to a localized shear pulse perturbation in brittle amorphous materials: On crack surface roughening, Int. J. Fract. 134, 1–22 (2005)

159

Active Materi 6. Active Materials

Guruswami Ravichandran

This chapter provides a brief overview of the mechanics of active materials, particularly those which respond to electro/magnetic/mechanical loading. The relative competition between mechanical and electro/magnetic loading, leading to interesting actuation mechanisms, has been highlighted. Key references provided within this chapter should be referred to for further details on the theoretical development and their application to experiments.

6.1

Background ......................................... 159 6.1.1 Mechanisms of Active Materials........................ 160

6.1.2 Mechanics in the Analysis, Design, and Testing of Active Devices ......... 160 6.2

Piezoelectrics ....................................... 161

6.3 Ferroelectrics ....................................... 6.3.1 Electrostriction............................. 6.3.2 Theory ........................................ 6.3.3 Domain Patterns .......................... 6.3.4 Ceramics .....................................

162 162 163 163 165

6.4 Ferromagnets....................................... 166 6.4.1 Theory ........................................ 166 6.4.2 Magnetostriction .......................... 167 References .................................................. 167

6.1 Background shown in Fig. 6.1 [6.1]. Active materials are widely used as sensors and actuators, including vibration damping, Work per volume (J/m3) 108

Experimental Theoretical

Shape memory alloy

7

10

Solid–liquid

106

Fatigued SMA

Thermopneumatic

Ferroelectric EM ES

Thermal expansion

105

PZT

Electromagnetic (EM)

104 Muscle Electrostatic (ES) EM

103 102 0 10

ES Microbubble

101

102

103

ZnO

104 105 106 107 Cycling frequency (Hz)

Fig. 6.1 Characteristics of common actuator systems (after

Krulevitch et al. [6.1])

Part A 6

Active materials in the context of mechanical applications are those that respond by changing shape to external stimuli such as electro/magnetic-mechanical loading, which is in general reversible in nature. The change in shape results in mechanical sensing and actuation that can be exploited in a variety of ways for practical applications. The mechanics of actuation depends on a wide range of mechanisms, which generally depend on some form of phase transition or motion of phase boundaries under external stimuli. There are also active materials which respond to thermomechanical loading such as shape-memory alloys, which are not discussed here. This chapter is confined to materials which respond to electro/magnetic-mechanical loading such as piezoelectric, ferroelectric, and ferromagnetic solids. A common feature of these materials is that cyclic actuation takes place, often accompanied by hysteresis. The common figures of merit used to characterize actuator performance are the work/unit volume and the cycling frequency. The characteristics of common actuator materials/systems in this parameter space are

160

Part A

Solid Mechanics Topics

micro/nanopositioning, ultrasonics, sonar, fuel injection, robotics, adaptive optics, active deformable structures, and micro-electromechanical systems (MEMS) devices such as micropumps and surgical tools.

a)

ε E

E

6.1.1 Mechanisms of Active Materials The mechanics of actuation for most solid-state active materials depend on one or a combination of the following effects, illustrated in Fig. 6.2a–c.

•

•

•

Piezoelectric effect (Fig. 6.2a) is a linear phenomenon in which the mechanical displacement (strain ε) is proportional to the applied field or voltage (E) and the sign of the displacement depends on the sign of the applied field. Electrostriction (Fig. 6.2b) is generally observed in dielectrics and most prominent in single-crystal ferroelectrics and ferroelectric polymers, where the displacement (strain ε) or actuation is a function of the square of the applied voltage (E) and hence the displacement is independent of the sign of the applied voltage. Magnetostriction (Fig. 6.2c) is similar to electrostriction except that the applied field is magnetic in nature, which is most common in ferromagnetic solids. The displacement (strain ε) in single-crystal magnetostrictive material is proportional to the square of the magnetic field (H).

b)

ε ±E E

c)

ε ±H H

Fig. 6.2 (a) Piezoelectricity described by the converse piezoelectric effect is a linear relationship between strain (ε) and applied electric field (E). (b) Electrostriction is a quadratic relationship between strain (ε) and electric field (E), or more generally, an electric-field-induced deformation that is independent of field polarity. (c) Magnetostriction is a quadratic relationship between strain (ε) and the applied magnetic field (H), or more generally, a magnetic-field-induced deformation that is independent of field polarity

Part A 6.1

trasonics, linear and rotary micropositioning devices, and sonar. Potential applications include microrobotics, active surgical tools, adaptive optics, and miniaturized actuators. The problems associated with active materials are multi-physics in nature and involve solving coupled boundary value problems. 6.1.2 Mechanics in the Analysis, Design, The formulation of boundary value problems in and Testing of Active Devices solid mechanics and the solution techniques have been discussed in Chap. 1 and will not be revisited here The quest for the design and analysis of efficient except to restate some of the governing equations inand compact devices for actuation in the form of volving linearized theory and the appropriate boundary micro/nano-electromechanical systems (MEMS/NEMS) conditions. The active materials of interest may undergo places considerable demands on the choice of mater- large deformations at the microstructural scale due ials, processing, and mechanics of actuation. Most of to various phase transformations (reorientation of unit the current applications of actuators except for a few cells); the macroscopic deformations generally do not specialized applications use piezoelectric materials. exceed a few percentage (ranging from 0.2% for piezoA detailed understanding of the various mechanisms electric, 1–6.5% for single-crystal ferroelectrics, and and mechanics of actuation in active materials will pave ≈ 0.1% for single-crystal magnetostrictive solids). Linthe way for the design of new actuation devices, ad- earized theory of elasticity is used throughout, which vancing further application of this promising class of suffices for most experimental design and applications. materials and other emerging multiferroic materials. Appropriate references for provided for readers who The current applications of these materials include ul- are interested in using rigorous large-strain (finiteThough the electrostrictive and magentostrictive effects are most evident in single-crystal materials, they also play an important role in polycrystalline solids, where these effects are influenced by the texture (orientation) of the various crystals in the solid.

Active Materials

deformation) formulations. For the basic notions of materials science such as the unit cell, crystallography, and texture, the reader is referred to Chap. 2. The parameters of interest in solid mechanics for the active materials (occupying a volume V , with surface denoted by S) include the Cauchy stress tensor (σ), the small-strain tensor ε(εij = 12 (u i, j + u j,i )), and the strain energy density W. The materials are assumed to be linearly elastic solids characterized by the fourth-order elastic moduli tensor C. The Cauchy stress is related to the strain through the elastic moduli σij = Cijkl εkl . The mechanical equilibrium of the stress state is governed by the following field equation σij, j + ρ0 bi = 0 in V .

(6.1)

The boundary conditions are characterized by the prescribed traction (force) vector (t 0 ) on S2 and/or the displacement vector (u0 ) on S1 . The traction on a surface is related to the stress through the Cauchy relation, t i = σij n j , where n is the unit outward normal to the surface. ρ0 is the mass density and b is the body force per unit mass.

6.2 Piezoelectrics

161

The parameters of interest in the electromechanics of solids include the polarization ( p) and the electrostatic potential (φ). The governing equation for the electrostatic potential is expressed by Gauss’s equation ∇ · (−ε0 ∇φ + p) = 0 in V ,

(6.2)

where ε0 is the permittivity. The boundary conditions on the electrostrictive solid are characterized by the conductors in the form of the electric field on  the electrodes (∇φ = 0 on C1 ), including the ground ( S ∂φ ∂n dS = 0 and φ = 0 on C2 ). The parameters of interest in the magnetomechanics of solids include the magnetization (m) and the induced magnetic field (H). The governing equations are given by ∇ × H = 0,

∇ · (H + 4πm) = 0 in V ,

(6.3)

An important aspect of electro(magnetic) active materials is that the electro(magnetic) field permeates the space (R3 ) surrounding the body that is polarized (magnetized).

6.2 Piezoelectrics

Di = dijk σ jk ,

(6.4)

where σ is the stress tensor and D is the electric displacement vector, which is related to the polarization p according to Di = pi + ε0 E i ,

(6.5)

where E is the electric field vector [6.3]. For materials with large spontaneous polarizations, such as ferroelectrics, the electric displacement is approximately equal to the polarization (D ≈ p). For actuators, a more common representation of piezoelectricity is the converse piezoelectric effect. This is a linear relationship between strain and electric field, as shown in Fig. 6.1a and in the following equation at constant stress, eij = dijk E k ,

(6.6)

where e is the strain tensor and d is the same as in (6.4). These relationships are often expressed in matrix notation as Di = dij σ j , e j = dij E i , σ j = sij E i ,

(6.7) (6.8) (6.9)

where s is the matrix of piezoelectric stress constants [6.2, 3]. The parameters commonly used to characterize the piezoelectric effect are the constants d3i (in particular, d33 ), which are measures of the coupling between the applied voltage and the resultant strain in the specimen.

Part A 6.2

Piezoelectricity is a property of ferroelectric materials, as well as many non-ferroelectric crystals, such as quartz, whose crystal structure satisfy certain symmetry criteria [6.2]. It also exists in certain ceramic materials that either have a suitable texture or exhibit a net spontaneous polarization. The most common piezoelectric materials which are widely used in applications include lead zirconate titanate (PZT, Pb(Zr,Ti)O3 ) and lead lanthanum zirconate titanate (PLZT, Pb(La,Zr,Ti)O3 ). Many polymers such as polyvinylidene fluoride (PVDF) and its copolymers with trifluoroethylene (TrFE) and tetrafluoroethylene (TFE) also exhibit the piezoelectric effect. The typical strain achievable in the common piezoelectric solids is in the range of 0.1–0.2%. The direct piezoelectric effect is defined as a linear relationship between stress and electric displacement or charge per unit area,

162

Part A

Solid Mechanics Topics

6.3 Ferroelectrics The term ferroelectric relates not to a relationship of the material to the element iron, but simply a similarity of the properties to those of ferromagnets. Ferroelectrics exhibit a spontaneous, reversible electrical polarization and an associated hysteresis behavior between the polarization and electric field [6.4–6]. Much of the terminology associated with ferroelectrics is borrowed from ferromagnets; for instance, the transition temperature below which the material exhibits ferroelectric behavior is referred to as the Curie temperature. The ferroelectric phenomenon was first discovered in Rochelle salt (NaKC4 H4 O6 · 4H2 O). Other common examples of ferroelectric materials include barium titanate (BaTiO3 ), lead titanate (PbTiO3 ), and lithium niobate (LiNbO3 ). Materials of the perovskite structure (ABO3 ) appear to have the largest electrostriction and spontaneous polarization.

6.3.1 Electrostriction Electrostriction, in its most general sense, means simply electric-field-induced deformation. However, the term is most often used to refer to an electric-field-induced deformation that is proportional to the square of the electric field, as illustrated in Fig. 6.1b, εij = Mijkl E k El .

(6.10)

Part A 6.3

This effect does not require a net spontaneous polarization and, in fact, occurs for all dielectric materials [6.5]. The effect is quite pronounced in some ferroelectric ceramics, such as Pb(Mgx Nb1−x )O3 (PMN) and (1 − x)[Pb(Mg1/3 Nb2/3 )O3 ] − xPbTiO3 (PMN-PT), which generate strains much larger than those of piezoelectric PZT. The term electrostriction will be defined in a more general sense as electric-field-induced deformation that is independent of electric field polarity. As mentioned earlier, piezoelectricity exists in polycrystalline ceramics which exhibit a net spontaneous polarization. For a ferroelectric ceramic, while each grain may be microscopically polarized, the overall material will not be, due to the random orientation of the grains [6.2, 7]. For this reason, the ceramic must be poled under a strong electric field, often at elevated temperature, in order to generate the net spontaneous polarization. The ceramic is exposed to a strong electric field, generating an average polarization. The most interesting property of a ferroelectric solid is that it can be depolarized by an electric field and/or stress. It is this property which can be

exploited effectively in achieving large electrostriction [6.6]. Poling involves the reorientation of domains within the grains. In the case of PZT it may also involve polarization rotations due to phase changes. PZT is a solid solution of lead zirconate and lead titanate that is often formulated near the boundary between the rhombohedral and tetragonal phases (the so-called morphotropic phase boundary). For these materials, additional polarization states are available as it can choose between any of the 100 polarized states of the tetragonal phase, the 111 polarized states of the rhombohedral phase, or the 11k polarized states of the monoclinic phase. The final polarization of each grain, however, is constrained by the mechanical and electrical boundary conditions presented by the adjacent grains. A typical polarization–electric field hysteresis curve for a ferroelectric material is shown in Fig. 6.3. The spontaneous polarization, P s , is defined by the extrapolation of the linear region at saturation back to the polarization axis. The remaining polarization when the electric field returns to zero is known as the remnant polarization P r . Finally, the electric field at which the polarization returns to zero is known as the coercive field E c [6.8]. Polarization

Ps Pr

Ec

Electric field

Fig. 6.3 Polarization–electric field hysteresis for ferro-

electric materials. The spontaneous polarization (Ps ) is defined by the line extrapolated from the saturated linear region to the polarization axis. The remnant polarization (Pr ) is the polarization remaining at zero electric field. The coercive field (E c ) is the field required to reduce the polarization to zero

Active Materials

6.3.2 Theory

163

W

Based on the concepts for ferroelectricity postulated by Ginzburg and Landau, Devonshire developed a theory in which he treated strain and polarization as order parameters or field variables, which is collectively known as the Devonshire–Ginzburg–Landau (DGL) model [6.2, 9]. This theory was enormously successful in organizing vast amounts of data and providing the basis for the basic studies of ferroelectricity. The adaptation of this theory following Shu and Bhattacharya [6.10] is the most amenable in the context of mechanics and is described below. Consider a ferroelectric crystal V at a fixed temperature subject to an applied traction t 0 on part of its boundary S2 and an external applied electric field E 0 . The displacement u and polarization p of the ferroelectric are those that minimize the potential energy,    1 ∇ p · A∇ p + W(x, ε, p) − E0 · p dx Φ( p, u) = 2 V   ε0 − t0 · u dS + (6.11) |∇φ|2 dx , 2 S2

6.3 Ferroelectrics

R3

W(θ, ε, p) = χij pi p j + ωijk pi p j pk + ξijkl pi p j pk pl + ψijklm pi p j pk pl pm + ζijklmn pi p j pk pl pm pn + Cijkl εij εkl + aijk εij pk + qijkl εij pk pl + · · · , (6.12) where χij is the reciprocal dielectric susceptibility of the unpolarized crystal, Cijkl is the elastic stiffness tensor, aijk is the piezoelectric constant tensor, qijkl is the electrostrictive constant tensor, and the coefficients are functions of temperature [6.8].

Fig. 6.4 The multiwell structure of the energy of a ferro-

electric solid with a tetragonal crystal structure as in the case of common perovskite crystals

The third and fourth terms in (6.11) are the potentials associated with the applied electric field and mechanical load, respectively. The final term is the electrostatic field energy that is generated by the polarization distribution. For any polarization distribution, the electrostatic potential φ is determined by solving Gauss’s equation (6.2) in all space, subject to appropriate boundary conditions, especially those on conductors. Thus, this last term is nonlocal. Ferroelectric crystals can be spontaneously polarized and strained in one of K crystallographically equivalent variants below their Curie temperature. Thus, if ε(i) , p(i) are the spontaneous strain and polarization of the i-th variant (i = 1, . . . , K ), then the stored energy W is minimum (zero without loss of generality) on the K [(ε(i) , p(i) )] and grows away from it as shown Z = ∪i=1 in the bottom right of Fig. 6.4.

6.3.3 Domain Patterns A region of constant polarization is known as a ferroelectric domain. The orientation of polarization and strain in ferroelectric crystals is determined by the possible variants of the underlying crystal structure. For example Fig. 6.5a shows the six possible variants that can form by the phase transformation of a perovskite (ABO3 ) crystal from the high-temperature cubic phase to the tetragonal phase when cooled below the Curie temperature. Domains are separated by 90◦ or 180◦ domain boundaries (the angle denotes the orientation between the polarization vectors in adjacent domains, Fig. 6.5b), which can be nucleated or moved by electric field or stress (the ferroelastic effect). Domain patterns are commonly visualized using polarized light microscopy and are shown in Fig. 6.5c for BaTiO3 . The process of changing the polarization direction of a domain by nucleation and growth or domain wall motion is known as domain switching. Electric field can induce both 90◦ or 180◦ switching, while stress

Part A 6.3

where A is a positive-definite matrix so that the first term above penalizes sharp changes in the polarization and may be regarded as the energetic cost of forming domain walls. The second term W is the stored energy density (the Landau energy density), which depends on the state variables or order parameters, the strain ε, and the polarization p, and also explicitly on the position x in polycrystals and heterogeneous media; W encodes the crystallographic and texture information, and may in principle be obtained from first-principles calculations based on quantum mechanics. It is traditional to take W to be a polynomial but one is not limited to this choice; for example, the energy density function W is assumed to be of the form,

ε, p

164

Part A

Solid Mechanics Topics

can induce only 90◦ switching [6.11]. The domain wall structures in the mechanical and electrical domain have recently been experimentally measured using scanning probe microscopy [6.12], which is typically in the range of tens of nanometers. In light of the multiwell structure of W, minimization of the potential energy in (6.11) leads to domain patterns or regions of almost constant strain and polarization close to the spontaneous values separated by domain walls. The width of the domain walls is proportional to the square root of the smallest eigenvalue of A. If this is small compared to the size of the crystal, as is typical, then the domain wall energy has a negligible effect on the macroscopic behavior and may be dropped [6.10]. This leads to an ill-posed problem as the minimizers may develop oscillations at a very fine scale (Fig. 6.5b), however there has been significant recent progress in studying such problems in recent years, motivated by active materials. a)

The minimizers of (6.11) for zero applied load and field are characterized by strain and polarization fields that take their values in Z. So one expects the solutions to be piecewise constant (domains) separated by jumps (domain walls). However the walls cannot be arbitrary. Instead, an energy-minimizing domain wall between variants i and j, i. e., an interface separating regions of strain and polarization (ε(i) , p(i) ) and (ε( j) , p( j) ) as shown in Fig. 6.5d, must satisfy two compatibility conditions [6.10], 1 ε( j) − ε(i) = (a ⊗ n + n ⊗ a) , 2 (6.13) ( p( j) − p(i) ) · n = 0 , where n is the normal to the interface. The first is the mechanical compatibility condition, which assures the mechanical integrity of the interface, and the second is the electrical compatibility conditions, which assures that the interface is uncharged and thus that energy minimizing. At first glance it appears impossible to solve these equations simultaneously: the first equation has at most two solutions for the vectors a and n, and there is no reason that these values of n should satisfy the second. It turns out however, that if the variants are related by two-fold symmetry, i. e., ε( j) = Rε(k) RT , 180◦

b)

90° boundary

Part A 6.3

180° boundary

c)

d)

n (ε(2), p (2))

(ε(1), p (1)) 100 μm

Fig. 6.5 (a) Variants of cubic-to-tetragonal phase transformation, the arrows indicate the direction of polarization. (b) Schematic of 90◦ and 180◦ domains in a ferroelectric crystal. (c) Polarized-light micrograph of the domain pattern in barium titanate. (d) Schematic of a domain wall in a ferroelectric crystal

ε( j) = Rp(k) ,

(6.14)

for some rotation R, then it is indeed possible to solve the two equations (6.14) simultaneously [6.10]. It follows that the only domain walls in a 001c polarized tetragonal phase are 180◦ and 90◦ domain walls, and that the 90◦ domain walls have a structure similar to that of compound twins with a rational {110}c interface and a rational 110c shear direction. The only possible domain walls in the 110c polarized orthorhombic phase are 180◦ domain walls, 90◦ domain walls having a structure like that of compound twins with a rational {100}c interface, 120◦ domain walls having a structure like that of type I twins with a rational {110}c interface, and 60◦ domain walls having a structure like that of type II twins with an irrational normal. The only possible domain walls in the 111c polarized rhombohedral phase are the 180◦ domain walls, and the 70◦ or 109◦ domain walls with a structure similar to that of compound twins. One can also use these ideas to study more-complex patterns involving multiple layers, layers within layers, and crossing layers. The potential energy (6.11) also allows one to study how applied boundary conditions affect the microstructure. The nonlocal electrostatic term is particularly interesting. For example, an isolated ferroelectric that

Active Materials

is homogeneously polarized generates an electrostatic field around it, and its energetic cost forces the ferroelectric to either become frustrated (form many domains at a small scale) or form closure domains or surface layers. In contrast, ferroelectrics shielded by electrodes can form large domains. Hard (high compliance, i. e., stiff) mechanical loading can force the formation of fine domain patterns [6.10], while soft (low compliance, dead loading being the most ideal example) loading leads to large domains. Therefore any strategy for actuation through domain switching must use electrodes to suitably shield the ferroelectric and soft loading in such a manner to create uniform electric and mechanical fields.

6.3 Ferroelectrics

165

by forming a (compatible) microstructure. However, when each pair of variants satisfy the compatibility conditions (6.13), the results of DeSimone and James [6.15] can be adopted to show that Z S equals the set of all possible averages of the spontaneous polarizations and strains of the variants  n  λi ε(i) , Z S = (ε, p) : ε∗ = i=1

p∗ =

n 

λi (2 f i − 1) p(i) , λi ≥ 0 ,

i=1

n 

 λi = 1, 0 ≤ f i ≤ 1 .

(6.15)

i=1

6.3.4 Ceramics

Z P ⊇ Z T = ∩ Z S (x) = x∈Ω

{(ε, p)|(R(x) ε RT (x), R(x) p) ∈ Z S (x), ∀x ∈ Ω} . (6.16)

This simple bound is easy to calculate and also a surprisingly good indicator of the actual behavior of the material, which has the following implications. A material that is cubic above the Curie temperature and 001c -polarized tetragonal below has a very small set of spontaneous polarizations and no set of spontaneous strains unless the ceramic has a 001c texture. Indeed, each grain has only three possible spontaneous strains so that it is limited to only two possible deformation modes. Consequently the grains simply constrain

Part A 6.3

A polycrystal is a collection of perfectly bonded single crystals with identical crystallography but different orientations [6.13]. The term ceramic in the case of ferroelectrics refers to a polycrystal with numerous small grains, each of which may have numerous domains. The functional (6.11) (with A = 0) describes all the details of the domain pattern in each grain, and thus is rather difficult to understand. Instead it is advantageous to replace the energy density W in the functional (6.1) with ¯ the effective energy density of the polycrystal. The W, energy density W describes the behavior at the smallest length scale, which has a multiwell structure as dis- ˆ x, ε, p cussed earlier. This leads to domains, and W is the energy density of the grain at x after it has formed a domain pattern with average strain η and average polarization p. Note that this energy is zero on a set Z S , which is larger than the set Z. Z S is the set of all possible average spontaneous or remnant strains and polarizations that a single crystal can have by formˆ and Z S can vary from ing domain patterns. However, W grain to grain. The collective behavior of the polycrys¯ W(ε, ¯ tal is described by the energy density, W. p) is the energy density of a polycrystal with grains and domain patterns when the average strain is ε∗ and the average polarization is p∗ . Notice that it is zero on the set Z P , which is the set of all possible average spontaneous or remnant strains and polarization of the polycrystal. The size of the set Z P is an estimate of the ease with which a ferroelectric polycrystal may be poled, and also the strains that one can expect through domain switching. A rigorous discussion and precise definitions are given by Li and Bhattacharya [6.14]. The set Z S is obtained as the average spontaneous polarizations and strains that a single crystal can obtain

This is the case in materials with a cubic nonpolar high-temperature phase, and tetragonal, rhombohedral or orthorhombic ferroelectric low-temperature phases; explicit formulas are given in [6.14]. This is not the case in a cubic–monoclinic transformation, but one can estimate the set in that case. In a ceramic, the grain x has its own set Z S (x), which is obtained from the reference set by applying the rotation R(x) that describes the orientation of the grain relative to the reference single crystal. The set Z p of the polycrystal may be obtained as the macroscopic averages of the locally varying strain and polarization fields, which take their values in Z S (x) in each grain x. An explicit characterization remains an open problem (and sample dependent). However, one can obtain an insight into the size of the set by the so-called Taylor bound Z T , which assumes that the (mesoscale) strain and polarization are equal in each grain. Z T is simply the intersection of all possible sets Z S (x) corresponding to the different grains as x varies over the entire crystal. It is a conservative estimate of the actual set Z p :

166

Part A

Solid Mechanics Topics

each other. Similar results hold for a material which is cubic above the Curie temperature and 111c -polarized rhombohedral below it unless the ceramic has a 111c texture. These results imply that tetragonal and rhombohedral materials will not display large strain unless they are single crystals or are textured. Furthermore, it shows that it is difficult to pole these materials. This is the situation in BaTiO3 at room temperature, PbTiO3 , or PZT away from the morphotropic phase boundary (MPB). The situation is quite different if the material is either monoclinic or has a coexistence of 001c -polarized

tetragonal and 111c -polarized rhombohedral states below the Curie temperature. In either of these situations, the material has a large set of spontaneous polarizations and at least some set of spontaneous polarizations irrespective of the texture. Thus these materials will always display significant strain and can easily be poled. This is exactly the situation in PZT at the MPB. In particular, this shows that PZT has large piezoelectricity at the MPB because it can easily be poled and because it can have significant extrinsic strains [6.14].

6.4 Ferromagnets

Part A 6.4

Magnetostrictive materials are ferromagnetics that are spontaneously magnetized and can be demagnetized by the application of external magnetic field and/or stress. Ferromagnets exhibit a spontaneous, reversible magnetization and an associated hysteresis behavior between magnetization and magnetic field. All magnetic materials exhibit magnetostriction to some extent and the spontaneous magnetization is a measure of the actuation strain (magnetostriction) that can be obtained. Application of external magnetic fields to materials such as iron, nickel, and cobalt results in a strain on the order of 10−5 –10−4 . Recent advances in materials development have resulted in large magnetostriction in ironand nickel-based alloys on the order of 10−3 –10−2 . The most notable examples of these materials include Tb0.3 Dy0.7 Fe2 (Terfenol-D) and Ni2 MnGa, a ferromagnetic shape-memory alloy [6.15,16]. A well-established approach that has been used to model magnetostrictive materials is the theory of micromagnetics due to Brown [6.17]. A recent approach known as constrained theory of magnetoelasticity proposed by DeSimone and James [6.15, 18] is presented here, which is more suited for applications related to experimental mechanics.

6.4.1 Theory The potential energy of a magnetostrictive solid can be written as [6.15, 18], Φ(ε, M, θ) = Φexch + Φmst + Φext + Φmel ,

(6.17)

where Φexch is the exchange energy, Φmst is the magnetostatic or stray-field energy, Φext is the energy associated with the external magnetomechanical loads consisting of a uniform prestress σ0 applied at the boundary of S, and of a uniform applied magnetic field H0 in V . The expression (6.17) is similar to the ex-

pression for the potential energy of a ferroelectric solid in (6.11). The arguments for the ferroelectric solid for neglecting the first two terms in the exchange energy and the stray-field energy are also applicable to the magnetostrictive solids under appropriate conditions, namely 1. that the specimen is much larger than the domain size so that the exchange energy associated with the domain walls can be neglected, and 2. that the sample is shielded so that energy associated with the stray fields is negligible. Such an approach enables one to explore the implications of the energy minimization of (6.17). However, shielding the sample is a challenge and needs careful attention. The energy associated with the external loading is written   (6.18) Φext = − t0 · u dS − H0 · m dV . S2

V

The magnetoelastic energy that accounts for the deviation of the magnetization from the favored crystallographic direction can be written  (6.19) Φmel = W(ε, m) dV . V

The magnetoelastic energy density W is a function of the elastic moduli, the magentostrictive constants, and the magnetic susceptibility, and is analogous to (6.12). For a given material, W can be minimized when evaluated on a pair consisting of a magnetization along an easy direction and of the corresponding stress-free strain. In view of the crystallographic symmetry, there will be several, symmetry-related energy-minimizing magnetization (m) and strain (ε) pairs, analogous to the

Active Materials

electrostrictive solids explored in Sect. 6.3.2. Assuming that the corresponding minimum value of Wis zero, one can define the set of energy wells of the material, K [(ε(i) , m(i) )], which is analogous to the energy Z = ∪i=1 wells for ferroelectric solids shown in Fig. 6.5; W increases steeply away from the energy wells. Necessary conditions for compatibility between adjacent domains in terms of jumps in strain and magnetization have been established and have a form similar to (6.13). Much of the discussion concerning domain patterns in ferroelectric solids presented in Sect. 6.3 is applicable to magentostrictive solids, with magnetization in place of electric polarization.

6.4.2 Magnetostriction The application of mechanical loading (stress) can demagnetize ferromagnets by reorienting the magnetization axis, which can be counteracted by an applied magnetic field. This competition between applied magnetic field and dead loading (constant stress) acting on the solid provides a competition, leading to nucleation and propagation of magnetic domains across a specimen that can give rise to large magnetic actuation. For Terfenol-D, the material with the largest known room-temperature magnetostriction, the energy wells comprising Z are eight symmetry-related variants. Experiments on Terfenol-D by Teter et al., illustrate the effect of crystal orientation and applied stress on magnetostriction and are reproduced in Fig. 6.6 [6.19]. Interesting features of the results

References

Magnetostriction (x 10 –3 ) 2.5 [111]

2 [112]

1.5 1 0.5 [110]

0 –2000

–1000

0

1000

2000 Field (Oe)

Fig. 6.6 Magnetostriction versus applied magnetic field for three mutually orthogonal directions in single-crystal Terfenol-D at 20 ◦ C and applied constant stress of 11 MPa. (after 6.19 with permission. Copyright 1990, AIP)

include the steep change in energy away from the minimum (th ereference state at zero strain) and the varying amounts of hysteresis for different orientations. The constrained theory of magnetoelasticity described above can reproduce qualitative features of the experiments based on energy-minimizing domain patterns. An important aspect of modeling magnetostriction hinges on the ability to measure the magnetoelastic energy density, W, and the associated constants.

6.2

6.3 6.4

6.5 6.6

P. Krulevitch, A.P. Lee, P.B. Ramsey, J.C. Trevino, J. Hamilton, M.A. Northrup: Thin film shape memory alloy microactuators, J. MEMS 5, 270–282 (1996) D. Damjanovic: Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Rep. Prog. Phys. 61, 1267–1324 (1998) L.L. Hench, J.K. West: Principles of Electronic Ceramics (Wiley, New York 1990) C.Z. Rosen, B.V. Hiremath, R.E. Newnham (Eds): Piezoelectricity. In: Key Papers in Physics (AIP, New York 1992) Y. Xu: Ferroelectric Materials and Their Applications (North-Holland, Amsterdam 1991) K. Bhattacharya, G. Ravichandran: Ferroelectric perovskites for electromechanical actuation, Acta Mater. 51, 5941–5960 (2003)

6.7

6.8 6.9 6.10

6.11

6.12

L.E. Cross: Ferroelectric ceramics: Tailoring properties for specific applications. In: Ferroelectric Ceramics, ed. by N. Setter, E.L. Colla (Monte Verita, Zurich 1993) pp. 1–85 F. Jona, G. Shirane: Ferroelectric Crystals (Pergamon, New York 1962), Reprint, Dover, New York (1993) A.F. Devonshire: Theory of ferroelectrics, Philos. Mag. Suppl. 3, 85–130 (1954) Y.C. Shu, K. Bhattacharya: Domain patterns and macroscopic behavior of ferroelectric materials, Philos. Mag. B 81, 2021–2054 (2001) E. Burcsu, G. Ravichandran, K. Bhattacharya: Large electrostrictive actuation of barium titanate single crystals, J. Mech. Phys. Solids 52, 823–846 (2004) C. Franck, G. Ravichandran, K. Bhattacharya: Characterization of domain walls in BaTiO3 using simultaneous atomic force and piezo response

Part A 6

References 6.1

167

168

Part A

Solid Mechanics Topics

6.13 6.14

6.15

force microscopy, Appl. Phys. Lett. 88, 1–3 (2006), 102907 C. Hartley: Introduction to materials for the experimental mechanist, Chapter 2 J.Y. Li, K. Bhattacharya: Domain patterns, texture and macroscopic electro-mechanical behavior of ferroelectrics. In: Fundamental Physics of Ferroelectrics 2001, ed. by H. Krakauer (AIP, New York 2001) p. 72 A. DeSimone, R.D. James: A constrained theory of magnetoelasticity, J. Mech. Phys. Solids 50, 283–320 (2002)

6.16

6.17 6.18

6.19

G. Engdahl, I.D. Mayergoyz (Eds.): Handbook of Giant Magnetostrictive Materials (Academic, New York 2000) W.F. Brown: Micromagnetics (Wiley, New York 1963) A. DeSimone, R.D. James: A theory of magnetostriction oriented towards applications, J. Appl. Phys. 81, 5706–5708 (1997) J.P. Teter, M. Wun-Fogle, A.E. Clark, K. Mahoney: Anisotropic perpendicular axis magnetostriction in twinned Tbx Dy1−x Fe1.95 , J. Appl. Phys. 67, 5004–5006 (1990)

Part A 6

169

Biological So 7. Biological Soft Tissues

Jay D. Humphrey

A better understanding of many issues of human health, disease, injury, and the treatment thereof necessitates a detailed quantification of how biological cells, tissues, and organs respond to applied loads. Thus, experimental mechanics can, and must, play a fundamental role in cell biology, physiology, pathophysiology, and clinical intervention. The goal of this chapter is to discuss some of the foundations of experimental biomechanics, with particular attention to quantifying the finite-strain behavior of biological soft tissues in terms of nonlinear constitutive relations. Towards this end, we review illustrative elastic, viscoelastic, and poroelastic descriptors of softtissue behavior and the experiments on which they are based. In addition, we review a new class of much needed constitutive relations that will help quantify the growth and remodeling processes within tissues that are fundamental to long-term adaptations and responses to disease, injury, and clinical intervention. We will see that much has

Constitutive Formulations – Overview .... 171

7.2

Traditional Constitutive Relations .......... 7.2.1 Elasticity ..................................... 7.2.2 Viscoelasticity .............................. 7.2.3 Poroelasticity and Mixture Descriptions .............. 7.2.4 Muscle Activation ......................... 7.2.5 Thermomechanics ........................

176 177 177

7.3

Growth and Remodeling – A New Frontier 7.3.1 Early Approaches.......................... 7.3.2 Kinematic Growth ........................ 7.3.3 Constrained Mixture Approach .......

178 178 179 180

7.4

Closure ................................................ 182

7.5

Further Reading ................................... 182

172 172 176

References .................................................. 183 been learned, yet much remains to be discovered about the wonderfully complex biomechanical behavior of soft tissues.

there was a need to await the development of a nonlinear theory of material behavior in order to quantify well that of soft tissues. It is not altogether surprising, therefore, that biomechanics did not truly come into its own until the mid-1960s. It is suggested that five independent developments facilitated this: (i) the post World War II renaissance in nonlinear continuum mechanics [7.2] established a general foundation needed for developing constitutive relations suitable for soft tissues; (ii) the development of computers enabled the precisely controlled experimentation [7.3] that was needed to investigate complex anisotropic behaviors and facilitated the nonlinear regressions [7.4] that were needed to determine best-fit material parameters from data; (iii) related to this technological advance, development of sophisticated numerical methods, particularly the

Part A 7

At least since the time of Galileo Galilei (1564–1642), there has been a general appreciation that mechanics influences the structure and function of biological tissues and organs. For example, Galileo studied the strength of long bones and suggested that they are hollow as this increases their strength-to-weight ratio (i. e., it increases the second moment of area given a fixed amount of tissue, which is fundamental to increasing the bending stiffness). Many other figures in the storied history of mechanics, including G. Borelli, R. Hooke, L. Euler, T. Young, J. Poiseuille, and H. von Helmholtz, contributed much to our growing understanding of biomechanics. Of particular interest herein, M. Wertheim, a very productive experimental mechanicist of the 19-th century, showed that diverse soft tissues do not exhibit the linear stress–strain response that is common to many engineering materials [7.1]. That is,

7.1

170

Part A

Solid Mechanics Topics

Mechanical stimuli ECM

α β

Cell-cell G-proteins

Ion channels

GF receptor Cadherins Cell membrane

Mechanobiologic factors

Cell-matrix Integrins FAC changes

Signaling pathways

CSK changes

Transcription factors Gene expression

Altered mechanical properties

Altered cell function

Fig. 7.1 Schema of cellular stimuli or inputs – chemical, mechanical, and of course, genetic – and possible mechanobio-

logic responses. The cytoskeleton (CSK) maintains the structural integrity of the cell and interacts with the extracellular matrix (ECM) through integrins (transmembrane structural proteins) or clusters of integrins (i. e., focal adhesion complexes or FACs). Of course, cells also interact with other cells via special interconnections (e.g., cadherins) and are stimulated by various molecules, including growth factors (GFs). Although the pathways are not well understood, mechanical stimuli can induce diverse changes in gene expression, which in turn alter cell function and properties as well as help control the structure and properties of the extracellular matrix. It is clear, therefore, that mechanical stimuli can affect many different aspects of cell function and thus that of tissues and organs

Part A 7

finite element method [7.5], enabled the solution of complex boundary value problems associated with soft tissues, including inverse finite element estimations of material parameters [7.6]; and (iv) the space race of the 1960s increased our motivation to study the response of the human body to applied loads, particularly high G-forces associated with lift-off and reentry as well as the microgravity environment in space. Finally, it is not coincidental that the birth of modern biomechanics followed closely (v) the birth of modern biology, often signaled by discoveries of the structure of proteins, by L. Pauling, and the structure of DNA, by J. Watson and F. Crick. These discoveries ushered in the age of molecular and cellular biology [7.7]. Indeed, in contrast to the general appreciation of roles of mechanics in biology in the spirit of Galileo, the last few decades have revealed a fundamentally more important role of mechanics and mechanical fac-

tors. Experiments since the mid-1970s have revealed that cells often alter their basic activities (e.g., their expression of particular genes) in response to even subtle changes in their mechanical environment. For example, in response to increased mechanical stretching, cells can increase their production of structural proteins [7.8]; in response to mechanical injury, cells can increase their production of enzymes that degrade the damaged structural proteins [7.9]; and, in response to increased flow-induced shear stresses, cells can increase their production of molecules that change permeability or cause the lumen to dilate and thereby to restore the shear stress towards its baseline value [7.10]. Cells that are responsive to changes in their mechanical loading are sometimes referred to as mechanocytes, and the study of altered cellular activity in response to altered mechanical loading is called mechanobiology (Fig. 7.1). Consistent with the mechanistic philosophy

Biological Soft Tissues

of R. Descartes, which motivated many of the early studies in biomechanics, it is widely accepted that cells are not only responsive to changes in their mechanical environment, they are also subject to the basic postulates of mechanics (e.g., balance of linear momen-

7.1 Constitutive Formulations – Overview

171

tum). As a result, basic concepts from mechanics (e.g., stress and strain) can be useful in quantifying cellular responses and there is a strong relationship between mechanobiology and biomechanics. Herein, however, we focus primarily on the latter.

7.1 Constitutive Formulations – Overview There are, of course, three general approaches to quantify complex mechanical behaviors via constitutive formulations: (i) theoretically, based on precise information on the microstructure of the material; (ii) experimentally, based directly on data collected from particular classes of experiments; and (iii) via trial and error, by postulating competing relations and selecting preferred ones based on their ability to fit data. Although theoretically derived microstructural relations are preferred in principle, efforts ranging from C. Navier’s attempt to model the behavior of metals to L. Treloar’s attempt to model the behavior of natural rubber reveal that that this is very difficult in practice, particularly for materials with complex microstructures such as soft tissues. Formulating constitutive relations directly from experimental data is thus a practical preference [7.11], but again history reveals that it has been difficult to identify and execute appropriate theoretically based empirical approaches. For this reason, trial-and-error phenomenological formulations, based on lessons learned over years of investigation and often motivated by limited microstructural information, continue to be common in biomechanics just as they are in more traditional areas of applied mechanics. Regardless of approach, there are five basic steps that one must follow in any constitutive formulation:

•

Specifically, the first step is to classify the behavior of the material under conditions of interest, as, for example, if the material exhibits primarily a fluid-like or a solid-like response, if the response is dissipative or not, if it is isotropic or not, if it is isochoric or not, and so forth. Once sufficient observations enable one to classify the behavior, one can then establish an appro-

Fibroblast Endothelial cell Collagen

Smooth muscle Adventitia

Elastin

Media Intima

Basal lamina

b)

Fig. 7.2 (a) Schema of the arterial wall, which consists of three basic layers: the intima, or innermost layer, the media, or middle layer, and the adventitia, or outermost layer. Illustrated too are the three primary cells types (endothelial, smooth muscle, and fibroblasts) and two of the key structural proteins (elastin and collagen). (b) For comparison, see the histological section of an actual artery: the dark inner line shows the internal elastic lamina, which separates the thin intima from the media; the lighter shade in the middle shows the muscle dominated media, and the darker shade in the outer layer shows the collagendominated adventitia

Part A 7.1

• • • •

delineate general characteristics of the material behavior, establish an appropriate theoretical framework, identify specific functional forms of the relations, calculate values of the material parameters, evaluate the predictive capability of the final relations.

a)

172

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Solid Mechanics Topics

priate theoretical framework (e.g., a theory of elasticity within the context of the definition of a simple material by W. Noll), with suitable restrictions on the possible constitutive relations (e.g., as required by the Clausius– Duhem inequality, material frame indifference, and so forth). Once a theory is available, one can then design appropriate experiments to quantify the material behavior in terms of specific functional relationships and best-fit values of the material parameters. Because of the complex behaviors, multiaxial tests are often preferred, including in-plane biaxial stretching of a planar specimen, extension and inflation of a cylindrical specimen, extension and torsion of a cylindrical specimen, or inflation of an axisymmetric membrane, each of which has a tractable solution to the associated finite-strain boundary value problem. The final step, of course, is to ensure that the relation has predictive capability beyond that used in formulating the relation. With regard to classifying the mechanical behavior of biological soft tissues, it is useful to note that they often consist of multiple cell types embedded within an extracellular matrix that consists of diverse proteins as well as proteoglycans (i. e., protein cores with polysaccharide branches) that sequester significant amounts of water. Figure 7.2 shows constituents in the arterial wall, an illustrative soft tissue. The primary structural proteins in arteries, as in most soft tissues, are elastin and different families of collagen. Elastin is perhaps the most elastic protein, capable of extensions of over 100% with little dissipation. Moreover, elastin is one of the most stable proteins in the body, with a normal halflife of decades. Collagen, on the other hand, tends to be very stiff and not very extensible, with the exception that it is often undulated in the physiologic state and thus can undergo large displacements until straightened.

The half-life of collagen varies tremendously depending on the type of tissue, ranging from a few days in the periodontal ligament to years in bone. Note, too, that the half-life as well as overall tissue stiffness is modulated in part by extensive covalent cross-links, which can be enzymatic or nonenzymatic (which occur in diabetes, for example, and contribute to the loss of normal function of various types of tissues, including arteries). In general, however, soft tissues can exhibit a nearly elastic response (e.g., because of elastin) under many conditions, one that is often nonlinear (due to the gradual recruitment of undulated collagen), anisotropic (due to different orientations of the different constituents), and isochoric (due to the high water content) unless water is exuded or imbibed during the deformation. Many soft tissues will also creep or stress-relax under a constant load or a constant extension, respectively. Hence, as with most materials, one must be careful to identify the conditions of interest. In the cardiovascular and pulmonary systems, for example, normal loading is cyclic, and these tissues exhibit a nearly elastic response under cyclic loading. Conversely, implanted prosthetic devices in the cardiovascular system (e.g., a coronary stent) may impose a constant distension, and thus induce significant stress relaxation early on but remodeling over longer periods due to the degradation and synthesis of matrix proteins. Finally, two distinguishing features of soft tissues are that they typically contain contractile cells (e.g., muscle cells, as in the heart, or myofibroblasts, as in wounds to the skin) and that they can repair themselves in response to local damage. That is, whether we remove them from the body for testing or not, we must remember that our ultimate interest is in the mechanical properties of tissues that are living [7.12].

Part A 7.2

7.2 Traditional Constitutive Relations At the beginning of this section, we reemphasize that constitutive relations do not describe materials; rather they describe the behavior of materials under welldefined conditions of interest. A simple case in point is that we do not have a constitutive relation for water; we have different constitutive relations for water in its solid (ice), liquid (water), or gaseous (steam) states, which is to say for different conditions of temperature and pressure. Hence, it is unreasonable to expect that any single constitutive relation can, or even should, describe a particular tissue. In other words, we should expect that

diverse constitutive relations will be equally useful for describing the behavior of individual tissues depending on the conditions of interest. Many arguments in the literature over whether a particular tissue is elastic, viscoelastic, poroelastic, etc. could have been avoided if this simple truth had been embraced.

7.2.1 Elasticity No biological soft tissue exhibits a truly elastic response, but there are many conditions under which the

Biological Soft Tissues

assumption of elasticity is both reasonable and useful. Toward this end, one of the most interesting, and experimentally useful, observations with regard to the behavior of many soft tissues is that they can be preconditioned under cyclic loading. That is, as an excised tissue is cyclically loaded and unloaded, the stress versus stretch curves tend to shift rightward, with decreasing hysteresis, until a near-steady-state response is obtained. Fung [7.12] suggested that this steadystate response could be modeled by separately treating the loading and unloading curves as nearly elastic; he coined the term pseudoelastic to remind us that the response is not truly elastic. In practice, however, except in the case of muscular tissues, the hysteresis is often small and one can often approximate reasonably well the mean response between the loading and the unloading responses using a single elastic descriptor, similar to what is done to describe rubber elasticity. Indeed, although mechanisms underlying the preconditioning of soft tissues are likely very different from those underlying the Mullin effect in rubber elasticity [7.13], in both cases initial cyclic loading produces stress softening and enables one to use the many advances in nonlinear elasticity. This and many other parallels between tissue and rubber elasticity likely result from the long-chain polymeric microstructure of both classes of materials, thus these fields can and should borrow ideas from one another (see discussion in [7.14] Chap. 1). For example, Cauchy membrane stress (g/cm) 120 RV epicardium equibiaxial stretch Circumferential Apex–to–base 60

1

1.16

1.32 Stretch

Fig. 7.3 Representative tension–stretch data taken, fol-

lowing preconditioning, from a primarily collagenous membrane, the epicardium or covering of the heart, tested under in-plane equibiaxial extension. Note the strong nonlinear response, anisotropy, and negligible hysteresis over finite deformations (note: membrane stress is the same as a stress resultant or tension, thus having units of force per length though here shown as a mass per length)

advances in rubber elasticity have taught us much about the importance of universal solutions, common types of material and structural instabilities, useful experimental approaches, and so forth [7.14–16]. Nonetheless, common forms of stress–strain relations in rubber elasticity – for example, neo-Hookean, Mooney–Rivlin, and Ogden – have little utility in soft-tissue biomechanics and at times can be misleading [7.17]. A final comment with regard to preconditioning is that, although we desire to know properties in vivo (literally in the body), it is difficult in practice to perform the requisite measurements without removing the cells, tissues, or organs from the body so that boundary conditions can be known. This process of removing specimens from their native environment necessarily induces a nonphysiological, often poorly controlled strain history. Because the mechanical behavior is history dependent, the experimental procedure of preconditioning provides a common, recent strain history that facilitates comparisons of subsequent responses from specimen to specimen. For this reason, the preconditioning protocol should be designed well and always reported. Whereas a measured linear stress–strain response implies a unique functional relationship, the nonlinear, anisotropic stress–strain responses exhibited by most soft tissues (Fig. 7.3) typically do not suggest a specific functional relationship. In other words, one must decide whether the observed characteristic stiffening over finite strains (often from 5% to as much as 100% strain) is best represented by polynomial, exponential, or more complex stress–strain relations. Based on one-dimensional (1-D) extension tests on a primarily collagenous membrane called the mesentery, which is found in the abdomen, Fung showed in 1967 that it can be useful to plot stiffness, specifically the change of the first Piola–Kirchhoff stress with respect to changes in the deformation gradient, versus stress rather than to plot stress versus stretch as is common [7.12]. Specifically, if P is the 1-D first Piola– Kirchhoff stress and λ is the associated component of the deformation gradient (i. e., a stretch ratio), then seeking a functional form P = P(λ) directly from data is simplified by interpreting dP/ dλ versus P. For the mesentery, Fung found a near-linear relation between stiffness and stress, which in turn suggested directly (i. e., via the solution of the linear first-order ordinary differential equation) an exponential stress–stretch relationship (with P(λ = 1) = 0):  α  β(λ−1) dP e −1 , = α+βP → P = dλ β

(7.1)

173

Part A 7.2

0

7.2 Traditional Constitutive Relations

174

Part A

Solid Mechanics Topics

where α and β are material parameters, which can be determined via nonlinear regressions of stress versus stretch data or more simply via linear regressions of stiffness versus stress data. Note, too, that one could expand and then linearize the exponential function to relate these parameters to the Young’s modulus of linearized elasticity if so desired. Albeit a very important finding, a 1-D constitutive relation cannot be extended to describe the multiaxial behavior that is common to many soft tissues, ranging from the mesentery to the heart, arteries, skin, cornea, bladder, and so forth. At this point, therefore, Fung made a bold hypothesis. Given that this 1-D first Piola–Kirchhoff versus stretch relationship was exponential, he hypothesized that the behavior of many soft tissues could be described by an exponential relationship between the second Piola–Kirchhoff stress tensor S and the Green strain tensor E, which are conjugate measures appropriate for large-strain elasticity. In particular, he suggested a hyperelastic constitutive relation of the form:   ∂W ∂Q whereby S = = c eQ , W = c eQ − 1 ∂E ∂E (7.2)

Part A 7.2

where W is a stored energy function and Q is a function of E. Over many years, Fung and others suggested, based on attempts to fit data, that a convenient form of Q is one that is quadratic in the Green strain, similar to the form of a stored energy function in terms of the infinitesimal strain in linearized elasticity. Fung argued that, because Q is related directly to ln W, material symmetry arguments were the same for this exponential stored energy function as they are for linearized elasticity. Q thus contains two, five, or nine nondimensional material parameters for isotropic, transversely isotropic, or orthotropic material symmetries, respectively, and this form of W recovers that for linearized elasticity as a special case. The addition of a single extra material parameter, c, having units of stress, is a small price to pay in going from linearized to finite elasticity, and this form of W has been used with some success to describe data on the biaxial behavior of skin, lung tissue, arteries, heart tissue, urinary bladder, and various membranes including the pericardium (which covers the heart) and the pleura (which covers the lungs). In some of these cases, this form of W was modified easily for incompressibility (e.g., by introducing a Lagrange multiplier) or for a twodimensional (2-D) problem. For example, a commonly used 2-D form in terms of principal Green strains

is     2 2 + c2 E 22 + 2c3 E 11 E 22 − 1 , W = c exp c1 E 11 (7.3)

where ci are material parameters. It is easy to show that physically reasonable behavior is ensured by c > 0 and ci > 0 and that convexity is ensured by the additional condition c1 c2 > c23 [7.18, 19]. Nonetheless, (7.2) and (7.3) are not without limitations. This 2-D form limits the degree of strain-dependent changes in anisotropy that are allowed, and experience has shown that it is typically difficult to find a unique set of material parameters via nonlinear regressions of data, particularly in three dimensions. Perhaps more problematic, experience has also revealed that convergent solutions can be difficult to achieve in finite element models based on the three-dimensional (3-D) relation except in cases of axisymmetric geometries. The later may be related to the recent finding by Walton and Wilber [7.20] that the 3-D Fung stored energy function is not strongly elliptic. Hence, despite its past success and usage, there is clearly a need to explore other constitutive approaches. Moreover, this is a good reminder that there is a need for a strong theoretical foundation in all constitutive formulations. The Green strain is related directly to the right Cauchy Green tensor C = F F, where F is the deformation gradient tensor, yet most work in finite elasticity has been based on forms of W that depend on C, thus allowing the Cauchy stress t to be determined via (as required by Clausius–Duhem, and consistent with (7.2) for the relation between the second Piola–Kirchhoff stress and Green strain): t=

∂W  2 F F . det F ∂C

(7.4)

For example, Spencer [7.21] suggested that materials consisting of a single family of fibers (i. e., exhibiting a transversely isotropic material symmetry) could be described by a W of the form ˆ II, III, IV, V) , W = W(I,

(7.5)

where I = tr C,

2II = (tr C)2 − tr C2 ,

IV = M · CM,

V = M·C M , 2

III = det C, (7.6)

and M is a unit vector that identifies the direction of the fiber family in a reference configuration. Incompress-

Biological Soft Tissues

ibility, III = 1, reduces the number of invariants by one while introducing an arbitrary Lagrange multiplier p, namely p ˜ II, IV, V) − (III − 1) , (7.7) W = W(I, 2 yet it is still difficult to impossible to rigorously determine specific functional forms directly from data. Indeed, this problem is more acute in cases of twofiber families, including orthotropy when the families are orthogonal. Hence, subclasses of this form of W have been evaluated in biomechanics. For example, Humphrey et al. [7.22] showed, and Sacks and Chuong [7.23] confirmed, that a stored energy funcˆ IV), determined directly from tion of the form W(I, in-plane biaxial tests (with I and IV separately maintained constant) on excised slabs of noncontracting heart muscle to be a polynomial function, described well the available in-plane biaxial stretching data. This 1990 paper [7.22] also illustrated the utility of performing nonlinear regressions of stress–stretch data using data sets that combined results from multiple biaxial tests as well as the importance of respecting Baker–Ericksen-type inequalities [7.2] in the parameter estimations. Holzapfel et al. [7.18] and others have similarly proposed specific forms of W for arteries based on a subclass of the two-fiber family approach of Spencer. Specifically, they propose a form of W that combines that of a neo-Hookean relation with a simple exponential form for two-fiber families, namely ˜ = c(tr C − 3) W  c1   exp b1 (M1 · CM1 − 1)2 − 1 + b1  c2   exp b2 (M2 · CM2 − 1)2 − 1 , + b2

(7.8)

175

depend on traditional invariants of C is not optimal, hence alternative invariant sets should be identified and explored [7.18, 24, 25]. Experiments designed based on these invariants remain to be performed, however. Preceding the one- and two-fiber family models were the microstructural models proposed by Lanir [7.26]. Briefly, electron and light microscopy reveal that the elastin and collagen fibers within many soft tissues have complex spatial distributions (notable exceptions being collagen fibers within tendons, which are coaxial, and those within the cornea of the eye, which are arranged in layered orthogonal networks). Moreover, it appears that, despite extensive cross-linking at the molecular level, these networks are often loosely organized. Consequently, Lanir suggested that a stored energy function for a tissue could be derived in terms of strain energies for straightened individual fibers if one accounted for the undulation and distribution of the different types of fibers, or alternatively that one could postulate exponential stored energy functions for the individual fibers (cf. (7.3)) and simply account for distribution functions for each type of fiber. For example, for a soft tissue consisting primarily of elastin and type I collagen (i. e., only two types of constituents), one could consider

   Ri (ϕ, θ) wif λif sin ϕ dϕ dθ , (7.9) φi W= i=1,2

φi

where and Ri are, respectively, the volume fraction and distribution function for constituent i (elastin or collagen), and wif is the 1-D stored energy function for a fiber belonging to constituent i. Clearly, the stress could be computed as in (7.1) provided that the fiber stretch can be related to the overall strain, which is easy if one assumes affine deformations. In principle, the distribution function could be determined from histology and the material parameters for the fiber stored energy function could be determined from straightforward 1-D tests, thus eliminating, or at least reducing, the need to find many free parameters via nonlinear regressions in which unique estimates are rare. In practice, however, it has been difficult to identify the distribution functions directly, thus they have often been assumed to be Gaussian or a similarly common distribution function. Although proposed as a microstructural model, the many underlying assumptions render this approach microstructurally motivated at best; that is, there is no actual modeling of the complex interactions (including covalent cross-links, van der Waals forces, etc.) between the many different proteins and proteoglycans that endow the tissue with its bulk properties,

Part A 7.2

where Mi (i = 1, 2) denote the original directions of the two-fiber families. Although neither of these forms is derived directly from precise knowledge of the microstructure nor inferred directly from experimental data, this form of W was motivated by the idea that elastin endows an artery with a nearly linearly elastic (neo-Hookean)-type response whereas the straightening of multiple families of collagen can be modeled by exponential functions in terms of fiber stretches. Thus far, this and similar forms of W have proven useful in large-scale computations and illustrates well the utility of the third approach to modeling noted above – trial and error based on experience with other materials or similar relations. It is important to note, however, that inferring forms of W for incompressible behaviors that

7.2 Traditional Constitutive Relations

176

Part A

Solid Mechanics Topics

which are measurable using standard procedures. Indeed, perhaps one reason that this approach did not gain wider usage is that it failed to predict material behaviors under simple experimental conditions, thus like competing phenomenological relations it had to rely on nonlinear regressions to obtain best-fit values of the associated material parameters. As noted earlier, the extreme complexity of the microstructure of soft tissues renders it difficult to impossible to derive truly microstructural relations. Nevertheless, as we discuss below, microstructurally motivated formulations can be a very useful approach to constitutive modeling provided that the relations are not overinterpreted. Among others, Bischoff et al. [7.27] have revisited microstructurally motivated constitutive models with a goal of melding them with phenomenological models. In summary, although no soft tissue is truly elastic in its behavior, hyperelastic constitutive relations have proven useful in many applications. We have reviewed but a few of the many different functional forms reported in the literature, thus the interested reader is referred to Fung [7.12], Maurel et al. [7.28], and Humphrey [7.19] for additional discussion.

7.2.2 Viscoelasticity Although the response of many soft tissues tends to be relatively insensitive to changes in strain rate over physiologic ranges, soft tissues creep under constant loads and they stress-relax under constant deformations. Among others, Fung [7.12] suggested that single-integral heredity models could be useful in biomechanics just as they are in rubber viscoelasticity [7.29]. For example, we recently showed that sets of strain-dependent stress relaxation responses of a collagenous membrane, before and after thermal damage, can be modeled via [7.30]

Part A 7.2

τ G(τ − s) t(τ) = − p(τ)I + 2F(τ) 0

  ∂ ∂W × (s) ds F (τ) , ∂s ∂C

(7.10)

where G is a reduced relaxation function that depends on fading time (τ − s) and W was taken to be an exponential function similar to (7.3); all other quantities are the same as before except with explicit dependence on the current time. Various forms of the reduced relaxation function can be used, including a simple form that

we found to be useful in thermal damage G(x) =

1− R +R, 1 + (x/τR )n

(7.11)

where n is a free parameter, τR is a characteristic time of relaxation, and R is the stress remainder, that is, the fraction of the elastic response that is left after a long relaxation (e.g., R = 0 for a viscoelastic fluid). If short-term responses are important, numerous models can be used; for example, the simple viscohyperelastic approach of Beatty and Zhou [7.31] is useful in modeling biomembranes [7.32]. Briefly, the Cauchy stress is assumed to be of the general form t = tˆ(B, D), where B = FF is the left Cauchy–Green tensor, with F the deformation gradient, and D = (L + L )/2 is the ˙ −1 . stretching tensor, with the velocity gradient L = FF Specifically, assuming incompressibility, the Cauchy stress has three contributions: a reaction stress, an elastic part, and a viscous part, namely t = − pI + 2F

˜  ∂W F + 2μD , ∂C

(7.12)

where p is again a Lagrange multiplier that enforces incompressibility, W is the same strain energy function that was used in the elastic-only description noted above, and μ is a viscosity. Hence, this description of short-term nonlinear viscohyperelasticity adds but one additional material parameter to the constitutive equation, and the elastic response can be quantified first via quasistatic tests, thereby reducing the number of parameters in each regression. Some have considered a synthesis of the short- and long-term models. For a discussion of other viscoelastic models in softtissue mechanics, see Provenzano et al. [7.33] or Haslach [7.34] and references therein.

7.2.3 Poroelasticity and Mixture Descriptions Not only do soft tissues consist of considerable water, every cell in these tissues is within ≈ 50 μm of a capillary, which is to say close to flowing blood. Clearly then, it can be advantageous to model tissues as solid–fluid mixtures under many conditions of interest. A basic premise of mixture theory [7.2] is that balance relations hold both for the mixture as a whole and for the individual constituents, with the requirement that summation of the balance relations for the constituents must yield the classical relations. Moreover, it is assumed that the constituent balance relations include additional constitutive relations, particularly those that model the

Biological Soft Tissues

exchanges of mass, momentum, or energy between constituents. The first, and most often used, approach to model soft tissues via mixture theory was proposed by Mow et al. [7.35]. Briefly, their linear biphasic theory treated cartilage as a porous solid (i. e., the composite response due to type II collagen, proteoglycans, etc.), which was assumed to exhibit a linearly elastic isotropic response, with an associated viscous fluid within. They proposed constitutive relations for the solid and fluid stresses of the form t (s) = −φ(s) pI + λs tr(ε)I + 2μs ε , t (f) = −φ(f) pI − 23 μf div v(f) I + 2μf D ,

(7.13)

and, for the momentum exchange between the solid and fluid, − p(f) = p(s) = p∇φ(f) + K (v(f) − v(s) ) ,

(7.14)

177

7.2.4 Muscle Activation Another unique feature of many soft tissues is their ability to contract via actin–myosin interactions within specialized cells called myocytes. Examples include the cardiac muscle of the heart, skeletal muscle of the arms and legs, and smooth muscle, which is found in many tissues including the airways, arteries, and uterus. The most famous equation in muscle mechanics is that postulated in 1938 by A. V. Hill to describe force–velocity relations. This relation, like many subsequent ones, focuses on 1-D behavior along the long axis of the myocyte or muscle; data typically comes from tests on muscle fibers or strips, or in some cases rings taken from arteries or airways. Although much has been learned, much remains to be learned particularly with respect to the multiaxial behavior. The interested reader is referred to Fung [7.12]. Zahalak et al. [7.42], and Rachev and Hayashi [7.43]. In addition, however, note that modeling muscle activity in the heart (i. e., the electromechanics) has advanced significantly and represents a great example of the synthesis of complex theoretical, experimental, and computational methods. Toward this end, the reader is referred to Hunter et al. [7.44, 45].

7.2.5 Thermomechanics The human body regulates its temperature to remain within a narrow range, and for this reason there has been little attention to constitutive relations for thermomechanical behaviors of cells, tissues, and organs. Nevertheless, advances in laser, microwave, highfrequency ultrasound, and related technologies have encouraged the development and use of heating devices to treat diverse diseases and injuries. For example, supraphysiologic temperatures can destroy cells and shrink collagenous tissue, which can be useful in treating cancer and orthopedic injuries, respectively. Laser-based corneal reshaping, or LASIK, is another prime example. Due to space limitations here, the interested reader is referred to Humphrey [7.46], and references therein, for a brief review of the growing field of biothermomechanics and insight into ways in which experimental mechanics and constitutive modeling can contribute. Also see Diller and Ryan [7.47] for information on the associated bioheat transfer. Of particular note, however, it has been shown that increased mechanical loading can delay the rate at which thermal damage accrues, hence there is a strong thermomechanical coupling and a pressing need for more mechanics-based studies – first experimental, then computational.

Part A 7.2

where the superscripts and subscripts ‘s’ and ‘f’ denote solid and fluid constituents, hence v(s) and v(f) are solid and fluid velocities, respectively. Finally, φ(i) are constituent fractions, μs and λs are the classical Lamé constants for the solid, μf is the fluid viscosity, and ε is the linearized strain in the solid. In some cases the fluid viscosity is neglected, thus allowing tissue viscoelasticity to be accounted for solely via the momentum exchange between the solid and diffusing fluid, where K is related to the permeability coefficient. Mow and colleagues have developed this theory over the years to account for additional factors, including the presence of diffusing ions [7.36, 37] (see also [7.38] for a related approach). Because of the complexity of poroelastic and mixture theories, as well as the inherent geometric complexities associated with most real initial–boundary value problems in soft tissues, finite element methods will continue to prove essential; see, for example, Spilker et al. [7.39] and Simon et al. [7.40] for such formulations. In summary, one can now find many different applications of mixtures in the literature on soft tissues (e.g., Reynolds and Humphrey [7.41] address capillary blood flow within a tissue using mixture theory) and, indeed, the past success and future promise of this approach mandates intensified research in this area, research that must not simply be application, but rather should include development and extension of past theories. Moreover, whereas many of the experiments in biomechanics have consisted of unconfined or confined uniaxial compression tests using porous indenters, there is a pressing need for new multiaxial tests.

7.2 Traditional Constitutive Relations

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Part A

Solid Mechanics Topics

7.3 Growth and Remodeling – A New Frontier As noted above, it has been thought at least since the time of Galileo that mechanical stimuli play essential roles in governing biological structure and function. Nevertheless an important step in our understanding of the biomechanics of tissues began with Wolff’s law for bone remodeling, which was put forth in the late 19-th century. Briefly, it was observed that the fine structure of cancellous (i. e., trabecular) bone within long bones tended to follow lines of maximum tension. That is, it appeared that the stress field dictated, at least in part, the way in which the microstructure of bone was organized. This observation led to the concept of functional adaptation wherein it was thought that bone functionally adapts so as to achieve maximum strength with a minimum of material. For a discussion of bone growth and remodeling, see Fung [7.12], Cowin [7.48], and Carter and Beaupre [7.49]. Although the general concept of functional adaptation appears to hold for most tissues, it is emphasized that bone differs significantly from soft tissues in three important ways. First, bone growth occurs on surfaces, that is, via appositional growth rather than via interstitial growth as in most soft tissues; second, most of the strength of bone derives from an inorganic component, which is not true in soft tissues; third, bone experiences small strains and exhibits a nearly linearly elastic, or poroelastic, behavior, which is very different from the nonlinear behavior exhibited by soft tissues over finite strains. Hence, let us consider methods that have been applied to soft tissues.

7.3.1 Early Approaches

Part A 7.3

Murray [7.50] suggested that biological “organization and adaptation are observed facts, presumably conforming to definite laws because, statistically at least, there is some sort of uniformity or determinism in their appearances. And let us assume that the best quantitative statement embodying the concept of organization is a principle which states that the cost of operation of physiological systems tends to be a minimum. . . ” Murray illustrated his ideas by postulating a cost function for “operating an arterial segment.” He proposed that the radius of a blood vessel results from a compromise between the advantage of increasing the lumen, which reduces the resistance to flow and thereby the workload on the heart, and the disadvantage of increasing overall blood volume, which increases the metabolic demand of maintaining the blood (e.g., red blood cells have a life-

span of a few months in humans, which necessitates a continual production and removal of cells). Murray’s findings suggest that “. . . the flow of blood past any section shall everywhere bear the same relation to the cube of the radius of the vessel at that point.” Recently, it has been shown that Murray’s ideas are consistent with the observation that the lumen of an artery appears to be governed, in part, so as to keep the wall shear stress at a preferred value – for a simple, steady, incompressible, laminar flow of a Newtonian fluid in a circular tube, the wall shear stress is proportional to the volumetric flow rate and inversely proportional to the cube of the radius [7.19]. Clearly, optimization approaches should be given increased attention, particularly with regard to the design of useful biomechanical experiments and the reduction of the associated data. Perhaps best known for inventing the Turing machine for computing, Turing also published a seminal paper on biological growth [7.51]. Briefly, he was interested in mathematically modeling morphogenesis, that is, the development of the form, or shape, of an organism. In his words, he sought to understand the mechanism by which “genes . . . may determine the anatomical structure of the resulting organism.” Turing recognized the importance of both mechanical and chemical stimuli in controlling morphogenesis, but he focused on the chemical aspects, especially the reaction kinetics and diffusion of morphogens, substances such as growth factors that regulate the development of form. For example, he postulated linear reaction– diffusion equations of the form ∂M1 = a (M1 − h) + b (M2 − g) + D1 ∇ 2 M1 , ∂t ∂M2 = c (M1 − h) + d (M2 − g) + D2 ∇ 2 M2 , ∂t

(7.15)

where M1 and M2 are concentrations of two morphogens, a, b, c, and d are reaction rates, and D1 and D2 are diffusivities; h and g are equilibrium values of M1 and M2 , and t represents time during morphogenesis. It was assumed that the local concentration of a particular morphogen tracked the local production or removal of tissue. Numerical examples revealed that solutions to such systems of equations could “develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances.” These solutions were proposed as possible descriptors of the morphogenesis. It was not until the

Biological Soft Tissues

1980s, however, that there was an increased interest in the use of reaction–diffusion models to study biological growth and remodeling, which now includes studies of wound healing, tumor growth, angiogenesis, and tissue engineering in addition to morphogenesis [7.52–56]. As noted by Turing, mechanics clearly plays an important role in such growth and remodeling, thus it is not surprising that there has been a trend to embed the reaction–diffusion framework within tissue mechanics (albeit often within the context of linearized elasticity or viscoelasticity). For example, Barocas and Tranquillo [7.53] suggested that reaction–diffusion models for spatial–temporal information on cells could be combined with a mixture theory representation of a tissue consisting of a fluid constituent and solid network. In this way, they studied mechanically stimulated cell migration, which is thought to be an early step towards mechanically stimulated changes in deposition of structural proteins. See the original paper for details. In summary, there has been significant attention to modeling the production, diffusion, and half-life of a host of molecules (growth factors, cytokines, proteases) and how they affect cell migration, mitosis, apoptosis, and the synthesis and reorientation of extracellular matrix. In some cases the reaction–diffusion models are used in isolation, but there has been a move towards combining such relations with those of mechanics (mass and momentum balance). Yet because of the lack of attention to the finite-strain kinematics and nonlinear material behavior characteristic of soft tissues [7.57], there is a pressing need for increased generalization if this approach is to become truly predictive. Moreover, there is a pressing need for additional biomechanical experiments throughout the evolution of the geometry and properties so that appropriate kinetic equations can be developed.

7.3.2 Kinematic Growth

being conserved, the overall mass density appeared to remain constant, thus focusing attention on changes in volume.) Tozeren and Skalak [7.59] suggested further that finite-strain growth and remodeling in a soft tissue (idealized as fibrous networks) could be described, in part, by considering that “The stress-free lengths of the fibers composing the network are not fixed as in an inert elastic solid, but are assumed to evolve as a result of growth and stress adaptation. Similarly, the topology of the fiber network may also evolve under the application of stress.” One of the remarkable aspects of Skalak’s work is that he postulated that, if differential growth is incompatible, then continuity of material may be restored via residual stresses. Residual stresses in arteries were reported soon thereafter (independently in 1983 by Vaishnav and Fung; see the discussion in [7.19]), and shown to affect dramatically the computed stress field in the arterial wall. The basic ideas of incompatible kinematic growth, residual stress, and evolving material symmetries and stress-free configurations were seminal contributions. Rodriguez et al. [7.60] built upon these ideas and put them into tensorial form – the approach was called “finite volumetric growth”, which is now described in brief. The primary assumption is that one models volumetric growth through a growth tensor Fg , which describes changes between two fictitious stress-free configurations: the original body is imagined to be fictitiously cut into small stress-free pieces, each of which is allowed to grow separately via Fg , with det Fg = 1. Because these growths need not be compatible, internal forces are often needed to assemble the grown pieces, via Fa , into a continuous configuration. This, in general, produces residual stresses, which are now known to exist in many soft tissues besides arteries. The formulation is completed by considering elastic deformations, via Fe , from the intact but residually stressed traction-free configuration to a current configuration that is induced by external mechanical loads. The initial–boundary value problem is solved by introducing a constitutive relation for the stress response to the deformation Fe Fa , which is often assumed to be isochoric and of the Fung type, plus a relation for the evolution of the stress-free configuration via Fg (actually Ug since the rotation Rg is assumed to be I). Thus, growth is assumed to occur in stress-free configurations and typically not to affect material properties. See, too, Lubarda and Hoger [7.61], who consider special cases of transversely isotropic and orthotropic growth. Among others, Taber [7.62] and Rachev et al. [7.63] independently embraced the concept of kinematic

179

Part A 7.3

In a seminal paper, Skalak [7.58] offered an approach very different from the reaction–diffusion approach, one that brought the analysis of biological growth within the purview of large-deformation continuum mechanics. He suggested that “any finite growth or change of form may be regarded as the integrated result of differential growth, i. e. growth of the infinitesimal elements making up the animal and plant.” His primary goal, therefore, was to “form a framework within which growth and deformation may be discussed in regard to the kinematics involved.” (Note: Although it was realized that mass may change over time, rather than

7.3 Growth and Remodeling – A New Frontier

180

Part A

Solid Mechanics Topics

growth and solved initial–boundary value problems relating to cardiac development, arterial remodeling in hypertension and altered flow, and aortic development. For the purposes of discussion, briefly consider the model of aortic growth by Taber [7.62]. The aortic wall was assumed to have material properties (given by Fung’s exponential relation) that remained constant during growth, which in turn was modeled via additional constitutive relations for time rates of change of the growth tensor Fg = diag[λgr , λgθ , 1], namely dλgr 1 ge  t¯θθ (s) − t¯θθ , = dt Tr dλgθ 1 ge  t¯θθ (s) − t¯θθ = dt Tθ 1 ge  + τ¯w (s) − τ¯w eα(R/Ri −1) , Tτ

(7.16)

where Ti are time constants, t¯jj and τ¯w are mean values of wall stress and flow-induced wall shear stress, respectively, the superscript ‘ge’ denotes growthequilibrium, α is a parameter that reflects the intensity of effects, at any undeformed radial location R, of growth factors produced by the cells that line the arterial wall and interact directly with the blood. Clearly, growth (i. e., the time rate of change of the stress-free configuration in multiple directions) continues until the stresses return to their preferred or equilibrium values. Albeit not in the context of vascular mechanics, Klisch et al. [7.64] suggested further that the concept of volumetric growth could be incorporated within the theory of mixtures (with solid constituents k = 1, . . . , n and fluid constituent f ) to describe growth in cartilage. The deformation of constituent k was given by Fk = Fke Fak Fkg , where Fkg was related to a scalar mass growth function m k via t

Part A 7.3

det Fkg (t) = exp

m k dτ,

(7.17)

0

and mass balance requires dk ρ k + ρk div vk = ρk m k , dt dfρf + ρ f div v f = 0 . dt

(7.18)

This theory requires evolution equations for Fkg (or similarly, m k ), which the authors suggested could depend on the stresses, deformations, growth of other constituents, etc., as well as constitutive relations for the mass growth

function. It is clear that such a function could be related to the reaction–diffusion framework of Turing, and thus chemomechanical stimulation of growth. Although it is reasonable, in principle, to consider a full mixture theory given that so many different constituents contribute to the overall growth and structural stability of a tissue or organ, it is very difficult in practice to prescribe appropriate partial traction boundary conditions and very difficult to identify the requisite constitutive relations for momentum exchanges. Indeed, it is not clear that there is a need to model such detail, such as the momentum exchange between different proteins that comprise the extracellular matrix or between the extracellular matrix and a migrating cell, for example, particularly given that such migration involves complex chemical reactions (e.g., degradation of proteins at the leading edge of the cell) not just mechanical interactions. For these and other reasons, let us now consider an alternative mixture theory.

7.3.3 Constrained Mixture Approach Although the theory of kinematic growth yields many reasonable predictions, we have suggested that it models consequences of growth and remodeling (G&R), not the processes by which they occur. G&R necessarily occur in stressed, not fictitious stress-free, configurations, and they occur via the production, removal, and organization of different constituents; moreover, G&R need not restore stresses exactly to homeostatic values. Hence, we introduced a conceptually different approach to model G&R, one that is based on tracking the turnover of individual constituents in stressed configurations [7.65]. Here, we illustrate this approach for 2-D (membrane-like) tissues [7.66]. Briefly, let a soft tissue consist of multiple types of structurally important constituents, each of which must deform with the overall tissue but may have individual material properties and associated individual natural (i. e., stress-free) configurations that may evolve over time. We employ the concept of a constrained mixture wherein constituents deform together in current configurations and tacitly assume that they coexist within neighborhoods over which a local macroscopic homogenization would be meaningful. Specifically, not only may different constituents coexist at a point of interest, the same type of constituent produced at different instants can also coexist. Because of our focus on thin soft tissues consisting primarily of fibrillar collagen, one can consider a constitutive relation for the principal Cauchy stress resultants for the tissue (i. e., constrained mixture) of the

Biological Soft Tissues

form 1 ∂wk T1 (t) = T10 (t) + , λ2 (t) ∂λ1 (t) k

1 ∂wk , T2 (t) = T20 (t) + λ1 (t) ∂λ2 (t)

(7.19)

k

where Ti0 (i = 1, 2) represent contributions by an amorphous matrix (e.g., elastin-dominated or synthetic/reconstituted in a tissue equivalent) that can degrade but cannot be produced, λi are measurable principal stretches that are experienced by the tissue, and wk is a stored energy function for collagen family k, which may be produced or removed over time. Note, too, that   2  2 k (7.20) λ (t) = λ1 cos α0k + λ2 sin α0k are stretches experienced by fibers in collagen family k relative to a common mixture reference configuration, with α0k the angle between fiber family k and the 1 coordinate axis. To account for the deposition of new collagen fibers within stressed configurations, however, we further assume the existence of a preferred (i. e., homeostatic) deposition stretch G kh , whereby the stretch experienced by fiber family k, relative to its unique natural configuration, can be shown to be λkn(τ) (t) = G kh λk (t)/λk (τ) ,

(7.21)

with t the current time and τ the past time at which family k was produced. Finally, to account for continual production and removal, let the constituent stored energies be [7.66] wk (t) =

0

(7.22)

where ρ is the mixture mass density, M k (0) is the 2-D mass density of constituent k at time 0, when G&R commences, Q k (t) ∈ [0, 1] is the fraction of constituent k that was produced before time 0 but survives to the current time t > 0, m k is the current mass density production of constituent k, W k λkn(τ) (t) is the strain energy function for a fiber family relative to its unique natural configuration, and q k is

181

an associated survival function describing that fraction of constituent k that was produced at time τ (after time 0) and survives to the current time t. Hence, consistent with (7.4), the principal Cauchy stress resultants of the constituents that may turnover are  k M (0)Q k (t)G kh ∂W k ∂λk (t) 1 T1k (t) = λ2 (t) ρλk (0) ∂λkn(τ) (t) ∂λ1 (t) t

m k (τ)q k (t − τ)G kh ρλk (τ) 0  ∂W k ∂λk (t) × k dτ , ∂λn(τ) (t) ∂λ1 (t)  k M (0)Q k (t)G kh ∂W k ∂λk (t) 1 T2k (t) = λ1 (t) ρλk (0) ∂λkn(τ) (t) ∂λ2 (t) +

t

m k (τ)q k (t − τ)G kh ρλk (τ) 0  ∂W k ∂λk (t) × k dτ . ∂λn(τ) (t) ∂λ2 (t) +

(7.23)

As in most other applications of biomechanics, the key challenge therefore is to identify specific functional forms for the requisite constitutive relations, particularly the individual mass density productions, the survival functions, and the strain energy functions for the individual fibers, not to mention relations for muscle contractility and its adaptation. Finally, there is also a need to prescribe the alignment of newly produced fibers, not just their rate of production and removal. These, too, will require contributions from experimental biomechanics. Illustrative simulations are found nonetheless in the original paper [7.66], which show that stable versus unstable growth and remodeling can result, depending on the choice of constitutive relation. Given that the biomechanics of growth and remodeling is still in its infancy, it is not yet clear which approaches will ultimately prove most useful. The interested reader is thus referred to the following as examples of alternate approaches [7.67–71]. Finally, it is important to emphasize that, regardless of the specific theoretical framework, the most pressing need at present is an experimental program wherein the evolving mechanical properties and geometries of cells, tissues, and organs are quantified as a function of time during adaptations (or maladaptations) in response to altered mechanical loading, and that such information must be

Part A 7.3

  M k (0) k Q (t)W k λkn(0) (t) ρ t k   m (τ) k + q (t − τ)W k λkn(τ) (t) dτ , ρ

7.3 Growth and Remodeling – A New Frontier

182

Part A

Solid Mechanics Topics

correlated with changes in the rates of production and removal of structurally significant constituents, which in turn depend on the rates of production, removal, and diffusion of growth factors, proteases, and related substances. Clearly, biomechanics is not simply mechanics

applied to biology; it is the extension, development, and application of mechanics to problems in biology and medicine, which depends on theoretically motivated experimental studies that seek to identify new classes of constitutive relations.

7.4 Closure In summary, much has been accomplished in our quest to quantify the biomechanical behavior of soft tissues, yet much remains to be learned. Fortunately, continuing technological developments necessary for advancing experimental biomechanics (e.g., optical tweezers, atomic force and multiphoton microscopes, tissue bioreactors) combined with traditional methods of testing (e.g., computer-controlled in-plane biaxial testing of planar specimens, inflation and extension testing of tubular specimens, and inflation testing of membranous specimens; see [7.19] Chap. 5) as well as continuing advances in theoretical and computational mechanics are helping us to probe deeper into the mechanobiology and biomechanics every day. Thus,

both the potential and the promise of engineering contributions have never been greater. It is hoped, therefore, that this chapter provided some background, and especially some motivation, to contribute to this important field. The interested reader is also referred to a number of related books, listed in the Bibliography, and encouraged to consult archival papers that can be found in many journals, including Biomechanics and Modeling in Mechanobiology, the Journal of Biomechanics, and the Journal of Biomechanical Engineering. Indeed, an excellent electronic search engine is NIH PubMed, which can be found via the National Institutes of Health web site (www.nih.gov); it will serve us well as we continue to build on past achievements.

7.5 Further Reading

Part A 7.5

Given the depth and breadth of the knowledge base in biomedical research, no one person can begin to gain all of the needed expertise. Hence, biomechanical research requires teams consisting of experts in mechanics (theoretical, experimental, and computational) as well as biology, physiology, pathology, and clinical practice. Nevertheless, bioengineers must have a basic understanding of the biological concepts. I recommend, therefore, that the serious bioengineer keep nearby books on (i) molecular and cell biology, (ii) histology, and (iii) medical definitions. Below, I list some books that will serve the reader well.

• • • •

H. Abe, K. Hayashi, M. Sato: Data Book on Mechanical Properties of Living Cells, Tissues, and Organs (Springer, New York 1996) Dorland’s Illustrated Medical Dictionary (Saunders, Philadelphia 1988) S.C. Cowin, J.D. Humphrey: Cardiovascular Soft Tissue Mechanics (Kluwer Academic, Dordrecht 2001) S.C. Cowin, S.B. Doty: Tissue Mechanics (Springer, New York 2007)

• • • • • • • • •

D. Fawcett: A Textbook of Histology (Saunders, Philadelphia 1986) Y.C. Fung: Biomechanics: Mechanical Properties of Living Tissues (Springer, New York 1993) F. Guilak, D.L. Butler, S.A. Goldstein, D.J. Mooney: Functional Tissue Engineering (Springer, New York 2003) G.A. Holzapfel, R.W. Ogden: Biomechanics of Soft Tissue in Cardiovascular Systems (Springer, Vienna 2003) G.A. Holzapfel, R.W. Ogden: Mechanics of Biological Tissue (Springer, Berlin, Heidelberg 2006) J.D. Humphrey, S.L. Delange: An Introduction to Biomechanics (Springer, New York 2004) V.C. Mow, R.M. Hochmuth, F. Guilak, R. TransSon-Tay: Cell Mechanics and Cellular Engineering (Springer, New York 1994) W.M. Saltzman: Tissue Engineering (Oxford Univ Press, Oxford 2004) L.A. Taber: Nonlinear Theory of Elasticity: Applications to Biomechanics (World Scientific, Singapore 2004)

Biological Soft Tissues

References

183

References 7.1

7.2

7.3

7.4 7.5 7.6

7.7

7.8

7.9

7.10

7.11

7.12 7.13

7.14

7.16 7.17

7.18

7.19

7.20

7.21 7.22

7.23

7.24

7.25

7.26 7.27

7.28

7.29

7.30

7.31

7.32

7.33

7.34

7.35

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7.48 7.49 7.50

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in compression: Theory and experiments, ASME J. Biomech. Eng. 102, 73–84 (1980) W.M. Lai, V.C. Mow, W. Zhu: Constitutive modeling of articular cartilage and biomacromolecular solutions, J. Biomech. Eng. 115, 474–480 (1993) V.C. Mow, G.A. Ateshian, R.L. Spilker: Biomechanics of diarthroidal joints: A review of twenty years of progress, J. Biomech. Eng. 115, 460–473 (1993) J.M. Huyghe, G.B. Houben, M.R. Drost, C.C. van Donkelaar: An ionised/non-ionised dual porosity model of intervertebral disc tissue experimental quantification of parameters, Biomech. Model. Mechanobiol. 2, 3–20 (2003) R.L. Spilker, J.K. Suh, M.E. Vermilyea, T.A. Maxian: Alternate Hybrid, Mixed, and Penalty Finite Element Formulations for the Biphasic Model of Soft Hydrated Tissues,. In: Biomechanics of Diarthrodial Joints, ed. by V.C. Mow, A. Ratcliffe, S.L.Y. Woo (Springer, New York 1990) pp. 401–436 B.R. Simon, M.V. Kaufman, M.A. McAfee, A.L. Baldwin: Finite element models for arterial wall mechanics, J. Biomech. Eng. 115, 489–496 (1993) R.A. Reynolds, J.D. Humphrey: Steady diffusion within a finitely extended mixture slab, Math. Mech. Solids 3, 147–167 (1998) G.I. Zahalak, B. de Laborderie, J.M. Guccione: The effects of cross-fiber deformation on axial fiber stress in myocardium, J. Biomech. Eng. 121, 376–385 (1999) A. Rachev, K. Hayashi: Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries, Ann. Biomed. Eng. 27, 459–468 (1999) P.J. Hunter, A.D. McCulloch, H.E.D.J. ter Keurs: Modelling the mechanical properties of cardiac muscle, Prog. Biophys. Mol. Biol. 69, 289–331 (1998) P.J. Hunter, P. Kohl, D. Noble: Integrative models of the heart: Achievements and limitations, Philos. Trans. R. Soc. London A359, 1049–1054 (2001) J.D. Humphrey: Continuum thermomechanics and the clinical treatment of disease and injury, Appl. Mech. Rev. 56, 231–260 (2003) K.R. Diller, T.P. Ryan: Heat transfer in living systems: Current opportunities, J. Heat. Transf. 120, 810–829 (1998) S.C. Cowin: Bone stress adaptation models, J. Biomech. Eng. 115, 528–533 (1993) D.R. Carter, G.S. Beaupré: Skeletal Function and Form (Cambridge Univ. Press, Cambridge 2001) C.D. Murray: The physiological principle of minimum work: I. The vascular system and the cost of blood volume, Proc. Natl. Acad. Sci. 12, 207–214 (1926) A.M. Turing: The chemical basis of morphogenesis, Proc. R. Soc. London B 237, 37–72 (1952)

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7.57

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7.61

7.62

7.63

7.64

7.65

7.66

7.67

R.T. Tranquillo, J.D. Murray: Continuum model of fibroblast-driven wound contraction: Inflammationmediation, J. Theor. Biol. 158, 135–172 (1992) V.H. Barocas, R.T. Tranquillo: An anisotropic biphasic theory of tissue equivalent mechanics: The interplay among cell traction, fibrillar network, fibril alignment, and cell contact guidance, J. Biomech. Eng. 119, 137–145 (1997) L. Olsen, P.K. Maini, J.A. Sherratt, J. Dallon: Mathematical modelling of anisotropy in fibrous connective tissue, Math. Biosci. 158, 145–170 (1999) N. Bellomo, L. Preziosi: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comput. Model. 32, 413–452 (2000) A.F. Jones, H.M. Byrne, J.S. Gibson, J.W. Dodd: A mathematical model of the stress induced during avascular tumour growth, J. Math. Biol. 40, 473–499 (2000) L.A. Taber: Biomechanics of growth, remodeling, and morphogenesis, Appl. Mech. Rev. 48, 487–545 (1995) R. Skalak: Growth as a finite displacement field, Proc. IUTAM Symposium on Finite Elasticity, ed. by D.E. Carlson, R.T. Shield (Martinus Nijhoff, The Hague 1981) pp. 347–355 A. Tozeren, R. Skalak: Interaction of stress and growth in a fibrous tissue, J. Theor. Biol. 130, 337– 350 (1988) E.K. Rodriguez, A. Hoger, A.D. McCulloch: Stressdependent finite growth in soft elastic tissues, J. Biomech. 27, 455–467 (1994) V.A. Lubarda, A. Hoger: On the mechanics of solids with a growing mass, Int. J. Solid Struct. 39, 4627– 4664 (2002) L.A. Taber: A model for aortic growth based on fluid shear and fiber stresses, ASME J. Biomech. Eng. 120, 348–354 (1998) A. Rachev, N. Stergiopulos, J.-J. Meister: A model for geometric and mechanical adaptation of arteries to sustained hypertension, ASME J. Biomech. Eng. 120, 9–17 (1998) S.M. Klisch, T.J. van Dyke, A. Hoger: A theory of volumetric growth for compressible elastic biological materials, Math. Mech. Solid 6, 551–575 (2001) J.D. Humphrey, K.R. Rajagopal: A constrained mixture model for growth and remodeling of soft tissues, Math. Model. Meth. Appl. Sci. 12, 407–430 (2002) S. Baek, K.R. Rajagopal, J.D. Humphrey: A theoretical model of enlarging intracranial fusiform aneurysms, J. Biomech. Eng. 128, 142–149 (2006) R.A. Boerboom, N.J.B. Driessen, C.V.C. Bouten, J.M. Huyghe, F.P.T. Baaijens: Finite element model of mechanically induced collagen fiber synthesis and degradation in the aortic valve, Ann. Biomed. Eng. 31, 1040–1053 (2003)

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Part A 7

187

Electrochemo 8. Electrochemomechanics of Ionic Polymer–Metal Composites

Sia Nemat-Nasser

The ionomeric polymer–metal composites (IPMCs) consist of polyelectrolyte membranes, with metal electrodes plated on both faces and neutralized with an amount of counterions, balancing the charge of anions covalently fixed to the membrane. IPMCs in the solvated state form soft actuators and sensors; they are sometimes referred to as artificial muscles. Here, we examine the nanoscale chemoelectromechanical mechanisms that underpin the macroscale actuation and sensing of IPMCs, as well as some of their electromechanical properties.

8.2.2 Pressure in Clusters ...................... 192 8.2.3 Membrane Stiffness ...................... 192 8.2.4 IPMC Stiffness .............................. 192 8.3 Voltage-Induced Cation Distribution ...... 193 8.3.1 Equilibrium Cation Distribution ...... 194 8.4 Nanomechanics of Actuation ................. 8.4.1 Cluster Pressure Change Due to Cation Migration ................ 8.4.2 Cluster Solvent Uptake Due to Cation Migration ................ 8.4.3 Voltage-Induced Actuation............

195 195 196 197

8.5 Experimental Verification ...................... 197 8.5.1 Evaluation of Basic Physical Properties ........... 197 8.5.2 Experimental Verification .............. 198

8.2 Stiffness Versus Solvation ...................... 191 8.2.1 The Stress Field in the Backbone Polymer .............. 191

8.6 Potential Applications ........................... 199 References .................................................. 199

Polyelectrolytes are polymers that carry covalentlybound positive or negative charges. They occur naturally, such as deoxyribonucleic acid (DNA) and ribonucleic acid (RNA), or they have been manuR factured for various applications, such as Nafion R  or Flemion , which consist of three-dimensionally structured backbone perfluorinated copolymers of polytetrafluoroethylene, having regularly spaced long perfluorovinyl ether pendant side-chains that terminate in ionic sulfonate (Nafion) or carboxylate (Flemion) groups. The resulting Nafion or Flemion membranes are permeable to water or other polar solvents and cations, while they are impermeable to anions. The ionomeric polymer–metal composites (IPMCs) consist of polyelectrolyte membranes, about 200 μm thick, with metal electrodes (5–10 μm thick) plated on both faces [8.1] (Fig. 8.1). The polyelectrolyte matrix is neutralized with an amount of counterions, balancing

the charge of anions covalently fixed to the membrane. When an IPMC in the solvated (e.g., hydrated) state is stimulated with a suddenly applied small (1–3 V, depending on the solvent) step potential, both the fixed anions and mobile counterions are subjected to an electric field, with the counterions being able to diffuse toward one of the electrodes. As a result, the composite undergoes an initial fast bending deformation, followed by a slow relaxation, either in the same or in the opposite direction, depending on the composition of the backbone ionomer, and the nature of the counterion and the solvent. The magnitude and speed of the initial fast deflection also depend on the same factors, as well as on the structure of the electrodes, and other conditions (e.g., the time variation of the imposed voltage). IPMCs that are made from Nafion and are neutralized with alkali metals or with alkyl-ammonium cations (except for tetrabutylammonium, TBA+ ), invariably first bend to-

Microstructure and Actuation................. 8.1.1 Composition ................................ 8.1.2 Cluster Size .................................. 8.1.3 Actuation ....................................

Part A 8

188 188 189 191

8.1

188

Part A

Solid Mechanics Topics

a)

b)

Au plating

Fig. 8.1a,b Cross section of (a) a Pt/Au-plated Nafion-117 membrane at electrode region; the length of the bar is 408 nm; and (b) an Auplated Flemion at electrode region; the length of the bar is 1 μm

Au plating

Platinum particles

Au 408 nm

Flemion

wards the anode under a step direct current (DC), and then relax towards the cathode while the applied voltage is being maintained, often moving beyond their starting position. In this case, the motion towards the anode can be eliminated by slowly increasing the applied potential at a suitable rate. For Flemion-based IPMCs, on the other hand, the initial fast bending and the subsequent relaxation are both towards the anode, for all counterions that have been considered. With TBA+ as the counterion, no noticeable relaxation towards the cathode has been recorded for either Nafionor Flemion-based IPMCs. Under an alternating electric

1μm

potential, cantilevered strips of IPMCs perform bending oscillations at the frequency of the applied voltage, usually no more than a few to a few tens of hertz, depending on the solvent. When an IPMC membrane is suddenly bent, a small voltage of the order of millivolts is produced across its faces. Hence, IPMCs of this kind can serve as soft actuators and sensors. They are sometimes referred to as artificial muscles [8.2, 3]. In this chapter, we examine the nanoscale chemoelectromechanical factors that underpin the macroscale actuation and sensing of IPMCs, as well as some of their electromechanical properties.

8.1 Microstructure and Actuation 8.1.1 Composition The Nafion-based IPMC, in the dry state, is about 180 μm thick and the Flemion-based one is about 160 μm thick (see [8.4–7] for further information on IPMC manufacturing). Samples consist of 1. backbone perfluorinated copolymer of polytetrafluoroethylene with perfluorinated vinyl ether sulfonate pendants for Nafion-based and perfluorinated b) Nanosized

a)

interconnected clusters

Cracked metal overlayer

Metal coating

Part A 8.1

Ionic polymer Naflon or Flamion

0.2 mm

Cluster group

Fig. 8.2 (a) A schematic representation of an IPMC and (b) a transmission electron microscopy (TEM) photo of the cluster structure (see Fig. 8.3 for more detail)

propyl ether carboxylate pendants for Flemionbased IPMCs, forming interconnected nanoscale clusters ([8.8]; primary physical data for fluorinated ionomers have been summarized in [8.9]) 2. electrodes, which in Nafion-based IPMCs consist of 3–10 nm-diameter platinum particles, distributed mainly within a 10–20 μm depth of both faces of the membrane, and usually covered with about 1 μmthick gold plating to improve surface conductivity, while in Flemion-based IPMCs, the electrodes are gold, with a dendritic structure, as shown in Fig. 8.1 3. neutralizing cations 4. a solvent Figure 8.2 shows a schematic representation of a typical IPMC, including a photograph of the nanostructure of the ionomer (Nafion, in this case), and a sketch of ionic polymer with metal coating. Figure 8.3 shows some additional details of a Nafion-based IPMC. The dark spots within the inset are the left-over platinum crystals. The ion exchange capacity (IEC) of an ionomer represents the amount of sulfonate (in Nafion) and carboxylate (in Flemion) group in the material, mea-

Electrochemomechanics of Ionic Polymer–Metal Composites

a) Metal overlayer

1μm

8.1 Microstructure and Actuation

b)

189

c)

O O

O

O

O O

Platinum diffusion zone (0 –25 μm)

Fig. 8.4 (a) Chemical structure of 18-Crown-6 (each node is CH2 ); (b) Na+ ion; and (c) K+ ion within a macrocyclic 18-Crown-6

ligand

8.1.2 Cluster Size Left-over platinum particles 20 nm

Fig. 8.3 Near-surface structure of an IPMC; the lower in-

set indicates the size of a typical cluster in Nafion

2  3NkT 3 NEWion Uela = − aI , ρ ∗ NA h 2  ρB + wρs ρ∗ = , 1+w

(8.1)

where N is the number of dipoles inside a typical cluster, k is Boltzmann’s constant, T is the temperature, h 2  is the mean end-to-end chain length, ρ∗ is the effective density of the solvated membrane, and NA is Avogadro’s number (6.023 × 1023 ). The electrostatic en-

Part A 8.1

sured in moles per unit dry polymer mass. The dry bare ionomer equivalent weight (EW) is defined as the weight in grams of dry ionomer per mole of its anion. The ion exchange capacity and the equivalent weight of Nafion are 0.91 meq/g and 1.100 g/mol, and those of Flemion are 1.44 meq/g and 694.4 g/mol, respectively. For neutralizing counterions, we have used Li+ , + Na , K+ , Rb+ , Cs+ , Mg2+ , and Al3+ , as well as alkylammonium cations TMA+ , TEA+ , TPA+ , and TBA+ . The properties of the bare ionomer, as well as those of the corresponding IPMC, change with the cation type for the same membrane and solvent. In addition to water, ethylene glycol, glycerol, and crown ethers have been used as solvents. Ethylene glycol or 1,2ethanediol (C2 H6 O2 ) is an organic polar solvent that can be used over a wide range of temperatures. Glycerol, or 1,2,3-propanetriol (C3 H8 O3 ) is another polar solvent with high viscosity (about 1000 times the viscosity of water). Crown ethers are cyclic oligomers of ethylene glycol that serve as macrocyclic ligands to surround and transport cations (Fig. 8.4). The required crown ether depends on the size of the ion. The 12-Crown-4 (12CR4) matches Li+ , 15-Crown-5 (15CR5) matches K+ , and 18-Crown-6 matched Na+ and K+ . For example, an 18-Crown-6 (18CR6) molecule has a cavity of 2.7 Å and is suitable for potassium ions of 2.66 Å diameter. A schematic configuration of this crown with sodium and potassium ions is shown in Fig. 8.4.

X-ray scanning of the Nafion membranes [8.10] has shown that, in the process of solvent absorption, hydrophilic regions consisting of clusters are formed within the membrane. Hydrophilicity and hydrophobicity are generally terms used for affinity or lack of affinity toward the polar molecule of water. In the present work we use these terms for interaction toward any polar solvent (e.g., ethylene–glycol). Cluster formation is promoted by the aggregation of hydrophilic ionic sulfonate groups located at the terminuses of vinyl ether sulfonate pendants of the polytetrafluoroethylene chain. While these regions are hydrophilic, the membrane backbone is hydrophobic and it is believed that the motion of the solvent takes place among these clusters via the connecting channels. The characteristics of these clusters and channels are important factors in IPMC behavior. The size of the solvated cluster radius aI depends on the cation form, the type of solvent used, and the amount of solvation. The average cluster size can be calculated by minimizing the free energy of the cluster formation with respect to the cluster size. The total energy for cluster formation consists of an elastic Uela , an electrostatic Uele , and a surface Usur component. The elastic energy is given by [8.11]

190

Part A

Solid Mechanics Topics

a13 (nm3) 15

10

5

0

Model x-ray 0

0.5

1

1.5

2

2.5

3 η 105

Fig. 8.5 Cluster size in Nafion membranes with different solvent uptakes

ergy is given by Uele = −g

N 2 m2 , 4πκe aI3

where n is the number of clusters present in the membrane. Minimizing  this energy with respect to cluster tot = 0 , gives the optimum cluster size at size, ∂U ∂aI which the free energy of the ionomer is minimum. In this manner Li and Nemat-Nasser [8.11] have obtained ⎡ γ h 2 EWion (w + ΔV ) aI = ⎣ 2RT ρB   −2 ⎤1/3 4πρ B ⎦ × 1− 3 ∗ , 3ρ (w + ΔV ) NA Vi ρd , (8.5) EWion where Vi is the volume of a single ion exchange site. Assuming that h 2  = EWion β [8.12], it can be seen that, ΔV =

(8.2)

aI3 =

  −2 EW2ion (w + ΔV ) 4πρB 3 η= 1− . ρB 3ρ∗ (w + ΔV )

where g is a geometric factor, m is the dipole moment, and κe is the effective permittivity within the cluster. The surface energy can be expressed as

(8.6)

Usur = 4πaI2 γ ,

(8.3)

where γ is the surface energy density of the cluster. Therefore, the total energy due to the presence of clusters in the ionomer is given by Utot = n(Uele + Uela + Usur ) , a)

γβ η, 2RT

(8.4)

(c) t = 4m 45s

Figure 8.5 shows the variation of the cluster size (aI3 in nm3 ) for different solvent uptakes. Data from Nafion-based IPMC samples with different cations and different solvent uptakes are considered. The model is compared with the experimental results on the cluster size based on x-ray scanning, shown as circles

b)

c) –

(d) t = 6 m 27s –

R

+

t = 3ms t = 210ms

+

t = 5ms

t= 129ms

1cm

(a) t = 0 (b) t = 0.6 s

t = 10ms

t = 54ms t = 36ms

t = 15ms t = 24ms

Fig. 8.6 (a) A Nafion-based sample in the thallium (I) ion form is hydrated and a 1 V DC signal is suddenly applied

Part A 8.1

and maintained during the first 5 min, after which the voltage is removed and the two electrodes are shorted. Initial fast bend toward the bottom ((a) to (b), anode) occurs during the first 0.6 s, followed by a long relaxation upward (towards the cathode (c)) over 4.75 min. Upon shorting, the sample displays a fast bend in the same upward direction (not shown), followed by a slow downward relaxation (to (d)) during the next 1.75 min. (b) A Nafion-based sample in the sodium ion form is solvated with glycerol, and a 2 V DC signal is suddenly applied and maintained. It deforms into a perfect circle, but its qualitative response is the same. (c) A Flemion-based sample in tetrabutylammonium ion form is hydrated, and a 3 V DC signal is suddenly applied and maintained, resulting in continuous bending towards the anode (no back relaxation)

Electrochemomechanics of Ionic Polymer–Metal Composites

in Fig. 8.5 for a Nafion ionomer in various cation forms and with water as the solvent [8.10]. We have set β = 1.547, γ = 0.15, and Vi = 68 × 10−24 cm3 to calculate the cluster size [8.12].

8.1.3 Actuation A Nafion-based IPMC sample in the solvated state performs an oscillatory bending motion when an alternating voltage is imposed across its faces, and it produces a voltage when suddenly bent. When the same strip is subjected to a suddenly imposed and sustained constant voltage (DC) across its faces, an initial fast displacement (towards the anode) is gen-

8.2 Stiffness Versus Solvation

191

erally followed by a slow relaxation in the reverse direction (towards the cathode). If the two faces of the strip are then shorted during this slow relaxation towards the cathode, a sudden fast motion in the same direction (towards the cathode) occurs, followed by a slow relaxation in the opposite direction (towards the anode). Figure 8.6 illustrates these processes for a hydrated Tl+ -form Nafion-based IPMC (left), Na+ -form with glycerol as solvent (middle), and hydrated Flemion-based in TBA+ -form (right), under 1, 2, and 3 V DC, respectively. The magnitudes of the fast motion and the relaxation that follows the fast motion change with the corresponding cation and the solvent.

8.2 Stiffness Versus Solvation To model the actuation of the IPMC samples in terms of the chemoelectromechanical characteristics of the backbone ionomer, the electrodes, the neutralizing cation, the solvent, and the level of solvation, it is first necessary to model the stiffness of the corresponding samples. This is discussed in the present section. A dry sample of a bare polymer or an IPMC placed in a solvent bath absorbs solvent until the resulting pressure within its clusters is balanced by the elastic stresses that are consequently developed within its backbone polymer membrane. From this observation the stiffness of the membrane can be estimated as a function of the solvent uptake for various cations. Consider first the balance of the cluster pressure and the elastic stresses for the bare polymer (no metal plating) and then use the results to calculate the stiffness of the corresponding IPMC by including the effect of the added metal electrodes. The procedure also provides a way of estimating many of the nanostructural parameters that are needed for the modeling of the actuation of the IPMCs.

8.2.1 The Stress Field in the Backbone Polymer

σr (r0 ) = − p0 + K [(r0 /a0 )−3 (w/w0 − 1) + 1]−4/3 , σθ (r0 ) = σϕ (r0 ) = − p0 + K [(r0 /a0 )−3 (w/w0 − 1) + 1]2/3 ,

(8.7)

where r0 measures the initial radial length from the center of the cluster, and w0 is the initial (dry) volume fraction of the voids. Theeffective elastic resistance of the (homogenized solvated) membrane balances the cluster’s pressure pc which is produced by the combined osmotic and electrostatic forces within the cluster.

Part A 8.2

The stresses within the backbone polymer may be estimated by modeling the polymer matrix as an incompressible elastic material [8.13, 14]. Here, it will prove adequate to consider a neo-Hookean model for the matrix material. In this model, the principal stresses σI are related to the principal stretches λI by σI = − p0 + K λ2I ,

where p0 is an undetermined parameter (pressure) to be calculated from the boundary data and K is an effective stiffness, approximately equal to a third of the overall Young’s modulus. The aim is to calculate K and p0 as functions of the solvent uptake w for various ionform membranes. To this end, examine the deformation of a unit cell of the solvated polyelectrolyte (bare membrane) by considering a spherical cavity of initial (i. e., dry state) radius a0 (representing a typical cluster), embedded at the center of a spherical matrix of initial radius R0 , and placed in a homogenized solvated membrane, referred to as the matrix. In micromechanics, this is called the double-inclusion model [8.15]. Assume that the stiffness of both the spherical shell and the homogenized matrix is the same as that of the (as yet unknown) overall effective stiffness of the hydrated membrane. For an isotropic expansion of a typical cluster, the two hoop stretches (and stresses) are equal, and using the incompressibility condition and spherical coordinates, it follows that

192

Part A

Solid Mechanics Topics

8.2.2 Pressure in Clusters For the solvated bare membrane or an IPMC in the M+ -ion form, and in the absence of an applied electric field, the pressure within each cluster pc consists of osmotic Π(M+ ) and electrostatic pDD components. The electrostatic component is produced by the ionic interaction within the cluster. The cation–anion conjugate pairs can be represented as uniformly distributed dipoles on the surface of a spherical cluster, and the resulting dipole–dipole (DD) interaction forces pDD calculated. The osmotic pressure is calculated by examining the difference between the chemical potential of the free (bath) solvent and that of the solvent within a typical cluster of known ion concentration within the membrane. In this manner, one obtains [8.16] pc =

νQ − B K0φ

w RT K0 = , F

±α2

1 −2 Q , 3κe B w2 ρB F , Q− B = EWion +

by the volume of solid), respectively. The membrane Young’s modulus may now be set equal to YB = 3K (w), assuming that both the elastomer and solvent are essentially incompressible under the involved conditions.

8.2.4 IPMC Stiffness To include the effect of the metal plating on the stiffness, note that for the Nafion-based IPMCs the gold plating is about a 1 μm layer on both faces of an IPMC strip, while platinum particles are distributed through the first 10–20 μm surface regions, with diminishing density. Assume a uniaxial stress state and average the axial strain and stress over the strip’s volume to obtain their average values, as ε¯ IPMC = f MH ε¯ B + (1 − f MH )¯εM , σ¯ IPMC = f MH σ¯ B + (1 − f MH )σ¯ M ,

f MH =

fM , 1+w (8.10)

(8.8)

where φ is the practical osmotic coefficient, α is the effective length of the dipole, F is the Faraday constant, ν is the cation–anion valence (ν = 2 for monovalent cations), R = 8.314 J/mol/K is the universal gas constant, T is the cluster temperature, ρB is the density of the bare ionomer, and κe is the effective permittivity.

where the barred quantities are the average values of the axial strain and stress in the IPMC, bare (solvated) membrane, and metal electrodes, respectively (indicated by the corresponding subscripts ‘B’ and ‘M’, respectively); and f M is the volume fraction of the metal plating in a dry sample. Setting σ¯ B = YB ε¯ B , σ¯ M = YM ε¯ M , and σ¯ IPMC = Y¯IPMC ε¯ IPMC , it follows that YM YB , BAB YM + (1 − BAB )YB (1 + w)(1 − fM) ¯ B= , w = w(1 ¯ − fM) , 1 + w(1 − fM) ¯

Y¯IPMC =

8.2.3 Membrane Stiffness The radial stress σr must equal the pressure pc in the cluster at r0 = a0 . In addition, the volume average of the stress tensor, taken over the entire membrane, must vanish in the absence of any externally applied loads. This is a consistency condition that to a degree accounts for the interaction among clusters. These conditions are sufficient to yield the undetermined pressure p0 and the stiffness K in terms of solvation volume fraction w and the initial dry void volume fraction w0 for each ion-form bare membrane, leading to the following final closed-form results: 1+w , w0 In − (w0 /w)4/3 1 + 2An 0 1 + 2A w − , A= −1 , In = w0 n 0 (1 + An 0 )1/3 (1 + A)1/3 K = pc

Part A 8.2

(8.9)

where n 0 and w0 = n 0 /(1 − n 0 ) are the initial (dry) porosity and initial void ratio (volume of voids divided

(8.11)

where AB is the concentration factor, giving the average stress in the bare polymer in terms of the average stress of the IPMC σ¯ IPMC . Here YB = 3K is evaluated from (8.9) at solvation w ¯ when the solvation of the IPMC is w. The latter can be measured directly at various solvation levels. Experimentally, the dry and solvated dimensions (thickness, width, and length) and masses of each bare or plated sample are measured. Knowing the composition of the ionomer, the neutralizing cation, and the composition of the electrodes, all physical quantities in equations (8.8) and (8.11) are then known for each sample, except for the three parameters α, κe , and AB , namely the effective distance between the neutralizing cation and its conjugate covalently fixed anion, the effective electric permittivity of the cluster, and the concentration factor that defines the fraction of axial stress

Electrochemomechanics of Ionic Polymer–Metal Composites

carried by the bare elastomer in the IPMC. The parameters α and κe are functions of the solvation level. They play critical roles in controlling the electrostatic and chemical interaction forces within the clusters, as will be shown later in this chapter in connection with the IPMC’s actuation. Thus, they must be evaluated with care and with due regard for the physics of the process. Estimate of κe and α The solvents are polar molecules, carrying an electrostatic dipole. Water, for example, has a dipole moment of about 1.87 D (Debyes) in the gaseous state and about 2.42 D in bulk at room temperature, the difference being due to the electrostatic pull of other water molecules. Because of this, water forms a primary and a secondary hydration shell around a charged ion. Its dielectric constant as a hydration shell of an ion (about 6) is thus much smaller than that in bulk (about 78). The second term in expression (8.8) for the cluster pressure pc , can change by a factor of 13 for water as the solvent, depending on whether the water molecules are free or constrained to a hydration shell. Similar comments apply to other solvents. For example, for glycerol and ethylene glycol, the room-temperature dielectric constants are about 9 and 8 as solvation shells, and 46 and 41 when in bulk. When the cluster contains both free and ion-bound solvent molecules, its effective electric permittivity can be estimated using a micromechanical model. Let κ1 = ε1 κ0 and κ2 = ε2 κ0 be the electric permittivity of the solvent in a solvation shell and in bulk, respectively, where κ0 = 8.85 × 10−12 F/m is the electric permittivity of free space. Then, using a double-inclusion

8.3 Voltage-Induced Cation Distribution

193

model [8.15] it can be shown that κ2 + 2κ1 + f (κ2 − κ1 ) κ1 , κe = κ2 + 2κ1 − f (κ2 − κ1 ) m w − CN EWion w f = , mw = (8.12) . mw Msolvent ρB ν Here, CN is the static solvation shell (equal to the coordination number), m w is the number of moles of solvent per mole of ion (cation and anion), and ν = 2. When m w is less than CN, then all solvent molecules are part of the solvation shell, for which κe = ε1 κ0 . On the other hand, when there are more solvent molecules, equation (8.12) yields the corresponding value of κe . Thus, κe is calculated in a cluster in terms of the cluster’s volume fraction of solvent, w. Similarly, the dipole arm α in (8.8), which measures the average distance between a conjugate pair of anion– cation, is expected to depend on the effective dielectric constant of the solvation medium. We now calculate the parameter ±α2 , as follows. As a first approximation, we let α2 vary linearly with w for m w ≤ CN, i. e., we set ± α 2 = a1 w + a2 ,

for

m w ≤ CN ,

(8.13)

and estimate the coefficients a1 and a2 from the experimental data. For m w > CN, furthermore, we assume that the distance between the two charges forming a pseudodipole is controlled by the effective electric permittivity of the their environment (e.g., water molecules), and hence is given by  2 κe (a1 w + a2 ) . (8.14) α2 = 10−20 κ1 Note that for m w > CN, we have a1 w + a2 > 0. An illustrative example is given in Sect. 8.5 where measured results are compared with model predictions.

8.3 Voltage-Induced Cation Distribution terized by [8.17–19], Ci Di ∂μi (8.15) + Ci υi , RT ∂x where Di is the diffusivity coefficient, μi is the chemical potential, Ci is the concentration, and υi is the velocity of species i. The chemical potential is given by Ji = −

μi = μ0 + RT ln(γi Ci ) + z i φF ,

(8.16)

where μ0 is the reference chemical potential, γi is the affinity of species i, z i is the species charge, and

Part A 8.3

When an IPMC strip in a solvated state is subjected to an electric field, the initially uniform distribution of its neutralizing cations is disturbed, as cations on the anode side are driven out of the anode boundary clusters while the clusters in the cathode side are supplied with additional cations. This redistribution of cations under an applied potential can be modeled using the coupled electrochemical equations that characterize the net flux of the species, caused by the electrochemical potentials (chemical concentration and electric field gradients). The total flux consists of cation migration and solvent transport. The flux Ji of species i is charac-

194

Part A

Solid Mechanics Topics

φ = φ(x, t) is the electric potential. For an ideal solution where γi = 1, and if there is only one type of cation, the subscript i may be dropped, as will be done in what follows. The variation in the electric potential field in the membrane is governed by the basic Poisson’s electrostatic equations [8.20, 21], ∂(κ E) = z(C − C − )F , ∂x

E=−

∂φ , ∂x

(8.17)

where E and κ are the electric field and the electric permittivity, respectively; C − is the anion concentration in moles per unit solvated volume; and C = C(x, t) is the total ion (cation and anion) concentration. Since the solvent velocity is very small, the last term in (8.15) may be neglected and, in view of (8.17) and continuity, it follows that

∂C(x, t) J(x, t) = −D ∂x   F − zC(x, t) E(x, t) , RT ∂C(x, t) ∂J(x, t) =− , ∂t  ∂x ∂E(x, t) F (8.18) =z (C(x, t) − C − ) . ∂x κ¯ Here κ¯ is the overall electric permittivity of the solvated IPMC sample that can be estimated from its measured effective capacitance. The above system of equations can be directly solved numerically, or they can be solved analytically using approximations. In the following sections, the analytic solution is considered. First, from (8.18) it follows that

∂ ∂(κ E) −D ∂x ∂t



∂ 2 (κ E) ∂x 2

−

C− F2 κ RT

 (κ E)

=0. (8.19)

Part A 8.3

This equation provides a natural length scale  and a natural time scale τ that characterize the ion redistribution,  2 κ¯ RT 1/2 , τ= (8.20) , = − 2 D C F where κ¯ is the overall electric permittivity of the IPMC. If Cap is the measured overall capacitance, then we set √ κ¯ = 2HCap. Since  is linear in κ¯ and κ¯ is proportional to the capacitance, it follows that  is proportional to the square root of the capacitance.

8.3.1 Equilibrium Cation Distribution To calculate the ion redistribution caused by the application of a step voltage across the faces of a hydrated strip of IPMC, we first examine the time-independent equilibrium case with J = 0. In the cation-depleted (anode) boundary layer the charge density is −C − F, whereas in the remaining part of the membrane the charge density is (C + − C − )F. Let the thickness of the cation-depleted zone be denoted by  and set Q(x, t) =

C+ − C− , C−

Q 0 (x) = lim Q(x, t) . t→∞

(8.21)

Then, it follows from (8.17) and boundary and continuity conditions that [8.16] the equilibrium distribution is given by ⎧  ⎪ ⎪ ⎨−1 for x ≤ −h +  Q 0 (x) ≡

F RT [B0 exp(x/) − B1 exp(−x/)] , ⎪ ⎪ ⎩ for − h +  < x < h ,

(8.22)

 2φ0 F   B1 = K 0 exp(−a ) , = −2 ,  RT 

  2 φ0 1  B2 = − K0 +1 +1 , 2 2    B0 = exp(−a) φ0 /2 + B1 exp(−a) + B2 , F K0 = (8.23) , RT where φ0 is the applied potential, a ≡ h/, and a ≡ (h −  )/. Since  is only 0.5–3 μm, a ≡ h/, a ≡ h  / 1, and hence exp(−a) ≈ 0 and exp(−a ) ≈ 0. The constants B0 and B1 are very small, of the order of 10−17 or even smaller, depending on the value of the capacitance. Therefore, the approximation used to arrive at (8.22) does not compromise the accuracy of the results. Remarkably, the estimated length of the anode boundary layer with charge √constant negative  density −C − F, i. e.,  = 2φ0 F/RT − 2 , depends only on the applied potential and the effective capacitance through the characteristic length. We use  as our length scale. From (8.23) it can be concluded that, over the most central part of the membrane, there is charge neutrality within all clusters, and the charge imbalance in clusters occurs only over narrow boundary layers, with the anode boundary layer being thicker

Electrochemomechanics of Ionic Polymer–Metal Composites

than the cathode boundary layer. The charge imbalance in the clusters is balanced by the corresponding electrode charges. Therefore, the charge imbalance defined by (8.23) applies to the clusters within each boundary layer and not to the boundary layer itself, since the combined clusters and the charged metal particles within the boundary layers are electrically balanced. We now examine the time variation of the charge distribution that results upon the application of a step voltage and leads to the equilibrium solution given above. To this end, rewrite equation (8.19) as  2 ∂ Q Q ∂Q − 2 =0, (8.24) −D ∂t ∂x 2  and set Q = ψ(x, t) exp(−t/τ) + Q 0 (x) to obtain the following standard diffusion equation for ψ(x, t): ∂ψ ∂2ψ =D 2 , ∂t ∂x

ψ(x, 0) = −Q 0 (x) ,

(8.25)

from which it can easily be concluded that, to a good degree of accuracy, one may use the following simple approximation in place of an exact infinite series

8.4 Nanomechanics of Actuation

195

solution [8.16]: Q(x, t) ≈ g(t)Q 0 (x) , g(t) = 1 − exp(−t/τ) . (8.26) Actually, when the electric potential is suddenly applied and then maintained, say, at a constant level φ0 , the cations of the anode clusters are initially depleted at the same rate as cations being added to the clusters within the cathode boundary layer. Thus, initially, the cation distribution is antisymmetric through the thickness of the membrane, as shown by Nemat-Nasser and Li [8.22], C(x) = C − +

κφ ¯ 0 sinh(x/) . 2F2 sinh(h/)

(8.27)

After a certain time period, say, τ1 , clusters near the anode face are totally depleted of their cations, so only the length of the anode boundary layer can further increase, while the cathode clusters continue to receive additional cations. Recognizing this fact, recently Nemat-Nasser and Zamani [8.23] have used one time scale for the first event and another time scale for the remaining process of cation redistribution. To simulate the actuation, however, these authors continue to use the one-time-scale approximation similar to (8.26).

8.4 Nanomechanics of Actuation changes in the pressure within the clusters in the anode and cathode boundary layers, and the resulting diffusive flow of solvent into or out of the corresponding clusters.

8.4.1 Cluster Pressure Change Due to Cation Migration The elastic pressure on a cluster can be calculated from (8.7) by simply setting r0 = aI , where aI is the initial cluster size just before the potential is applied, and using the spatially and temporally changing volume fraction of solvent and ion concentration w(x, t) and ν(x, t), since the osmotic pressure depends on the cluster’s ion concentration and it is reduced in the anode and is increased in the cathode clusters. The corresponding osmotic pressure can be computed from Π(x, t) =

φQ − C + (x, t) B K0 +1 , ν(x, t) , ν(x, t) = w(x, t) C− (8.28)

C + (x, t)

where is the cation concentration; note from (8.21) that ν(x, t) = Q(x, t) + 2. The pressure pro-

Part A 8.4

The application of an electric potential produces two thin boundary layers, one near the anode and the other near the cathode electrodes, while maintaining overall electric neutrality in the IPMC strip. The cation imbalance within the clusters of each boundary layer changes the osmotic, electrostatic, and elastic forces that tend to expand or contract the corresponding clusters, forcing the solvents out of or into the clusters, and producing the bending motion of the sample. Thus, the volume fraction of the solvent within each boundary layer is controlled by the effective pressure in the corresponding clusters produced by the osmotic, electrostatic, and elastic forces. These forces can even cause the cathode boundary layer to contract during the back relaxation that is observed for Nafion-based IPMCs in various alkali metal forms, expelling the extra solvents onto the IPMC’s surface while cations continue to accumulate within the cathode boundary layer. This, in fact, is what has been observed in open air during the very slow back relaxation of IPMCs that are solvated with ethylene glycol or glycerol [8.24]. To predict this and other actuation responses of IPMCs, it is thus necessary to model the

196

Part A

Solid Mechanics Topics

duced by the electrostatic forces among interacting fixed anions and mobile cations within each cluster varies as the cations move into or out of the cluster in response to the imposed electric field. As the cations of the clusters within the anode boundary layer are depleted, the dipole–dipole interaction forces diminish. We model this in the anode boundary layer by calculating the resulting pressure, pADD , as a function of the cation concentration, as follows:  2  + 1 C (x, t) −2 α(x, t) pADD (x, t) = . Q 3κ(x, t) B w(x, t) C− (8.29)

Parallel with the reduction in the dipole–dipole interaction forces is the development of electrostatic interaction forces among the remaining fixed anions, which introduces an additional pressure, say pAA , 1 2 pAA (x, t) = Q− B 18κ(x, t)  R02 C + (x, t) × 1− , (8.30) C− [w(x, t)]4/3 where R0 is the initial (unsolvated, dry) cluster size. This expression is obtained by considering a spherical cluster with uniformly distributed anion charges on its surface. The total pressure within a typical anode boundary layer cluster hence is tA (x, t) = −σr (a0 , t) + Π(x, t) + pAA (x, t) + pADD (x, t) .

(8.31)

Part A 8.4

Consider now the clusters in the cathode boundary layer. In these clusters, in addition to the osmotic pressure we identify two forms of electrostatic interaction forces. One is repulsion due to the cation–anion pseudodipoles already present in the clusters, and the other is due to the extra cations that migrate into the clusters and interact with the existing pseudo-dipoles. The additional stresses produced by this latter effect may tend to expand or contract the clusters, depending on the distribution of cations relative to fixed anions. We again model each effect separately, although in fact they are coupled. The dipole–dipole interaction pressure in the cathode boundary layer clusters is assumed to be reduced as  1 α(x, t) 2 2 Q− pCDD (x, t) = 3κ(x, t) B w(x, t)  C− × (8.32) , C + (x, t)

while at the same time new dipole–cation interaction forces are being developed as additional cations enter the clusters and disturb the pseudo-dipole structure in the clusters. The pressure due to these latter forces is represented by pDC (x, t) =

 2 2Q − R0 α(x, t) C(x, t) B − 1 . 9κ(x, t) [w(x, t)]5/3 C− (8.33)

However, for sulfonates in a Nafion-based IPMC, we expect extensive restructuring and redistribution of the cations. It appears that this process underpins the observed reverse relaxation of the Nafion-based IPMC strip. Indeed, this redistribution of the cations within the clusters in the cathode boundary layer may quickly diminish the value of pCD to zero or even render it negative. To represent this, we modify (8.33) by a relaxation factor, and write 2Q − R0 αC (x, t) B pDC (x, t) ≈ 9κC (x, t) [wC (x, t)]5/3  + C (x, t) × − 1 g1 (t) , C− g1 (t) = [r0 + (1 − r0 ) exp(−t/τ1 )], 2

r0 < 1 , (8.34)

where τ1 is the relaxation time and r0 is the equilibrium fraction of the dipole–cation interaction forces. The total stress in clusters within the cathode boundary layer is now approximated by tC = −σr (a0 , t) + ΠC (x, t) + pCDD (x, t) + pDC (x, t) . (8.35)

8.4.2 Cluster Solvent Uptake Due to Cation Migration The pressure change in clusters within the anode and cathode boundary layers drives solvents into or out of these clusters. This is a diffusive process that can be modeled using the continuity equation     ∂υ x, t w ˙ x, t  + (8.36) =0, ∂x 1 + w x, t and a constitutive model to relate the gradient of the solvent velocity, ∂υ(x, t)/∂x, to the cluster pressure. We may use a linear relation and obtain     w ˙ i x, t   = DBL ti x, t , i = A, C , (8.37) 1 + wi x, t

Electrochemomechanics of Ionic Polymer–Metal Composites

where DBL is the boundary-layer diffusion coefficient, assumed to be constant here.

8.4.3 Voltage-Induced Actuation As is seen from Fig. 8.6, an IPMC strip can undergo large deflections under an applied electric potential. Since the membrane is rather thin (0.2 mm) even if it deforms from a straight configuration into a circle of radius, say 1 cm still the radius-to-thickness ratio would be 50, suggesting that the maximum axial strain in the membrane is very small. It is thus reasonable to use a linear strain distribution through the thickness and use the following classical expression for the maximum strain: εmax ≈ ±

H , Rb

(8.38)

8.5 Experimental Verification

197

where Rb is the radius of the curvature of the membrane and H is half of its thickness. The strain in the membrane is due to the volumetric changes that occur within the boundary layers, and can be estimated from εv (x, t) = ln[1 + w(x, t)] .

(8.39)

We assume that the axial strain is one-third of the volumetric strain, integrate over the thickness, and obtain L L = Rb 2H 3 (3Y IPMC − 2YB ) h × YBL (w(x, t))x ln[1 + w(x, t)] dx , (8.40) −h

where all quantities have been defined before and are measurable.

8.5 Experimental Verification We now examine some of the experimental results that have been used to characterize this material and to verify the model results.

8.5.1 Evaluation of Basic Physical Properties Both the bare ionomer and the corresponding IPMC can undergo large dimensional changes when solvated. The phenomenon is also affected by the nature of the neutralizing cation and the solvent. This, in turn, substantially changes the stiffness of the material. Therefore, techniques have been developed to measure the dimensions and the stiffness of the ma-

terial in various cation forms and at various solvation stages. The reader is referred to Nemat-Nasser and Thomas [8.25, 26], Nemat-Nasser and Wu [8.27], and Nemat-Nasser and Zamani [8.23] for the details of various measurements and extensive experimental results. In what follows, a brief account of some of the essential features is presented. One of the most important quantities that characterize the ionic polymers is the material’s equivalent weight (EW), defined as dry mass in grams of ionic polymer in proton form divided by the moles of sulfonate (or carboxylate) groups in the polymer. It is expressed in grams per mole, and is measured by neutralizing the same sample sequentially with deferent

Table 8.1 Measured stiffness of dry and hydrated Nafion/Flemion ionomers and IPMCs in various cation forms Thickness

⎧ ⎨ Bare

Nafion

⎧ ⎨ Flemion IPMC

⎩

Nafion-based Flemion-based

K+ Cs+ Na+ K+ Cs+ Cs+ Cs+

Stiffness

Thickness

Water-saturated form Density Stiffness

(μm)

(g/cm3 )

(MPa)

(μm)

(g/cm3 )

(MPa)

Hydration volume (%)

182.1 178.2 189.1 149.4 148.4 150.7 156.0 148.7

2.008 2.065 2.156 2.021 2.041 2.186 3.096 3.148

1432.1 1555.9 1472.2 2396.0 2461.2 1799.2 1539.5 2637.3

219.6 207.6 210.5 167.6 163.0 184.2 195.7 184.1

1.633 1.722 1.836 1.757 1.816 1.759 2.500 2.413

80.5 124.4 163.6 168.6 199.5 150.6 140.4 319.0

71.3 50.0 41.4 42.0 34.7 53.7 54.1 58.1

Part A 8.5

⎩

Na+

Dry form Density

198

Part A

Solid Mechanics Topics

Part A 8.5

cations, and measuring the changes in the weight of the sample. This change is directly related to the number of anion sites within the sample and the difference in the atomic weight of the cations. Another important parameter is the solvent uptake w, defined as the volume of the solvent divided by the volume of the dry sample. Finally, it is necessary to measure the weight fraction of the metal in an IPMC sample, which is usually about 40%. The uniaxial stiffness of both the bare ionomer and the corresponding IPMC changes with the solvent uptake. It is also a function of the cation form, the solvent, and the backbone material. Table 8.1 gives some data on bare and metal-plated Nafion and Flemion in the indicated cation forms. Note that the stiffness can vary by a factor of ten for the same sample depending on whether it is dry or fully hydrated. Surface conductivity is an important electrical property governing an IPMC’s actuation behavior. When applying a potential across the sample’s thickness at the grip end, the bending of the cantilever is affected by its surface resistance, which in turn is dependent on the electrode morphology, cation form, and the level of solvation. An applied electric field affects the cation distribution within an IPMC membrane, forcing the cations to migrate towards the cathode. This change in the cation distribution produces two thin layers, one near the anode and the other near the cathode boundaries. In time, and once an equilibrium state is attained, the anode boundary layer is essentially depleted of its cations, while the cathode boundary layer has become cation rich. If the applied constant electric potential is V and the corresponding total charge that is accumulated within the cathode boundary layer once the equilibrium state is attained is Q, then the effective electric capacitance of the IPMC is defined as C = Q/V . From this, one obtains the corresponding areal capacitance, measured in mF/cm2 , by dividing by the area of the sample. Usually, the total equilibrium accumulated charge can be calculated by time integration of the measured net current, and using the actual dimensions of the saturated sample. For alkali-metal cations, one may have capacitance of 1–50 mF/cm2 for an IPMC.

8.5.2 Experimental Verification As an illustration of the model verification, consider first the measured and modeled stiffness of the bare and metal-plated Nafion-based samples (Fig. 8.7). Here, the

Stiffness (MPa) 2500 Experimental_bare Naflon Experimental_IPMC (Sh2) Experimental_IPMC (Sh5) Model_bare Naflon Model_Naflon IPMC

2000 1500 1000 500 0

0

10

20

30

40

50

60

70

80

Hydration (%)

Fig. 8.7 Uniaxial stiffness (Young’s modulus) of bare Nafion-117 (lower data points and the solid curve, model) and an IPMC (upper data points and solid curve) in the Na+ -form versus hydration water

lower data points and the solid curve are for the bare ionomer in Na+ -form, and the upper data points and the solid curve are for the corresponding IPMC. The model results have been obtained for the bare Nafion using (8.9) and for the IPMC from (8.11). In (8.9), φ is taken to be 1 and n 0 to be 1%. The dry density of the bare membrane is measured to be 2.02 g/cm3 , the equivalent weight for Nafion-117 in Na+ -form (with 23 atomic weight for sodium) is calculated to be 1122, the electric permittivity is calculated from (8.12) with CN = 4.5, the temperature is taken to be 300 K (room temperature), and κe is calculated using (8.12), with α2 Tip displacement/gauge length 0.12 Experiment Model

0.08 0.04 0 –0.04 –0.08 –0.12 –0.16 –10

0

10

20

30

40

50

60

Time (s)

Fig. 8.8 Tip displacement of a 15 mm cantilevered strip of a Nafion-based IPMC in Na+ -form, subjected to a 1.5 V step potential for about 32 s, then shorted; the solid curve is the model and the geometric symbols are experimental points

Electrochemomechanics of Ionic Polymer–Metal Composites

being calculated from (8.13) or (8.14) depending on the level of hydration, i. e., the value of w. The only free parameters are then a1 and a2 , which are calculated to be a1 = 1.5234 × 10−20 and a2 = −0.0703 × 10−20 , using two data points. In (8.11), f M is measured to be 0.0625, YM is 75 GPa, and AB (the only free parameter) is set equal to 0.5. To check the model prediction of the observed actuation response of this Nafion-based IPMC, consider a hydrated cantilevered strip subjected to a 1.5 V step potential across its faces that is maintained for 32 s and then shorted. Figure 8.8 shows (geometric symbols) the measured tip displacement of a 15 mm-long cantilever strip that is actuated by applying a 1.5 V step potential across its faces, maintaining the voltage for about 32 s and then removing the voltage while the two faces are shorted. The initial water uptake is wIPMC = 0.46, and the volume fraction of metal plating is 0.0625. Hence, the initial volume fraction of water in the Nafion part of the IPMC is given by wI = wIPMC /(1 − f M ) = 0.49. The formula weight of sodium is 23, and the dry density of the bare membrane is 2.02 g/cm3 . With EWNa+ = 1122 g/mol, the initial value of C − for the

References

199

bare Nafion becomes 1208. The thickness of the hydrated strip is measured to be 2H = 224 μm, and based on inspection of the microstructure of the electrodes, we set 2h ≈ 212 μm. The effective length of the anode boundary layer for φ0 = 1.5 V is L A = 9.78, and in order to simplify the model calculation of the solvent flow into or out of the cathode clusters, an equivalent uniform boundary layer is used near the cathode whose thickness is then estimated to be L C = 2.84, where  = 0.862 μm for φ0 = 1.5 V. The electric permittivity and the dipole length are calculated using a1 = 1.5234 × 10−20 and a2 = −0.0703 × 10−20 , which are the same as for the stiffness modeling. The measured capacitance ranges from 10 to 20 mF/cm2 , and we use 15 mF/cm2 . Other actuation model parameters are: DA = 10−2 (when pressure is measured in MPa), τ = 1/4 and τ1 = 4 (both in seconds), and r0 = 0.25. Since wI ≈ (a/R0 )3 , where a is the cluster size at wa−1/3 ter uptake wI , we set R0 ≈ w0 a, and adjust a to fit the experimental data; here, a = 1.65 nm, or an average cluster size of 3.3 nm, prior to the application of the potential. The fraction of cations that are left after shorting is adjusted to r = 0.03.

8.6 Potential Applications tors in space-based applications [8.2, 37], as their lack of multiple moving parts is ideal for any environment where maintenance is difficult. Besides these, IPMCs find applications in other disciplines, including fuel-cell membranes, electrochemical sensing [8.38], and electrosynthesis [8.2, 37, 39], as their lack of multiple moving parts is ideal for any environment where maintenance is difficult. Besides these, IPMCs find applications in other disciplines, including fuelcell membranes, electrochemical sensing [8.38], and electrosynthesis [8.39]. The commercial availability of Nafion has enabled alternate uses for these materials to be found. Growth and maturity in the field of IPMC actuators will require exploration beyond these commercially available polymer membranes, as the applications for which they have been designed are not necessarily optimal for IPMC actuators.

References 8.1

P. Millet: Noble metal-membrane composites for electrochemical applications, J. Chem. Ed. 76(1), 47– 49 (1999)

8.2

Y. Bar-Cohen, T. Xue, M. Shahinpoor, J.O. Simpson, J. Smith: Low-mass muscle actuators using elec-

Part A 8

The ultimate success of IPMC materials depends on their applications. Despite the limited force and frequency that IPMC materials can offer, a number of applications have been proposed which take advantage of IPMCs’ bending response, low voltage/power requirements, small and compact design, lack of moving parts, and relative insensitivity to damage. Osada and coworkers have described a number of potential applications for IPMCs, including catheters [8.28, 29], elliptic friction drive elements [8.30], and ratchet-and-pawl-based motile species [8.31]. IPMCs are being considered for applications to mimic biological muscles; Caldwell has investigated artificial muscle actuators [8.32, 33] and Shahinpoor has suggested applications ranging from peristaltic pumps [8.34] and devices for augmenting human muscles [8.35] to robotic fish [8.36]. Bar-Cohen and others have discussed the use of IPMC actua-

200

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8.3

8.4

8.5

8.6

8.7

8.8

8.9

8.10

8.11

8.12

8.13 8.14 8.15

8.16

8.17

Part A 8

8.18 8.19

troactive polymers (EAP), Proc. SPIE. 3324, 218–223 (1998) Y. Bar-Cohen, S. Leary, M. Shahinpoor, J.O. Harrison, J. Smith: Electro-active polymer (EAP) actuators for planetary applications, SPIE Conference on Electroactive Polymer Actuators, Proc. SPIE. 3669, 57–63 (1999) R. Liu, W.H. Her, P.S. Fedkiw: In situ electrode formation on a Nafion membrane by chemical platinization, J. Electrochem. Soc. 139(1), 15–23 (1992) M. Homma, Y. Nakano: Development of electrodriven polymer gel/platinum composite membranes, Kagaku Kogaku Ronbunshu 25(6), 1010–1014 (1999) T. Rashid, M. Shahinpoor: Force optimization of ionic polymeric platinum composite artificial muscles by means of an orthogonal array manufacturing method, Proc. SPIE 3669, 289–298 (1999) M. Bennett, D.J. Leo: Manufacture and characterization of ionic polymer transducers with non-precious metal electrodes, Smart Mater. Struct. 12(3), 424–436 (2003) C. Heitner-Wirguin: Recent advances in perfluorinated ionomer membranes – structure, properties and applications, J. Membr. Sci. 120(1), 1–33 (1996) R.E. Fernandez: Perfluorinated ionomers. In: Polymer Data Handbook, ed. by J.E. Mark (Oxford Univ. Press, New York 1999) pp. 233–238 T.D. Gierke, C.E. Munn, P.N. Walmsley: The morphology in Nafion perfluorinated membrane products, as determined by wide- and small-angle X-ray studies, J. Polym. Sci. Polym. Phys. Ed. 19, 1687–1704 (1981) J.Y. Li, S. Nemat-Nasser: Micromechanical analysis of ionic clustering in Nafion perfluorinated membrane, Mech. Mater. 32(5), 303–314 (2000) W.A. Forsman: Statistical mechanics of ion-pair association in ionomers, Proc. NATO Adv. Workshop Struct. Properties Ionomers (1986) pp. 39–50 L.R.G. Treolar: Physics of Rubber Elasticity (Oxford Univ. Press, Oxford 1958) R.J. Atkin, N. Fox: An Introduction to the Theory of Elasticity (Longman, London 1980) S. Nemat-Nasser, M. Hori: Micromechanics: Overall Properties of Heterogeneous Materials, 1st edn. (North-Holland, Amsterdam 1993) S. Nemat-Nasser: Micromechanics of actuation of ionic polymer-metal composites, J. Appl. Phys. 92(5), 2899–2915 (2002) J. O’M Bockris, A.K.N. Reddy: Modern Electrochemistry 1: Ionics, Vol. 1 (Plenum, New York 1998) P. Shewmon: Diffusion in Solids, 2nd edn. (The Minerals Metals & Materials Society, Warrendale 1989) M. Eikerling, Y.I. Kharkats, A.A. Kornyshev, Y.M. Volfkovich: Phenomenological theory of electro-osmotic effect and water management in polymer electrolyte proton-conducting membranes, J. Electrochem. Soc. 145(8), 2684–2699 (1998)

8.20 8.21 8.22

8.23

8.24

8.25

8.26

8.27

8.28

8.29

8.30

8.31

8.32

8.33 8.34

8.35

8.36

J.D. Jackson: Classical Electrodynamics (Wiley, New York 1962) W.B. Cheston: Elementary Theory of Electric and Magnetic Fields (Wiley, New York 1964) S. Nemat-Nasser, J.Y. Li: Electromechanical response of ionic polymer-metal composites, J. Appl. Phys. 87(7), 3321–3331 (2000) S. Nemat-Nasser, S. Zamani: Modeling of electrochemo-mechanical response of ionic-polymermetal composites with various solvents, J. Appl. Phys. 38, 203–219 (2006) S. Nemat-Nasser, S. Zamani, Y. Tor: Effect of solvents on the chemical and physical properties of ionic polymer-metal composites, J. Appl. Phys. 99, 104902 (2006) S. Nemat-Nasser, C. Thomas: Ionomeric polymermetal composites. In: Electroactive Polymer (EAP) Actuators as Artificial Muscles, ed. by Y. Bar-Cohen (SPIE, Bellingham 2001) pp. 139–191 S. Nemat-Nasser, C. Thomas: Ionomeric polymermetal composites. In: Electroactive Polymer (EAP) Actuators as Artificial Muscles 2nd edn, ed. by Y. BarCohen (SPIE, Bellingham 2004) pp. 171–230 S. Nemat-Nasser, Y. Wu: Comparative experimental study of Nafion-and-Flemion-based ionic polymermetal composites (IPMC), J. Appl. Phys. 93(9), 5255– 5267 (2003) S. Sewa, K. Onishi, K. Asaka, N. Fujiwara, K. Oguro: Polymer actuator driven by ion current at low voltage, applied to catheter system, Proc. IEEE Ann. Int. Workshop Micro Electro Mech. Syst. 11th (1998) pp. 148–153 K. Oguro, N. Fujiwara, K. Asaka, K. Onishi, S. Sewa: Polymer electrolyte actuator with gold electrodes, Proc. SPIE 3669, 64–71 (1999) S. Tadokoro, T. Murakami, S. Fuji, R. Kanno, M. Hattori, T. Takamori, K. Oguro: An elliptic friction drive element using an ICPF actuator, IEEE Contr. Syst. Mag. 17(3), 60–68 (1997) Y. Osada, H. Okuzaki, H. Hori: A polymer gel with electrically driven motility, Nature 355(6357), 242– 244 (1992) D.G. Caldwell: Pseudomuscular actuator for use in dextrous manipulation, Med. Biol. Eng. Comp. 28(6), 595 (1990) D.G. Caldwell, N. Tsagarakis: Soft grasping using a dextrous hand, Ind. Robot. 27(3), 194–199 (2000) D.J. Segalman, W.R. Witkowski, D.B. Adolf, M. Shahinpoor: Theory and application of electrically controlled polymeric gels, Smart Mater. Struct. 1(1), 95–100 (1992) M. Shahinpoor: Ionic polymeric gels as artificial muscles for robotic and medical applications, Iran. J. Sci. Technol. 20(1), 89–136 (1996) M. Shahinpoor: Conceptual design, kinematics and dynamics of swimming robotic structures using active polymer gels, Act. Mater. Adapt. Struct. Proc. ADPA/AIAA/ASME/SPIE Conf. (1992) pp. 91–95

Electrochemomechanics of Ionic Polymer–Metal Composites

8.37

8.38

Y. Bar-Cohen, S. Leary, M. Shahinpoor, J.O. Harrison, J. Smith: Electro-active polymer (EAP) actuators for planetary applications, SPIE Conf. Electroactive 8.39 Polymer Actuators Proc. SPIE 3669, 57–63 (1999) D.W. DeWulf, A.J. Bard: Application of Nafion/platinum electrodes (solid polymer electrolyte structures) to voltammetric investigations of highly resistive

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solutions, J. Electrochem. Soc. 135(8), 1977–1985 (1988) J.M. Potente: Gas-phase Electrosynthesis by Proton Pumping Through a Metalized Nafion Membrane: Hydrogen Evolution and Oxidation, Reduction of Ethene, and Oxidation of Ethane and Ethene. Ph.D. Thesis (North Carolina State University, Raleigh 1988)

Part A 8

203

Wendy C. Crone

The expanding and developing fields of micro-electromechanical systems (MEMS) and nano-electromechanical (NEMS) are highly interdisciplinary and rely heavily on experimental mechanics for materials selection, process validation, design development, and device characterization. These devices range from mechanical sensors and actuators, to microanalysis and chemical sensors, to micro-optical systems and bioMEMS for microscopic surgery. Their applications span the automotive industry, communications, defense systems, national security, health care, information technology, avionics, and environmental monitoring. This chapter gives a general introduction to the fabrication processes and materials commonly used in MEMS/NEMS, as well as a discussion of the application of experimental mechanics techniques to these devices. Mechanics issues that arise in selected example devices are also presented.

9.1

Background ......................................... 203

9.2

MEMS/NEMS Fabrication ......................... 206

9.3 Common MEMS/NEMS Materials and Their Properties ............................. 206 9.3.1 Silicon-Based Materials ................ 207 9.3.2 Other Hard Materials .................... 208

9.3.3 Metals ........................................ 9.3.4 Polymeric Materials ...................... 9.3.5 Active Materials ........................... 9.3.6 Nanomaterials ............................. 9.3.7 Micromachining ........................... 9.3.8 Hard Fabrication Techniques ......... 9.3.9 Deposition .................................. 9.3.10 Lithography ................................. 9.3.11 Etching .......................................

208 208 209 209 210 211 211 211 212

9.4 Bulk Micromachining versus Surface Micromachining .............. 213 9.5 Wafer Bonding ..................................... 214 9.6 Soft Fabrication Techniques................... 215 9.6.1 Other NEMS Fabrication Strategies .. 215 9.6.2 Packaging ................................... 216 9.7

Experimental Mechanics Applied to MEMS/NEMS .......................... 217

9.8 The Influence of Scale ........................... 9.8.1 Basic Device Characterization Techniques .................................. 9.8.2 Residual Stresses in Films .............. 9.8.3 Wafer Bond Integrity .................... 9.8.4 Adhesion and Friction...................

217 218 219 220 220

9.9 Mechanics Issues in MEMS/NEMS ............. 221 9.9.1 Devices ....................................... 221 9.10 Conclusion ........................................... 224 References .................................................. 225

9.1 Background The acronym MEMS stands for micro-electromechanical system, but MEMS generally refers to microscale devices or miniature embedded systems involving one or more micromachined component that enables higher-level functionality. Similarly NEMS, nanoelectromechanical system, refers to such nanoscale devices or nanodevices. MEMS and NEMS are fab-

ricated microscale and nanoscale devices that are often made in batch processes, usually convert between some physical parameter and a signal, and may be incorporated with integrated circuit technology. The field of MEMS/NEMS encompasses devices created with micromachining technologies originally developed to produce integrated circuits, as well as

Part A 9

A Brief Introd 9. A Brief Introduction to MEMS and NEMS

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Part A 9.1

Table 9.1 Sample applications of MEMS/NEMS Sensors Actuators

Passive structures

Accelerometers, biochemical analyzers, environmental assay devices, gyroscopes, medical diagnostic sensors, pressure sensors Data storage, drug-delivery devices, drug synthesis, fluid regulators, ink-jet printing devices, micro fuel cells, micromirror devices, microphones, optoelectric devices, radiofrequency devices, surgical devices Atomizers, fluid spray systems, fuel injection, medical inhalers

non-silicon-based devices created by the same micromachining or other techniques. They can be classified as sensors, actuators, and passive structures (Table 9.1). Sensors and actuators are transducers that convert one physical quantity to another, such as electromagnetic, mechanical, chemical, biological, optical or thermal phenomena. MEMS sensors commonly measure pressure, force, linear acceleration, rate of angular motion, torque, and flow. For instance, to sense pressure an intermediate conversion step, such as mechanical stress, can be used to produce a signal in the form of electrical energy. The sensing or actuation conversion can use a variety of methods. MEMS/NEMS sensing can employ change in electrical resistance, piezoresistive, piezoelectric, change in capacitance, and magnetoresistive methods (Table 9.2). MEMS/NEMS actuators provide the ability to manipulate physical parameters at the micro/nanoscale, and can employ electrostatic, thermal, magnetic, piezoelectric, piezoresistive, and shape-memory transformation methods. Passive MEMS structures such as micronozzles are used in atomizers, medical inhalers, fluid spray systems, fuel injection, and ink-jet printers.

MEMS have a characteristic length scale between 1 mm and 1 μm, whereas NEMS devices have a characteristic length scale below 1 μm (most strictly, 1–100 nm). For instance a digital micromirror device has a characteristic length scale of 14 μm, a quantum dot transistor has components measuring 300 nm, and molecular gears fall into the 10–100 nm range [9.2]. Additionally, although an entire device may be mesoscale, if the functional components fall in the microscale or nanoscale regime it may be referred to as a MEMS or NEMS device, respectively. MEMS/NEMS inherently have a reduced size and weight for the function they carry out, but they can also carry advantages such as low power consumption, improved speed, increased function in one package, and higher precision. There is no distinct MEMS/NEMS market, instead there is a collection of niche markets where MEMS/NEMS become attractive by enabling a new function, bringing the advantage of reduced size, or lowering cost [9.1]. Despite this characteristic, the MEMS industry is already valued in tens of billions of dollars and growing rapidly. The Small Times Tech Business DirectoryTM

Table 9.2 Physical quantities used in MEMS/NEMS sensors and actuators (after Maluf [9.1]) Method

Description

Physical and material parameters

Order of energy density (J/cm3 )

Electrostatic

Attractive force between two components carrying opposite charge Certain materials that change shape under an electric field Thermal expansion or difference in coefficient of thermal expansion Electric current in a component surrounded by a magnetic field gives rise to an electromagnetic force Certain materials that undergo a solid–solid phase transformation producing a large shape change

Electric field, dielectric permittivity

≈ 0.1

Electric field, Young’s modulus, piezoelectric constant Coefficient of expansion, temperature change, Young’s modulus Magnetic field, magnetic permeability

≈ 0.2

Transformation temperature

≈ 10

Piezoelectric Thermal Magnetic

Shape memory

≈5 ≈4

A Brief Introduction to MEMS and NEMS

ples include biofluidic chips for biochemical analyses, biosensors for medical diagnostics, environmental assays for toxin identification, implantable pharmaceutical drug delivery, DNA and genetic code analysis, imaging, and surgery. NEMS is often associated with biotechnology because this size scale allows for interaction with biological systems in a fundamental way. BioNEMS may be used for drug delivery, drug synthesis, genome synthesis, nanosurgery, and artificial organs comprised of nanomaterials. The sensitivity of such bioNEMS devices can be exquisite, selectively binding and detecting a single biomolecule. More complete background information on microfluidic devices can be found in Beeby [9.7], Koch [9.9] and Kovacs [9.10]. Semiconductor NEMS devices can offer microwave resonance frequencies, exceptionally high mechanical quality factors, and extraordinarily small heat capacities [9.11, 12]. Examples of NEMS devices also include transducers, radiating energy devices, nanoscale integrated circuits, and optoelectronic devices [9.13, 14]. NEMS manufacturing is being further enabled by the drive towards nanometer feature sizes in the microelectronics industry. Terascale computational ability will require nanotransistors, nanodiodes, nanoswitces, and nanologic gates [9.15]. NEMS also opens the door for fundamental science at the nanometer scale investigating phonon-mediated mechanical processes [9.16] and quantum behavior of mesoscopic mechanical systems [9.17]. Although there is some discussion as to whether the NEMS definition requires a characteristic length scale below 1000 nm or 100 nm, there is no argument that the field of NEMS is in its infancy. Existing commercial devices are limited at this point, but research on NEMS is extremely active and highly promising. Many challenges remain, including assembly of nanoscale devices and mass production capabilities. In the long term, a number of issues must be addressed in analysis, design, development, and fabrication for high-performance MEMS/NEMS to become ubiquitous. Of most relevance to the focus of this handbook, advanced materials must be well characterized and MEMS/NEMS testing must be further developed. Additionally for commercialization, MEMS/NEMS design must consider issues of market (need for product, size of market), impact (enabling new systems, paradigm shift for the field), competition (other macro and micro/nanoproducts existing), technology (available capability and tools), and manufacturing (manufacturability in volume at low cost) [9.18].

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lists more than 700 manufacturers/fabricators of microsystems and nanotechnologies [9.3]. High-volume production with lucrative sales have been achieved by several companies making devices such as accelerometers for automobiles (Analog Devices, Motorola, Bosch), micromirrors for digital projection displays (Texas Instruments), and pressure sensors for the automotive and medical industries (NovaSensor). Currently, the MEMS markets with the largest commercial value are ink-jet printer heads, optical MEMS (which includes the Digital Micromirror DeviceTM discussed below), and pressure sensors, followed by microfluidics, gyroscopes, and accelerometers [9.4]. The MEMS market was reported to be US $5.1 billion in 2005 and projected to reach US $9.7 billion by 2010 [9.4]. The NEMS industry, while still young, has been growing in value. The market research firm Report Buyer recently released a market report on Nanorobotics and NEMS indicating that the global market for NEMS increased from US $29.5 million to US $34.2 million between 2004 and 2005 and projecting that the market will reach US $830.4 million by 2011 [9.5]. Microfluidic and nanofluidic devices also fall under the umbrella of MEMS/NEMS and are often classified as bioMEMS/bioNEMS devices when involving biological materials. These devices incorporate channels with at least one microscale or nanoscale dimension in which fluid flows. The small scale of these devices allow for smaller sample size, faster reactions, and higher sensitivity. Microfluidic devices commonly use both hard and soft fabrication techniques to produce channels and other fluidic structures [9.6]. The common feature of these devices is that they allow for flow of gas and/or liquid and use components such as pumps, valves, nozzles, and mixers. Commercial and defense applications include automotive controls, pneumatics, environmental testing, and medical devices. The advantages of the microscale in these applications include high spatial resolution, fast time response, small fluid volumes required for analysis, low leakage, low power consumption, low cost, appropriate compatibility of surfaces, and the potential for integrate signal processing [9.7]. At the microscale, pressure drop over a narrow channel is high and fluid flow generated by electric fields can be substantial. The micrometer and nanometer length scales are particularly relevant to biological materials because they are comparable to the size of cells, molecules, diffusions length for molecules, and electrostatic screening lengths of ionic conducting fluids [9.8]. Device exam-

9.1 Background

206

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Part A 9.3

The field of MEMS/NEMS is highly multidisciplinary, often involving expertise from engineering, materials science, physics, chemistry, biology, and medicine. Because of the breadth of the field and the range of activities that fall under the scope of MEMS/NEMS, a comprehensive review is not possible in this chapter. After providing general background, the focus will be on mechanics and specifically experimental mechanics as it is applied to MEMS/NEMS.

Mechanics is critical to the design, fabrication, and performance of MEMS/NEMS. A broad range of experimental tools has been applied to MEMS/NEMS. This chapter will provide an overview of such work. Additional information on the application of mechanics to MEMS/NEMS can be found in the proceedings of the annual symposium held by the MEMS and Nanotechnology Technical Division of the Society for Experimental Mechanics (see, for example, [9.19]).

9.2 MEMS/NEMS Fabrication Traditionally MEMS/NEMS are thought of in the context of microelectronics fabrication techniques which utilize silicon. This approach to MEMS/NEMS brings with it the momentum of the integrated circuits industry and has the advantage of ease of integration with semiconductor devices, but fabrication is expensive in both the infrastructure and equipment required and the time investment needed to create a working prototype. An alternative approach that has seen significant success, especially in its application to microfluidic devices, is the use of soft materials such as polydimethylsiloxane (PDMS). Soft MEMS/NEMS fabrication can often be conducted with bench-top techniques with no need for the clean-room facilities used in microelectronics fabrication. Additionally, polymers offer a range of properties not available in siliconbased materials such as mechanical shock tolerance, biocompatibility, and biodegradability. However, polymers can carry disadvantages for certain applications because of their viscoelastic behavior and low thermal stability. Ultimately a combination of function and

economics decides the medium of choice for device construction. Whether we talk about hard or soft MEMS/NEMS, the basic approach to device construction is similar. Material is deposited onto a substrate, a lithographic step is used to produce a pattern, and material removal is conducted to create a shape. For traditional microelectronics fabrication, the substrate is often silicon, material deposition is achieved by vapor deposition or sputtering, lithography involves patterning of a chemically resistant polymer, and material is removed by a chemical etch. Alternatively, for soft MEMS/NEMS materials, fabrication often utilizes a glass or plastic substrate, material in the form of a monomer is flowed into a region, a lithographic mask allows exposure of a pattern to ultraviolet (UV) radiation triggering polymerization, and the unpolymerized monomer is removed with a flushing solution. For both hard and soft MEMS/NEMS fabrication there are a number of variations on these basic steps which allow for a wide array of structures and devices to be constructed.

9.3 Common MEMS/NEMS Materials and Their Properties Materials used in MEMS/NEMS must simultaneously satisfy a range requirements for chemical, structural, mechanical, and electrical properties. For biomedical and bioassay devices, material biocompatibility and bioresistance must also be considered. Most MEMS/NEMS devices are created on a substrate. Common substrate materials include singlecrystal silicon, single-crystal quartz, fused quartz, gallium arsenide, glass, and plastics. Devices are made with a range of methods by machining into the substrate and/or depositing additional material on top of the

substrate. The additional materials may be structural, sacrificial, or active. Although traditionally MEMS in particular have relied on silicon, the materials used in MEMS/NEMS are becoming more heterogeneous. Selected properties are given in Table 9.3 for comparative purposes, but an extensive list of properties for the wide range of materials used in MEMS/NEMS cannot be included here. It should be noted, however, that the constitutive behavior of materials used in MEMS/NEMS applications can be sensitive to fabrication method, processing parame-

A Brief Introduction to MEMS and NEMS

9.3 Common MEMS/NEMS Materials and Their Properties

Property

Si

SiO2

Si3 N4

Quartz

SiC

Si(111)

Stainless steel

Al

Young’s modulus (GPa) Yield strength (GPa) Poisson’s ratio Density (g/cm3 ) Coefficient of thermal expansion (10−6 /◦ C) Thermal conductivity at 300 K (W/cm · K) Melting temperature (◦ C)

160 7 0.22 2.4 2.6

73 8.4 0.17 2.3 0.55

323 14 0.25 3.1 2.8

107 9 0.16 2.65 0.55

450 21 0.14 3.2 4.2

190 7 0.22 2.3 2.3

200 3 0.3 8 16

70 0.17 0.33 2.7 24

1.57

0.014

0.19

0.0138

5

1.48

0.2

2.37

1415

1700

1800

1610

2830

1414

1500

660

ters, and thermal history due to the relative similarity between characteristic length scales and device dimensions. A good resource compiling characterization data from a number of sources is the material database at http://www.memsnet.org/material/ [9.20]. The following books, used as references for the discussion here, are valuable resources for more extensive information: Senturia [9.18], Maluf [9.1], and Beeby [9.7].

9.3.1 Silicon-Based Materials Silicon, Polysilicon, and Amorphous Silicon Silicon-based materials are the most common materials currently used in MEMS/NEMS commercial production. MEMS/NEMS devices often exploit the mechanical properties of silicon rather than its electrical properties. Silicon can be used in a number of different forms: oriented single-crystal silicon, amorphous silicon, or polycrystal silicon (polysilicon). Single-crystal silicon, which has cubic crystal structure, exhibits anisotropic behavior which is evident in its mechanical properties such as Young’s modulus. A high-purity ingot of single-crystal silicon is grown, sawn to the desired thickness, and polished to create a wafer. Single-crystal silicon used for MEMS/NEMS are usually the standard 100 mm (4 inch diameter, 525 μm thickness) or 150 mm (6 inch diameter, 650 μm thickness) wafers. Although larger 8-inch and 12-inch wafers are available, they are not used as prevalently for MEMS fabrication. The properties of the wafer depend on both the orientation of crystal growth and the dopants added to the silicon (Fig. 9.1). Impurity doping has a significant impact on electrical properties but does not generally impact the mechanical properties if the concentration is approximately < 1020 cm−3 . Silicon is a group IV semiconductor. To create a p-type material, dopants

from group III (such as boron) create mobile charge carriers that behave like positively charged species. To create an n-type material, dopants from group V (such as phosphorous, arsenic, and antimony) are used to create mobile charge carriers that behave like negatively charged electrons. Doping of the entire wafer can be accomplished during crystal growth. Counter-doping can be accomplished by adding dopants of the other type to an already doped substrate using deposition followed by ion implantation and annealing (to promote diffusion and relieve residual stresses). For instance, p-type into n-type creates a pn-junction. Amorphous and polysilicon films are usually deposited with thicknesses of < 5 μm, although it is also possible to create thick polysilicon [9.21]. The residual stress in deposited polysilicon and amorphous silicon thin films can be large, but annealing can be used to provide some relief. Polysilicon has the disadvantage of a somewhat lower strength and lower piezoresistivity than single-crystal silicon. Additionally, Young’s

(100) n-type

Secondary flat

(111) n-type

Primary flat

(100) p-type

Primary flat

Secondary flat

Primary flat

(111) p-type

Primary flat

Secondary flat

Fig. 9.1 Flats on standard commercial silicon wafers used

to identify crystallographic orientation and doping (after Senturia [9.18])

Part A 9.3

Table 9.3 Properties of selected materials (after Maluf [9.1] and Beeby [9.7])

207

208

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Part A 9.3

modulus may vary significantly because the diameter of a single grain may comprise a large fraction of a component’s width [9.22]. Silicon, polysilicon, and amorphous silicon are also piezoresistive, meaning that the resistivity of the material changes with applied stress. The fractional change in resistivity, Δρ/ρ, is linearly dependent on the stress components parallel and perpendicular to the direction of the resistor. The proportionality constants are the piezoresistive coefficients, which are dependent on the crystallographic orientation, and the dopant type/concentration in single-crystal silicon. This property can be used to create a strain gage. Silicon Dioxide The success of silicon is heavily based on its ability to form a stable oxide which can be predictably grown at elevated temperature. Dry oxidation produces a higherquality oxide layer, but wet oxidation (in the presence of water) enhances the diffusion rate and is often used when making thicker oxides. Amorphous silicon dioxide can be used as a mask against etchants. It should be noted that these films can have large residual stresses. Silicon Nitride Silicon nitride can be deposited by chemical vapor deposition (CVD) as an amorphous film which can be used as a mask against etchants. It should be noted that these films can have large residual stresses. Silicon Carbide Silicon carbide is an attractive material because of its high hardness, good thermal properties, and resistance to harsh environments. Additionally, silicon carbide is piezoresistive. Although it can be produced as a bulk polycrystalline material it is generally grown or deposited on a silicon substrate by epitaxial growth (single crystal) or by chemical vapor deposition (polycrystal).

Quartz Single-crystal quartz, which has a hexagonal crystal structure, can be used in natural or synthesized form. Like silicon, it can be etched selectively but the results are less ideal than in silicon because of unwanted facets and poor edge definition. Single-crystal quartz can be used as substrate material in a range of cuts which have different temperature sensitivities for piezoelectric or mechanical properties. Detailed information about quartz cuts can be found in Ikeda [9.25]. Quartz is also piezoelectric, meaning that there is a relationship between strain and voltage in the material. Fused quartz (silica) is a glassy noncrystalline material that is also occasionally used in MEMS/NEMS devices. Glass Glasses such as phosphosilicate and borosilicate (Pyrex) can be used as a substrate or in conjunction with silicon and other materials using wafer bonding (discussed below). Diamond Diamond is also attractive because of its high hardness, high fracture strength, low thermal expansion, low heat capacity, and resistance to harsh environments. Diamond is also piezoresistive and can be doped to produce semiconducting and metal-like behavior [9.26]. Because of its hardness, diamond is particularly attractive for parts exposed to wear. The most promising synthetic forms are amorphous diamond-like carbon, nanocrystalline diamond, and ultra-nanocrystalline diamond films created by pulsed laser deposition or chemical vapor deposition [9.27–31].

9.3.3 Metals

Silicon on Insulator (SOI) Silicon on insulator (SOI) wafers are also used for MEMS sensors and actuators [9.23]. Different SOI materials are distinguished by their properties. Buried oxide layers can be produced either through ion implantation or wafer bonding processes; these techniques are discussed further below [9.24].

Metals are usually deposited as a thin film by sputtering, evaporation or chemical vapor deposition (CVD). Gold, nickel and iron can also be electroplated. Aluminum is the most common metal used in MEMS/NEMS, and is often used for light reflection and electrical conduction. Gold is used for electrochemistry, infrared (IR) light reflection, and electrical conduction. Chromium is often used as an adhesion layer. Alloys of Ni, such as NiTi and PermalloyTM , can be used for actuation and are discussed in more detail below.

9.3.2 Other Hard Materials

9.3.4 Polymeric Materials

Gallium Arsenide Gallium arsenide (GaAs) is a III–V compound semiconductor which is often used to create lasers, optical devices, and high-frequency components.

Photoresists Polymeric photoresist materials are generally used as a spin-cast film as part of a photolithographic process. The film is modified by exposure to radiation

A Brief Introduction to MEMS and NEMS

Polydimethylsiloxane Polydimethylsiloxane (PDMS) is an elastomer used as both a structural component in MEMS devices and a stamping material for creating micro- and nanoscale features on surfaces. PDMS is a common silicone rubber and is used extensively because of its processibility, low curing temperature, stability, tunable modulus, optical transparency, biocompatibility, and adaptability by a range functional groups that can be attached [9.32,33].

9.3.5 Active Materials There are several types of active materials that successfully perform sensing and actuation functions at the microscale. Several examples of active materials are given below. NiTi Near-equiatomic nickel titanium alloy can be deposited as a thin film and used an as active material. This material is of particular interest to MEMS because the actuation work density of NiTi is more than an order of magnitude higher than the work densities of other actuation schemes. These shape-memory alloys (SMAs) undergo a reversible phase transformation that allows the material to display dramatic and recoverable stress- and temperature-induced transformations. The behavior of NiTi SMA is governed by a phase transformation between austenite and martensite crystal structures. Transformation between the austenite (B2) and martensite (B19) phases in NiTi can be produced by temperature cycling between the hightemperature austenite phase and the low-temperature martensite phase (shape-memory effect), or loading and unloading the material to favor either the highstrain martensite phase or the low-strain austenite phase (superelasticity). Thus both stress and temperature produce the transformation between the austenite and martensite phases of the alloy. The transfor-

mation occurs in a temperature window, which can be adjusted from −100 ◦ C to +160 ◦ C by changing the alloy composition and heat treatment processing [9.34]. PermalloyTM PermalloyTM , Nix Fe y , displays magnetoresistance properties and is used for magnetic transducing. Multilayered nanostructures of this alloy give rise to a giant-magnetoresistance (GMR) phenomenon which can be used to detect magnetic fields. It has been widely applied to read the state of magnetic bits in data storage media. Lead Zirconate Titanate (PZT) Lead zirconate titanate (PZT) is a ceramic solid solution of lead zirconate (PbZrO3 ) and lead titanate (PbTiO3 ). PZT is a piezoelectric material that can be deposited in thin film form by sputtering or using a sol–gel process. In addition to natural piezoelectric materials such as quartz, other common synthetic piezoelectric materials include polyvinylidene fluoride (PVDF), zinc oxide, and aluminum nitride. Actuation performed by piezoelectrics has the advantage of being capable of achieving reasonable displacements with fast response, but the material processing is complex. Hydrogels Hydrogels, such as poly(2-hydroxyethyl methacrylate (HEMA)) gel, with volumetric shape-memory capability are now being employed as actuators, fluid pumps, and valves in microfluidic devices. In an aqueous environment, hydrogels will undergo a reversible phase transformation that results in dramatic volumetric swelling and shrinking upon exposure and removal of a stimulus. Hydrogels have been produced that actuate when exposed to such stimuli as pH, salinity, electrical current, temperature, and antigens. Since the rate of swelling and shrinking in a hydrogel is diffusion limited, the temporal response of hydrogel structures can be reduced to minutes or even seconds in microscale devices.

9.3.6 Nanomaterials Nanostructuring of materials can produce unique mechanical, electrical, magnetic, optical, and chemical properties. The materials themselves range from polymers to metals to ceramics, it is their nanostructured nature that gives them exciting new behaviors. Increased hardness with decreasing grain size allows for

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such as visible light, ultraviolet light, x-rays or electrons. Exposure is usually conducted through a mask so that a pattern is created in the photoresist layer and subsequently on the substrate through an etching or deposition process. Resists are either positive or negative depending on whether the radiation exposure weakens or strengthens the polymer. In the developer step, chemicals are used to remove the weaker material, leaving a patterned photoresist layer behind. Important photoresist properties include resolution and sensitivity, particularly as feature sizes decrease.

9.3 Common MEMS/NEMS Materials and Their Properties

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Table 9.4 Micromachining processes and their applications (after Kovacs [9.10]) Process

Example applications

Lithography Thin-film deposition

Photolithography, screen printing, electron-beam lithography, x-ray lithography Chemical vapor deposition (CVD), plasma-enhanced chemical vapor deposition (PECVD), physical vapor deposition (PVD) such as sputtering and evaporation, spin casting, sol–gel deposition Blanket and template-delimited electroplating of metals Electroplating, LIGA, stereolithography, laser-driven CVD, screen printing, microcontact printing, dip-pen lithography Plasma etching, reactive-ion enhanced etching (RIE), deep reactive-ion etching (DRIE), wet chemical etching, electrochemical etching Drilling, milling, electrical discharge machining (EDM), focused ion beam (FIB) milling, diamond turning, sawing Direct silicon-fusion bonding, fusion bonding, anodic bonding, adhesives Wet chemical modification, plasma modification, self assembled monolayer (SAM) deposition, grinding, chemomechanical polishing Thermal annealing, laser annealing

Electroplating Directed deposition Etching Machining Bonding Surface modification Annealing

hard coatings and protective layers, lower percolation threshold impacts conductivity, and narrower bandgap with decreasing grain size enable unique optoelectronics [9.35]. Hundreds of different synthesis routes have been created for manufacturing nanostructured materials. (See, for example, the proceedings of the International Conferences on Nanostructured Materials [9.36].) A few examples of such materials are given below. Carbon Nanotubes and Fullerenes Carbon nanotubes (CNTs) and fullerenes (buckyballs, e.g., C60 ) are self-assembled carbon nanostructures. CNTs are cylindrical graphene structures of single- or multiwall form which are extremely strong and flexible. They possess metallic or semiconducting electronic behavior depending on the details of the structure (chirality). They can be created in an arc plasma furnace, laser ablation, or grown by chemical vapor deposition (CVD) on a substrate using catalyst particles [9.37]. Quantum Dots, Quantum Wires, and Quantum Films Quantum behavior occurs in semiconductor materials (such as GaAs) when electrons are confined to nanoscale dimensions. The confined space forces electrons to have energy states that are clustered around specific peaks, producing fundamentally different electrical and optical properties than would be found in the same material in bulk form. The number of directions free of confinement is used to classify structures, thus two-dimensional (2-D) confinement leads to a quan-

tum film, one-dimensional (1-D) confinement leads to a quantum wire, and zero-dimensional (0-D) confinement leads to a quantum dot. The dimension of the confined direction(s) is so small that the energy states are quantized in that direction [9.37]. Nanowires A variety of methods have been developed for making nanowires of a wide range of metals, ceramics, and polymers. Examples include gold nanowires made by a solution method [9.38], palladium nanowires created by electroplating on a stepped surface [9.39], and zinc oxide nanowires created by a vapor/liquid/solid method [9.40]. In one popular technique, electroplating is conducted inside a nanoporous template of alumina or polycarbonate to direct the growth of nanowires [9.41, 42]. The template can be chemically removed, leaving the nanowires behind. In another application, lithographically patterned metal is used as a catalyst for silicon nanowire growth, creating predefined regions of nanowires on a surface [9.43, 44]. Using various combinations of metal catalysts and gases, a wide range of nanowire compositions can be created from chemical vapor deposition methods.

9.3.7 Micromachining Micromachining is a set of material removal and forming techniques for creating microscale movable features and complex structures, often from silicon. The micromachining processes listed in Table 9.4 can be applied to other materials such as glasses, ce-

A Brief Introduction to MEMS and NEMS

9.3.8 Hard Fabrication Techniques Hard MEMS utilizes enabling technologies for fabrication and design from the microelectronics industry. The MEMS industry has modified advanced techniques, leveraging well beyond the capability to fabricate integrated circuits. Micromachining involves three fundamental processes: deposition, lithography, and etching. Deposition may employ oxidation, chemical vapor deposition, physical vapor deposition, electroplating, diffusion, or ion implantation. Lithography methods include optical and electron-beam techniques. Etching methods include wet and dry chemical etches, which can be either isotropic (uniform etching in all directions, resulting in rounded features) or anisotropic (etching in one preferential direction, resulting in well-defined features).

9.3.9 Deposition Physical Vapor Deposition (PVD) Physical vapor deposition (PVD) includes evaporation and sputtering. The evaporation method is used to deposit metals on a surface from vaporized atoms removed from a target by heating with an electron beam. This technique is performed under high vacuum and produces very directional deposition and can create

shadows. Sputtering of a metallic or nonmetallic material is accomplished by knocking atoms off a target with a plasma of an inert gas such as argon. Sputtering is less directional and allows for higher deposition rates. Chemical Vapor Deposition (CVD) In chemical vapor deposition (CVD), precursor material is introduced into a heated furnace and a chemical reaction takes place on the surface of the wafer. The CVD process is generally performed under low-pressure conditions and is sometimes explicitly referred to as low-pressure CVD (LPCVD). A range of materials can be deposited by CVD, including films of silicon (formed by decomposition of silane (SiH4 )), silicon nitride formed by reacting dichlorosilane (SiH2 Cl2 ) with ammonia (NH3 )), and silicon oxide (formed by silane with an oxidizing species). LPCVD can produce amorphous inorganic dielectric films and polycrystalline polysilicon and metal films. Epitaxy is a CVD process where temperature and growth rate are controlled to achieve ordered crystalline growth in registration with the substrate. PECVD is a plasma-enhanced CVD process. Electroplating A variety of electroplating techniques are used to make micro- and nanoscale components. A mold is created into which metal is plated. Gold, copper, chromium, nickel, and iron are common plating metals. Spin Casting Spin casting is used to create films from a solution. The most common spin-cast material is polymeric photoresist. Sol–Gel Deposition A range of sol–gel processes can be used to make films and particles. The general technique involves a colloidal suspension of solid particles in a fluid that undergo a reaction to generate a gelatinous network. After deposition of the gel, the solvent can be removed to transform the network into a solid phase which is subsequently sintered. Piezoelectric materials such as PZT can be deposited with this method.

9.3.10 Lithography Most of the micromachining techniques discussed below utilize lithography, or pattern transfer, at some point in the manufacturing process. Depending on the resolution required to produce the desired feature sizes and the aspect ratio necessary, lithography is either per-

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ramics, polymers, and metals, but silicon is favored because of its widespread use and the availability of design and processing techniques. Other advantages of silicon include the availability of relatively inexpensive pure single-crystal substrate wafers, its desirable electrical properties, its well-understood mechanical properties, and ease of integration into a circuit for control and signal processing [9.7]. Although often performed in batch processes, micromachining for MEMS application may make large-aspect-ratio features and incorporation of novel or active materials a higher priority than batch manufacturing. This opens the door for a wider range of fabrication techniques such as focused ion-beam milling, laser machining, and electron-beam writing [9.22, 45, 46]. A brief overview of micromachining is provided below. The following books, used as references for the discussion here, are valuable resources for more extensive information [9.1, 2, 10, 18, 22, 33]. Additional information can be found in Taniguchi [9.47] and Evans [9.48] on microfabrication technology, Bustillo [9.49] on surface micromachining, and Gentili [9.50] on nanolithography.

9.3 Common MEMS/NEMS Materials and Their Properties

Solid Mechanics Topics

Part A 9.3

formed with ultraviolet light, an ion beam, x-rays, or an electron beam. X-ray lithography can produce features down to 10 nm and electron beams can be focused down to less than 1 nm [9.50]. Optical lithography allows aspect ratios of up to three whereas x-ray lithography can produce aspect ratios > 100. This large depth of focus, lack of scattering effects, and insensitivity to organic dust make x-ray lithography very attractive for NEMS production. Electron-beam lithograph has the attractive feature that a pattern can be directly written onto a resist, as well as the fact that it produces lower defect densities with a large depth of focus, but the process must be performed in vacuum. In most cases a mask that carries either a positive or negative image of the features to be created must first be produced. Masks are commonly made with a chromium layer on fused silica. Photoresist covering the chromium is exposed with an optical pattern generated from a sequence of small rectangles used to draw out the pattern desired. Other mask production techniques include photographic emulsion on quartz, electron-beam lithography with electron-beam resist, and high-resolution ink-jet printing on acetate or mylar film. Photolithographic fabrication techniques have a long history of use with ceramics, plastics, and glasses. In the case of silicon fabrication, the wafer is coated with a polymeric photoresist layer sensitive to ultraviolet light. Exposure of the photoresist layer is conducted through a mask. Depending on whether a positive or negative photoresist is used, the light either weakens the polymer or strengthens the polymer. In the developer step, chemicals are used to remove the weaker material, leaving a patterned photoresist layer behind. The photoresist acts as a protective layer when etching is conducted. Contact lithography produces a 1:1 ratio of the mask size and feature size. Proximity lithography also gives a 1:1 ratio with slightly lower resolution because a gap is left between the mask and the substrate to minimize damage to the mask. A factor of 5–10 reduction is common for projection step-and-repeat lithography. Because this technique allows for the production of feature sizes smaller then the mask, only a small region is exposed at one time and the mask must be stepped across the substrate.

9.3.11 Etching A number of wet and dry etchants have been developed for silicon. Important properties include orientation

dependence, selectivity, and the geometric details of the etched feature (Fig. 9.2). A common isotropic wet etchant for silicon is HNA (a combination of HF, HNO3 , and CH3 COOH), while anisotropic wet etchants include KOH, which etches {100} planes 100 times faster than {111} planes, tetramethylammonium hydroxide (called TMAH or (CH3 )4 NOH), which etches {100} planes 30–50 times faster than {111} planes but leaves silicon dioxide and silicon nitride unetched, and ethylenediamine pyrochatechol (EDP), which is very hazardous but does not etch most metals. Wet etchants such as HF for silicon dioxide, H3 PO4 for silicon nitride, KCl for gold, and acetone for organic layers, can be performed in batch processes with little cost [9.51]. An important feature of an etchant is its selectivity; for example, the etch rate of an oxide by HF is 100 nm/min compared to 0.04 nm/min for silicon nitride [9.51]. The etching reaction can be either reaction rate controlled or mass transfer limited. Because wet etchants act quickly, making it hard to control depth of the etch, electrochemical etching is sometimes employed using an electric potential to moderate the reaction along with a precision thickness epitaxial layer used for etch stop. The challenge comes with drying after the wet etching process is complete. Capillary forces can easily draw surfaces together, causing damage and stiction. Supercritical drying, where the liquid is converted to a gas, can be used to prevent this. Alternatively, application of a hydrophobic passivation layer such as a fluorocarbon polymer can be used to prevent stiction. Chemically reactive vapors and plasmas are highly effective dry etchants. Xenon difluoride (XeF2 ) is a commercially important highly selective vapor etchant for silicon. Dry etchants such as CHF3 + O2 for silicon dioxide, SF6 for silicon nitride, Cl2 + SiCl4 for

Wet etch

Plasma (dry) etch

Isotropic

Part A

Anisotropic

212

{111}

Fig. 9.2 Trench profiles produced by different etching processes (after Maluf [9.1])

A Brief Introduction to MEMS and NEMS

all parts of the wafer. Ion milling refers to selective sputtering and can be done uniformly over a wafer or with focusing electrodes by focused ion-beam milling (FIB). FIB is also becoming a more important technique for test sample production and the application of gratings used for interferometry [9.52]. In addition to FIB, techniques such as scanning probe microscope (SPM) lithography and molecular-beam epitaxy can also be used to create micro- and nanoscale gratings [9.53]. Beyond the use of etching as part of the initial fabrication of a device, some small adjustments may be required after the device is fabricated due to small variations that occur in processing. Compensation can be performed by trimming resistors and altering mechanical dimensions via techniques such as laser ablation and FIB milling. Calibration can be performed electronically with correction coefficients.

9.4 Bulk Micromachining versus Surface Micromachining The processes for silicon micromachining fall into two general categories: bulk (subtraction of substrate material) and surface (addition of layers to the substrate). Other techniques used on a range of materials include surface micromachining, wafer bonding, thin film screen printing, electroplating, lithography galvanoforming molding (LIGA, from the German Lithografie-Galvanik-Abformung), injection molding,

electric-discharge machining (EDM), and focused ion beam (FIB). Figure 9.3 provides a basic comparison of bulk micromachining, surface micromachining, and LIGA. Bulk Micromachining Removal of significant regions of substrate material in bulk micromachining is accomplished through

a)

b)

c)

Bulk micromachining

Surface micromachining

LIGA Resist structure Base plate

Deposition of sacrificial layer

Deposition of silica layers on Si Membrane face

Metal structure Electroforming

Patterning with mask

Gate plate

Patterning with mask and etching of Si to produce cavity Silicon

Silica

Lithography

Mold insert Mold fabrication

Deposition of microstructure layer Molding mass

Mold filling

Etching of sacrificial layer to produce freestanding structure Silicon

Polysilicon

Sacrificial material

Plastic structure

Unmolding

Fig. 9.3a–c Schematic diagrams depicting the processing steps required for (a) bulk micromachining (b) surface micromachining, and (c) LIGA. All views are shown from the side (after Bhushan [9.2] Chap. 50)

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aluminum, and O2 for organic layers, are used as a plasma [9.51]. The process is conducted in a specially designed system that generates a chemically reactive plasma species of ion neutrals and accelerates them towards a substrate with an electric or magnetic field. Plasma etching is the spontaneous reaction of neutrals with the substrate materials, while reactive-ion etching involves a synergistic role between the ion bombardment and the chemical reaction. Deep reactive-ion etching (DRIE) allows for the creation of high-aspectratio features. DRIE involves periodic deposition of a protective layer to shield the sidewalls either through condensation of reactant gasses produced by cryogenic cooling of the substrate or interim deposition cycles to put down a thin polymer film. Ions can also be used to sputter away material. For example, argon plasma will remove material from

9.4 Bulk Micromachining versus Surface Micromachining

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anisotropic etching of a silicon single-crystal wafer. The fabrication process includes deposition, lithography, and etching. Bulk micromachining is commonly used for high-volume production of accelerometers, pressure sensors, and flow sensors.

are used to produce a free-standing structure. Surface micromachining is attractive for integrating MEMS sensors with electronic circuits, and is commonly used for micromirror arrays, motors, gears, and grippers.

Surface Micromachining Alternating structural and sacrificial thin film layers are built up and patterned in sequence for surface micromachining. The process used by Sandia National Laboratory uses up to five structural polysilicon and five sacrificial silicon dioxide layers, whereas Texas Instrument’s digital micromirror device (discussed below) is made from a stack of structural metallic layers and sacrificial polymer layers [9.1, 54]. Deposition methods include oxidation, chemical vapor deposition (CVD), and sputtering. Annealing must sometimes be used to relax the mechanical stresses that build up in the films. Lithography and etching

LIGA Lithography galvanoforming molding (LIGA, from the German Lithografie-Galvanik-Abformung) is a lithography and electroplating method used to create very high-aspect-ratio structures (aspect ratios of more than 100 are common). The use of x-rays in the lithography process takes advantage of the short wavelength to create a larger depth of focus compared to photolithography [9.14]. Devices can be up to 1 mm in height with another dimension being only a few microns and are commonly made of materials such as metals, ceramics, and polymers. See Guckel [9.55], Becker [9.56], and Bley [9.57] for additional details.

9.5 Wafer Bonding Although microelectronics fabrication processes allow stacking layers of films, structures are relatively two dimensional. Wafer bonding provides an opportunity for a more three-dimensional structure and is commonly used to make pressure sensors, accelerometers, and microfluidic devices (Fig. 9.4). Anodic and direct bonding are the most common techniques, but

2.5μm LPCVD polysilicon mask

Sheat Sample inlet injector holes

Fused silica substrate Electrochemical-discharge machined through-holes

Two substrates thermally bonded together

Fig. 9.4 Schematic diagram depicting a wafer bonding process used to create a microfluidic channel for flow cytometry (after Kovacs [9.10])

bonding can also be achieved by using intermediate layers such as polymers, solders, and thin-film metals. Anodic (electrostatic) bonding can be used to bond silicon to a sodium-containing glass substrate (with a matched coefficient of thermal expansion) using an applied electric field. This is accomplished with the application of a large voltage at elevated temperature to make positive Na+ ions mobile. The positively charged silicon is held to the negatively charged glass by electrostatic attraction. Direct (silicon-fusion) bonding requires two flat, clean surfaces in intimate contact. Direct bonding of a silicon/glass stack can be achieved by applying pressure. Direct wafer bonding allows joining of two silicon surfaces or silicon and silicon dioxide surfaces and is used extensively to create SOI wafers. After treatment of the surfaces to produce hydroxyl (OH) groups, intimate contact allows van der Waals forces to make the initial bond followed by an annealing step to create a chemical reaction at the interface. Grinding and polishing is sometimes needed to thin a bonded wafer. Annealing must be performed afterwards to remove defects incurred during grinding. Alternatively, chemomechanical polishing can be used to combine chemical etching with the mechanical action of polishing.

A Brief Introduction to MEMS and NEMS

9.6 Soft Fabrication Techniques

Self-Assembly Partly because of the high cost of nanolithography and the time-consuming nature of atom-by-atom placement using probe microscopy techniques, self-assembly is an important bottom-up approach to NEMS fabrication [9.59]. To offset the time it takes to build unit by unit to create a useful device, massive parallelism and autonomy is required. The advantage of self-assembly is that it occurs at thermodynamic minima, relying on naturally occurring phenomena that govern at the nanoscale and create highly perfect assemblies [9.58]. The atoms, molecules, collections of molecules, or nanoparticles self-organize into functioning entities using thermodynamic forces and kinetic control [9.60]. Such self-organization at the nanoscale is observed naturally in liquid crystals, colloids, micelles, and self-assembled monolayers [9.61]. Reviews of self-assembly can be found in [9.62–65]. At the nanoparticle level, a variety of methods have been used to promote self-assembly. Three basic requirements must be met: there must be some sort of bonding force present between particles or the particles and a substrate, the bonding must be selective, and the particles must be in random motion to facilitate chance interactions with a relatively high rate of occurrence. Additionally, for the technique to be practical, the particles must be easily synthesized. Selectivity can be facilitated by micromachining the substrate including patterns with geometric designs that allow for only certain orientations of the mating particle. Particularly powerful are self-assembly methods using complementary pairs and molecular building blocks (analogous to DNA replication). Complementary pairs can bind electrostatically or chemically (using functional groups with couple monomers). Molecular Table 9.5 Techniques for creating patterned SAMs [9.58] Method

Scale of features

Microcontact printing Micromachining Microwriting with pen Photolithography/lift-off Photochemical patterning Photo-oxidation Focused ion-beam writing Electron-beam writing Scanning tunneling microscope writing

100 nm – some cm 100 nm – some μm ≈ 10–100 μm > 1 μm > 1 μm > 1 μm ≈ some μm 25–100 nm 15–50 nm

building blocks can use a number of different bonds and linkages (ionic bonds, hydrogen bonds, transition metal complex bonds, amide linkages, and ester linkages) to create building blocks for three-dimensional (3-D) nanostructures and nanocrystals such as quantum dots. Self-assembled monolayers (SAMs) can be produced in patterned form by several techniques that produce features in a range of micro- and nanoscale sizes (Table 9.5). Combined with lithography, defined areas of self-assembly on a surface can be created. Applications of SAMs include fundamental studies of wetting and electrochemistry, control of adhesion, surface passivation (to protect from corrosion, control oxidation, or use as resist), tribology, directed assembly, optical systems, colloid fabrication, and biologically active surfaces for biotechnology [9.58]. Soft Lithography The term soft lithography encompasses a number of techniques that can be used to fabricate microand nanoscale structures using replica molding and self-assembly. These techniques include microcontact printing, replica molding, microtransfer molding, micromolding in capillaries, and solvent-assisted micromolding [9.66]. As an example, microcontact printing uses a selfassembled monolayer as ink in a stamping operation that transfers the SAM to a surface (Fig. 9.5). The stamp is fabricated from of an elastomeric material such as PDMS by casting onto a master with surface features. The master can be produced with a range of photolithographic techniques. The polymeric replica mold is used as a stamp to enable physical pattern transfer. The advantages of microcontact printing are its simplicity, conformal contact with a surface, the reusability of the stamp, and the ability to produce multiple stamps from one master. Although defect density and registration of patterns over large scales can be issues, the flexibility of the stamp can be use to make small features (≈ 100 nm) using compression or pattern transfer onto curved surfaces [9.58]. The aspect ratio of features is a constraint with PDMS however. Ratios between 0.2 and 2 must be used to ensure defect-free stamps and molds [9.67].

9.6.1 Other NEMS Fabrication Strategies Nanoscale structures can be created from both topdown and bottom-up approaches. Because of the push

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9.6 Soft Fabrication Techniques

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to miniaturize commercial electronics, many top-down methods are refinements of micromachining techniques with the goal of achieving manufacturing accuracy on the nanometer scale. Bottom-up methods rely on additive atomic and molecular techniques, such as self-organization, self-assembly, and templating, using building blocks similar and size to those used in nature [9.68]. A brief review of some additional examples is provided below. Nanomachining Scanning probe microscopes (SPMs) are a valuable set of tools for NEMS characterization, but these tools Photoresist

Si Photolithography is used to create a master Photoresist pattern (1–2 μm thickness)

Si

PDMS is poured over master and cured

PDMS Photoresist pattern

Si PDMS is peeled away from the master

PDMS PDMS is exposed to a solution containing HS(CH2)15CH3

PDMS

Alkanethiol

Stamping onto gold substrate transfers thiol to form SAM

SAMs (1–2 nm) Au (5–2000 nm)

Si

Ti (5–10 nm) Metal not protected by SAM is removed by exposure to selective chemical etchant

Si

Fig. 9.5 Schematic diagram depicting the processing steps required for microcontact printing. All views are shown from the side (after Wilbur [9.58])

can also be used for NEMS manufacturing. These microscopes share the common feature that they employ a nanometer-scale probe tip in the proximal vicinity of a surface. They are many times more powerful than scanning electron microscopes because their resolution is not determined by wavelength for the interaction with the surface under investigation. The scanning tunneling microscope (STM) can be used to create a strong electric field in the vicinity of the probe tip to manipulate individual atoms. Atoms can be induced to slide over a surface in order to move them into a desired arrangement by mechanosynthesis [9.69]. Resolution is effectively the size of a single atom but practically the process is exceptionally time consuming and the sample must be held at very low temperature to prevent movement of atoms out of place [9.70]. With slightly less resolution but still less than 100 nm, an STM can also be used to write on a chemically amplified negative electron-beam resist. Nanolithography Surface micromachining can be conducted at the nanoscale using electron-beam lithography to create free-standing or suspended mechanical objects. Although the general approach parallels standard lithography (see above), the small-scale ability of this technique is enabled by the fact that an electron beam with energy in the keV range is not limited by diffraction. The electron beam can be scanned to create a desired pattern in the resist [9.8]. Nanoscale resolution can also be obtained using alternative lithographic techniques such as dip-pen nanolithography (DPN) [9.71]. This technique employs an atomic force microscope (AFM) probe tip to deposit a layer of material onto a surface, much as a pen writes on paper. A pattern can be drawn on a surface using a wide range of inks such as thiols, silanes, metals, sol–gel precursors, and biological macromolecules. Although the DPN process is inherently slower that standard mask lithographic techniques, it can be used for intricate functions such as mask repair and the application of macromolecules in biosensor fabrication, or it can be parallelized to increase speed [9.72]. This and other nanofabrication techniques using AFM to modify and pattern surfaces are reviewed by Tang [9.73].

9.6.2 Packaging Packaging of a MEMS/NEMS device provides a protective housing to prevent mechanical damage, minimize stresses and vibrations, guard against contamination,

A Brief Introduction to MEMS and NEMS

considerations include thickness of the device, wafer dicing (separation of the wafer into separate dice), sequence of final release, cooling of heat-dissipating devices, power dissipation, mechanical stress isolation, thermal expansion matching, minimization of creep, protective coatings to mitigate damaging environmental effects, and media isolation for extreme environments [9.1]. In the die-attach process, each individual die is mounted into a package, by bonding it to a metal, ceramic or plastic platform with a metal alloy solder or an adhesive. For silicon and glass, a thin metal layer must be placed over the surface prior to soldering to allow for wetting. Electrical interconnects can be produced with wire bonding (thermosonic gold bonding with ultrasonic energy and elevated temperature) and flip-chip bonding (using solder bumps between the die and package pads). Fluid interconnects are created by insertion of capillary tubes, mating of self-aligning fluid ports, and laminated layers of plastic [9.1].

9.7 Experimental Mechanics Applied to MEMS/NEMS With a basic understanding of the materials and processes used to make MEMS/NEMS devices, the role of mechanics in materials selection, process validation, design development, and device characterization can now be discussed. The remainder of this chapter will

focus on the forces and phenomena dominant at the micrometer and nanometer scales, basic device characterization techniques, and mechanics issues that arise in MEMS/NEMS devices.

9.8 The Influence of Scale To gain perspective on the micrometer and nanometer size scales, consider that the diameter of human hair is 40–80 μm and a DNA molecule is 2–3 nm wide. The weight of a MEMS structure can be about 1 nN and that of a NEMS components about 10−20 N. Compare this to the mass of a drop of water (10 μN) or an eyelash (100 nN) [9.2]. The minuscule size of forces that influence behavior at these small scales is hard to imagine. For instance, if you take a 10 cm length of your hair and hold it like a cantilever beam, the amount of force placed on the tip of the cantilever to deflect it by 1 cm is on the order of 1 pN. That piece of hair is 40–80 μm in diameter, which is large compared to most MEMS/NEMS components. In dealing with micro- and nanoscale devices, engineering intuition developed through experience with

macroscale behavior is often misleading. It should be noted that many macroscale techniques can be applied at the micro- and nanoscales, but advantages come not from miniaturization but rather working at the relevant size scale using the uniqueness of the scale. The balance of forces at these scales differs dramatically from the macroscale (Table 9.6). Compared to a macroscale counterpart of the same aspect ratio, the structural stiffness of a microscale cantilever increases relative to inertially imposed loads. When the length scale changes by a factor of a thousand, the area decreases by a factor of a million and the volume by a factor of a billion. Surface forces, proportional to area, become a thousand times large than forces that are proportional to volume, thus inertial and electromagnet forces become negligible. At small scales, adhesion, friction, stiction

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protect from harsh environmental conditions, dissipate heat, and shield from electromagnetic interference [9.15]. Packaging is critical because it enables the usefulness, safety, and reliability of the device. Hermetic packaging made of metal, ceramic, glass or silicon is used to prevent the infiltration of moisture, guard against corrosion, and eliminate contamination. The internal cavity is evacuated or filled with an inert gas. For MEMS/NEMS the packaging may also be required to provide access to the environment through electrical and/or fluid interconnects and optically transparent windows. In these cases, the devices are left more vulnerable in order for them to interact with the environment to perform their function. Although there are well-established techniques for packaging of common microelectronics devices, packaging of MEMS/NEMS presents particular challenges and may account for 75–95% of the overall cost of the device [9.1]. Packaging design must be conducted in parallel with design of the MEMS/NEMS component. Design

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Table 9.6 Scaling laws and the relative importance of phenomena as they depend on linear dimension, i (after Madou [9.22]) Importance at small scale Diminished

Increased

Phenomena

Power of linear dimension

Flow Gravity Inertial force Magnetic force Thermal emission Electrostatic force Friction Pressure Piezoelectricity Shape-memory effect Velocity Surface tension Diffusion van der Waal force

l4 l3 l3 l 2 , l 3 or l 4 l 2 or l 4 l2 l2 l2 l2 l2 l l l 1/2 l 1/4

(static friction), surface tension, meniscus forces, and viscous drag often govern. Acceleration of a small object becomes rapid. At the nanoscale, phenomena such as quantum effects, crystalline perfection, statistical time variation of properties, surface interactions, and interface interactions govern behavior and materials properties [9.35]. Additionally, the highly coupled nature of thermal transport properties at the microscale can be either an advantage or disadvantage depending on the device. Enhanced mass transport due to large surface-to-volume ratio can be a significant advantage for applications such as capillary electrophoresis and gas chromatography. However, purging air bubbles in microfluidic systems can be extremely difficult due to capillary forces. The interfacial surface tension force will cause small bubble less than a few millimeters in diameter to adhere to channel surfaces because the mass of liquid in a capillary tube produces an insubstantial inertial force compared to the surface tension [9.10]. Some scaling effects favor particular micro- and nanoscale situations but others do not. For instance, large surface-to-volume ratio in MEMS devices can undermine device performance because of the retarding effects of adhesion and friction. However, electrostatic force is a good example of a phenomena that can have substantial engineering value at small scales. Transla-

tional motion can be achieved in MEMS by electrostatic force because this scales as l 2 as compared to inertial force which scales as l 3 . Microactuation using electrostatic forces between parallel plates is used in comb drives, resonant microstructures, linear motors, rotary motors, and switches. In relation to MEMS testing, gripping of a tension sample can be achieved using an electrostatic force between a sample and the grip [9.74]. It is also important to note that, as the size scale decreases, breakdown in the predictions of continuumbased theories can occur at various length scales. In the case of electrostatics, electrical breakdown in the air gap between parallel plates separated by less than 5 μm does not occur at the predicted voltage [9.75]. In optical devices, nanometer-scale gratings can produce an effective refractive index different from the natural refractive index of the material because the grating features are smaller than the wavelength of light [9.76]. For resonant structures continuum mechanics predictions break down when the structure’s dimensions are on the order of tens of lattice constants in cross section [9.11]. Detailed discussions of issues related to size scale can be found in Madou [9.22] and Trimmer [9.77].

9.8.1 Basic Device Characterization Techniques A range of mechanical properties are needed to facilitate design, predict allowable operating limits, and conduct quality control inspection for MEMS. As with any macroscale device or component, structural integrity is critical to MEMS/NEMS. Concerns include friction/stiction, wear, fracture, excessive deformation, and strength. Properties required for complete understanding of the mechanical performance of MEMS/NEMS materials include elastic modulus, strength, fracture toughness, fatigue strength, hardness, and surface topography. In MEMS devices the minimum feature size is on the order of 1 μm, which is also the natural length scale for microstructure (such as the grain size, dislocation length, or precipitate spacing) in most materials. Because of this, many of the mechanical properties of interest are size dependent, which precipitates the need for new testing methods given that knowledge of material properties is essential for predicting device reliability and performance. A detailed discussion about micro- and nanoscale testing can be found in Part B of this handbook as well as in references such as Sharpe [9.78], Srikar [9.79], Haque and Saif [9.80], Bhushan [9.2], and Yi [9.81]. The following sections

A Brief Introduction to MEMS and NEMS

9.8.2 Residual Stresses in Films Many MEMS/NEMS devices involve thin films of materials. Properties of thin-film material often differ from their bulk counterparts due to the high surfaceto-volume ratio of thin films and the influence of surface properties. Additionally, these films must have good adhesion, low residual stress, low pinhole density, good mechanical strength, and good chemical resistance [9.22]. These properties often depend on deposition and processing details. The stress state of a thin film is a combination of external applied stress, thermal stress, and intrinsic residual stress that may arise due to factors such as doping (in silicon), grain boundaries, voids, gas entrapment, creep, and shrinkage with curing (in polymeric materials). Stresses that develop during deposition of thin-film material can be either tensile or compressive and may give rise to cracking, buckling, blistering, delaminating, and void formation, all of which degrade device performance. Residual stresses can arise because of coefficient of thermal expansion mismatch, lattice mismatch, growth processes, and nonuniform plastic deformation. Residual stresses that do not cause mechanical failure may still significantly affect device performance by causing warping of released structures, changes in resonant frequency of resonant structures, and diminished electrical characteristics. In some instances, however, residual stresses can be used productively, such as in shape setting of shape-memory alloy films or stress-modulated growth and arrangement of quantum dots. There are numerous techniques for measuring residual stresses in thin films. Fundamental techniques rely on the fact that stresses within a film will cause bending in its substrate (tension causing concavity, compression causing convexity). Simple displacement measurements can be conducted on a circular disk or a micromachined beam and stress calculated from the radius of curvature of the bent substrate or the deflection of a cantilever. Strain gages may also be made directly in the film and used to make local measurements. Freestanding portions of the thin film can be created by micromachining so that the films stresses can be explored by applied

pressure, external probe, critical length for buckling, or resonant frequency measurements. For instance, the critical stress to cause buckling in a doubly supported beam can be estimated from: π 2t2 , K L2 where K is a constant determined by the boundary conditions (3 for a doubly supported beam), E is Young’s modulus, t is the beam thickness, and L is the shortest length of beam displaying buckling [9.10]. The stress or strain gradient over a region of a film can be found by measuring deflections in a simple cantilever. The upward or downward deflection along the length of the beam can be measured by optical methods and used to estimate the internal bending moment M from the expression: σCR = E

(1 − ν2 ) δ (x) =K+ Mx , x 2E I where δ(x) is the vertical deflection at a distance x from the support, E is Young’s modulus, ν is Poisson’s ratio, I is the moment of the beam cross section about the axis of bending, and K is a constant determined by the boundary conditions at the support [9.83]. A number of techniques have been developed for determining residual stresses including an American Society for Testing and Materials (ASTM) standard involving optical interferometry [9.84]. The bulge test is a basic technique for measuring residual stress in a freestanding thin film [9.85]. The bulge test structure can be easily created by micromachining with well-defined boundary conditions. The M-test is an on-chip test that uses bending of an integrated free-standing prismatic beam [9.86]. The principle of an electrostatic actuator is used to conduct the test to find the onset of instability in the structure. The wafer curvature test is regularly used for residual stress measurement in nonintegrated film structures, and can be used even when the film thickness is much smaller than the substrate thickness [9.87]. Dynamic testing can be used to measure resonant frequency and extract information about residual stress and modulus. Resonant frequency increases with tension and decreases in compression [9.88, 89]. Air damping can significantly impact theses measurements, however, so they must be conducted in vacuum [9.18]. Other established techniques that can be employed to measure residual stresses in films include passive strain sensors, Raman spectroscopy, and nanoindentation [9.79]. More recently, nanoscale gratings created by focused ion-beam (FIB) milling have been used to

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provide a review of some mechanics issues that arise at the device level. The following sources, used as references for the discussion below, should be consulted for additional background on mechanics, metrology, and MEMS: Trimmer [9.77], Madou [9.22], Bhushan [9.2], and Gorecki [9.82].

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produce moiré interference between the grating on the specimen surface and raster scan lines of a scanning electron microscope (SEM) image [9.90]. This technique can be used to provide details of residual strains in microscale structures as they evolve with etching of the underlying sacrificial layer [9.52]. Digital image correlation (DIC) has also been applied to SEM and atomic force microscopy (AFM) images in combination with FIB. DIC is used to capture deformation fields while nearby FIB milling of the specimens releases residual stresses, allowing very local evaluation [9.91].

9.8.3 Wafer Bond Integrity Wafer bonding is often an essential device fabrication step, particularly for microfluidic devices, microengines, and microscale heat exchangers. Although direct bonding of silicon can achieve strengths comparable to bulk silicon, the process is sensitive to bonding parameters such as temperature and pressure. The appearance of voids and bubbles at the interface is particularly undesirable for both strength and electrical conductivity [9.92]. An important nondestructive technique for assessing the bond quality of bonded silicon wafers is infrared transmission. At IR wavelengths of about 1.1 μm silicon is transparent [9.1]. Quantification of bond strength can be conducted with techniques such as the pressure burst test, tensile/shear test, knife-edge test, or four-point bend-delamination test [9.93]. Although a range of techniques and processes can be employed to bond both similar and dissimilar materials, the stresses and deformation of the wafers that develop are consistent. The residual stress stored in the bonded wafers is important because it may provide the elastic strain energy to drive fracture. Details of the wafer geometry can impact the final shape of the bonded pair and the integrity of the bond interface [9.94].

9.8.4 Adhesion and Friction Adhesion is both essential and problematic for MEMS/NEMS. For multilayered devices, good adhesion between layers is critical for overall performance and reliability, where delamination under repetitive applied mechanical stresses must be avoided. Adhesion between material layers can be enhanced by improved substrate cleanliness, increased substrate roughness, increased nucleation sites during deposition, and addition of a thin adhesion-promoting layer. Standard tests for film adhesion include: the scotch-tape test, abrasion,

scratching, deceleration using ultrasonic and ultracentrifuge techniques, bending, and pulling [9.95]. In situ testing of adhesion can also be conducted by pressurizing the underside of a film until initiation of delamination. This method also allows the determination of the average work of adhesion. Adhesion can be problematic if distinct components or a component and the nearby substrate come into contact, causing the device to fail. For example, although the mass in an accelerometer device is intended to be free standing at all points of operation, adhesion can occur in the fabrication process. Commonly with freestanding portions of MEMS structures, the capillary forces present during the drying of a device after etching to remove sacrificial material are large enough to cause collapse of the structure and failure due to adhesion [9.96]. To avoid this problem, supercritical drying is used. Contacting surfaces that must move relative to one another in MEMS/NEMS are minimized or eliminated altogether, the reason being that friction and adhesion at these scales can overwhelm the other forces at play. Because silicon readily oxidizes to form a hydrophilic surface, it is much more susceptible to adhesion and accumulation of static charge [9.97]. When contacting surfaces are involved, lubricant films and hydrophobic coatings with low surface energy can be applied to minimize wear and stiction (the large lateral force required to initiate relative motion between two surfaces). For instance, Analog Devices uses a nonpolar silicone coating in its accelerometers to resist charge buildup and stiction [9.98]. Processing plays a major role in surface properties such as friction and adhesion. Polishing will dramatically affect roughness, as in the case of polysilicon where roughness can be reduced by an order of magnitude from the as-deposited state [9.2]. The doping process can also lead to higher roughness. Organic monolayer films show promise for lubrication of MEMS to reduce friction and prevent wear. The atomic force microscope and the surface force apparatus used to quantify friction and MEMS test structures such as those developed at Sandia National Laboratory are aiding the development of detailed mechanics models addressing friction [9.99–102]. Flow Visualization Flow in the microscale domain occurs in a range of MEMS devices, particularly in bioMEMS, microchannel networks, ink-jet printer heads, and micropropulsion systems. The different balance of forces at micro-

A Brief Introduction to MEMS and NEMS

flow fields in microfluidic devices, where micron-scale spatial resolution is critical [9.103]. Microparticle image velocimetry (μPIV) has been used to characterize such things as microchannel flow [9.104] and microfabricated ink-jet printer head flow [9.105]. For the high-velocity, small-length-scale flows found in microfluidics, high-speed lasers and cameras are used in conjunction with a microscope to image the particles seeded in the flow. With μPIV techniques, the flow boundary topology can be measured to within tens of nanometers [9.106].

9.9 Mechanics Issues in MEMS/NEMS 9.9.1 Devices A wide range of MEMS/NEMS devices is discussed in the literature, both as research and commercialized devices. These devices are commonly planar in nature and employ structures such as cantilever beams, fixed–fixed beams, and springs that are loaded in bending and torsion. A range of mechanics calculations are needed for device characterization, including the effective stiffness of composite beams, deflection analysis of beams, modal analysis of a resonant structures, buckling analysis of a compressively loaded beams, fracture and adhesion analysis of structures, and contact mechanics calculations for friction and wear of surfaces. A substantial literature is available on the application of mechanics to MEMS/NEMS devices. The selected MEMS/NEMS examples presented below were chosen for their illustrative nature. Digital Micromirror Device Optical MEMS devices range from bar-code readers to fiber-optic telecommunication, and use a range of wideband-gap materials, nonlinear electro-optic polymers, and ceramics [9.107]. (See Walker and Nagel [9.108] for more information on optical MEMS.) A wellestablished commercial example of an optical MEMS device is the Digital Micromirror DeviceTM (DMD) by Texas Instruments used for projection display (Fig. 9.6) [9.109]. These devices have superior resolution, brightness, contrast, and convergence performance compared to conventional cathode ray tube technology [9.2]. The DMD contains a surface micromachined array of half a million to two million independently controlled, reflective, hinged micromirrors that have a mechanical switching time of 15 μs [9.110]. This de-

vice steers a reflected beam of light with each individual mirrored aluminum pixel. Pixel motion is driven by an electrostatic field between the yoke and an underlying electrode. The yoke rotates and comes to rest on mechanical stops and its position is restored upon release by torsional hinge springs [9.111]. Almost all commercial MEMS structures avoid any contact between structural members in the operation of the device, and sliding contact is avoided completely because of stiction, friction, and wear. The DMD is currently the only commercial device where structural components come in and out of contact, with contact occurring between the mirror spring tips and the underlying mechanical stops, which act as landing sites. To prevent adhesion problems in the DMD, a self healing perfluorodecanoic acid coating is used on the structural aluminum components [9.112]. Other challenges for the DMD include creep and fatigue behavior in the hinge, shock and vibration, and sensitivity to debris within the package [9.2]. The primary failure mechanisms are surface contamination and hinge memory due to creep in the metallic alloy resulting in a residual tilt angle [9.1]. Heat transfer, which contributes to the creep problem, is also an issue for micromirrors. When the reflection coefficient is less than 100% some of the optical power is absorbed as heat and can cause changes in the flatness of the mirror, damage to the reflective layer, and alterations in the dynamic behavior of the system [9.113]. Micromirrors for projection display involve rotating structures and members in torsion. Such torsional springs must be well characterized and their mechanics well modeled. For production devices extensive finite element models are developed to optimize performance [9.114]. For initial design calculations however,

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scopic length scales can influence fluid flow to produce counterintuitive behavior in microscopic flows. Additionally, the breakdown in continuum laws for fluid flow begins to occur at the microscale. For instance, the no-slip condition no longer applies and the friction factor starts to decrease with channel reduction. Particle image velocimetry (PIV) is a technique commonly used at macroscopic length scales to measure velocity fields through the use of particles seeded in the fluid. The technique has been adapted to measure

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some closed-form solutions for mechanics analysis can be employed. For instance, an appropriate material can be chosen or the basic dimensional requirements can be found from calculation of the maximum shear stress τmax in a beam of elliptical cross section in torsion (with a and b the semi-axis lengths) using: 2Gαa2 b , for a > b , a 2 + b2 where G is the shear modulus and α in the angular twist [9.107]. Mechanical integrity of the DMD relies on low stresses in the hinge, thus the tilt angle is limited to ±10◦ [9.1]. τmax =

atomic force microscopy [9.117, 118], and magnetic beads [9.119], but these techniques have the disadvantage of requiring external probes, labeling, and/or optical excitation. Alternatively, there are several methods using molecular recognition and the small-scale forces created by events such as DNA hybridization and receptor–ligand binding to produce bending in cantilevers to create sensors with high selectivity and resolution [9.115, 120]. Microcantilever sensors have been used for some time to detect changes in relative humidity, temperature, pressure, flow, viscosity, sound, natural gas, mercury vapor, and ultraviolet and infrared radiation. More re-

Biomolecular Recognition Device Biological molecules can be probed by external methods using techniques such as optical tweezers [9.116], a)

100 μm Oligonucleotide

x

b)

Mirror –10 deg Mirror +10 deg

Hybridization

Δx Hinge Yoke Landing tip

CMOS substrate

Fig. 9.6 (a) SEM image of yoke and hinges of one pixel with mirror removed (b) Schematic of two tilted pixels with mirrors (shown as transparent) (reprinted with permission, Hornbeck [9.111], Bhushan [9.2])

Fig. 9.7 SEM image of a portion of the cantilever sensor

array and schematics illustrating functionalized cantilevers with selective sensing capability (reprinted with permission, Fritz [9.115])

A Brief Introduction to MEMS and NEMS

Δm ≈ 2

Meff Δω ω0

where Meff is the effective vibratory mass of the resonator, and ω0 is the resonance frequency of the device [9.11]. The mass sensitivity of NEMS devices with micromachined cantilevers can be as small as a single small molecule (in the range of a single Dalton). In a device such as that shown in Fig. 9.7, a liquid medium, which contains molecules that dock to a layer of receptor molecules attached to one side of the cantilever, is injected into the device. Sensitizing an array of cantilevers with different receptor allows docking of different substances in the same solution [9.115]. Hybridization can be done with short strands of single-stranded DNA and proteins known a)

to recognize antibodies. When docking occurs, the increase in the molecular packing density leads to surface stress, causing bending (10–20 nm of deflection). This deflection can be measured by a laser beam reflected off of the end of the cantilever [9.115]. Alternatively, simple geometric interference by interdigitated cantilevers that act as diffraction gratings can be used to provide output of a binding event [9.120]. Thermomechanical Data Storage Device Much of the drive to nanometer-scale devices originates in the desire for higher density and faster computational devices. Magnetic data storage has been pushed into the nanoscale regime, but limitations have prompted the development of alternative methods for data storage such as the NEMS device known as the Millipede, developed by IBM. The Millipede, or scanning probe array memory device, is an array of individually addressable scanning probe tips (similar to atomic force microscope probe tips) that makes precisely positioned indentations in a polymer thin film. The Millipede is scanned to address a large area for data storage. The indentations are bits of digital information. A polymer thin film (50 nm thick) of polymethyl methacrylate (PMMA) is used for write, read, erase, and rewrite operations. Each individual bit is a nanoscale feature, which allows the Millipede to extend storage density to the Tbit/in2 range with a bit size of 30–40 nm [9.123]. The device uses multiple cantilever probe tips equipped with integrated heaters which allow for data transfer rates of up to a few Mb/s [9.124].

Multiplex driver

2-D cantilever array chip

b) Highly doped silicon cantilever leg

Nickel bridge

x

Metal 1 (Gold) Metal 2 (Nickel)

z3 Low-doped Schottky diode area y

Polymer storage media on x/y/z scanner

z1 z2

Stress-controlled nitride

Silicon cantilever

Heater platform

Fig. 9.8 (a) Schematic illustration of the millipede device, (b) with a detail of one cantilever cell (reprinted with permission, Despont [9.122])

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cently micromachined cantilevers have been used to interact and probe material at the molecular level. Devices employing these micromachined cantilevers can be dynamic, which are sensitive to mass changes down to 10–21 g (the single molecule level), or static, which are sensitive to surface stress changes in the low mN/m range (changes in Gibbs free energy caused by binding site-analyte interactions) [9.121]. In this case adhesion is required between the device and the material to be detected. In a functionalized cantilever array device produced to measure biomechanical forces created by DNA hybridization or receptor–ligand binding, detection of the mass change is accomplished by measuring a shift in resonant frequency. The responsiveness of the device to a change in mass is given by the expression:

9.9 Mechanics Issues in MEMS/NEMS

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Substrate Polymer Cantilever Write current Pit: 25nm deep 40nm wide (maximum)

Erasure current

Sensing current

Inscribed pins Data stream 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 Output signal 0

The Millipede device is a massively parallel structure with a large array of thousands of probe tips (100 cantilevers/mm2 ), each of which is able to address a region of the substrate where it produces indentations for use as data storage bits (Fig. 9.8) [9.125]. As illustrated in Fig. 9.9, the probes writes a bit by heating and a mechanical force applied between a can-

Fig. 9.9 Schematic of the writing, erasing, and reading op-

erations in the Millipede device. Data are mechanically stored in pits on a surface (reprinted with permission, Vettiger [9.127])

tilever tip and a polymer film. Erasure of a bit is also conducted with heating by placing a small pit just adjacent to the bit to be erased or using the spring back of the polymer when a hot tip is inserted into a pit. Reading is also enabled by heat transfer since the sensing relies on a thermomechanical sensor that exploits temperature-dependent resistance [9.126]. The change in temperature of a continuously heated resistor is monitored while the tip is scanned over the film and relies on the change in resistance that occurs when a tip moves into a bit [9.123]. Scanning x, y manipulation is conducted magnetically with the entire array at once. The data storage substrate is suspended above the cantilever array with leaf springs which enables the nanometer-scale scanning tolerances required. The cantilevers are precisely curved using residual stress control of a silicon nitride layer in order to minimize the distance between the heating platform of the cantilever and the polymer film while maximizing the distance between the cantilever array and the film substrate to ensure that only the tips come into contact [9.123]. Fabrication details are given in Despont [9.125]. Thermal expansion is a major hurdle for this device since a shift of ≈ 30 nm can cause misalignment of the data storage substrate and the cantilever array. A 10 nm tip position accuracy of a 3 mm × 3 mm silicon area requires that temperature of the device be controlled to 1 ◦ C using several sensors and heater elements [9.123]. Tip wear due to contact between the tip and the underlying silicon substrate is an issue for device reliability. Additionally, the PMMA is prone to charring at the temperatures necessary for device operation (around 350 ◦ C) [9.128] so new polymeric formulations had to be developed to minimize this problem. The feasibility of using thin-film NiTi shape-memory alloy (SMA) for thermomechanical data storage as an alternative to the polymer thin film has also been shown [9.129].

9.10 Conclusion The sensors, actuators, and passive structures developed as MEMS and NEMS devices require a highly interdisciplinary approach to their analysis, design, de-

velopment, and fabrication. Experimental mechanics plays a critical role in design development, materials selection, prediction of allowable operating lim-

A Brief Introduction to MEMS and NEMS

MEMS/NEMS testing must be further developed. This chapter has provided a brief review of the fabrication processes and materials commonly used and experimental mechanics as it is applied to MEMS and NEMS.

References 9.1 9.2 9.3

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9.10 9.11

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N. Maluf: An Introduction to Microelectromechanical Systems Engineering (Artech House, Boston 2000) B. Bhushan (Ed.): Springer Handbook of Nanotechnology (Springer, Berlin, Heidelberg 2006) Small Tech Business DirectoryTM Guide, Small Times Media, http://www.smalltechdirectory.com/ directory/directory.asp. Accessed July 20, 2007. J.C. Eloy: The MEMS industry: Current and future trends, OnBoard Technol., 22–24 (2006), http://www.onboard-technology.com/ pdf_settembre2006/090604.pdf (Accessed July 20, 2007) NewswireToday: Nanorobotics and NEMS to Reach $830.4 million by 2011, 05/31/2007, http://www.newswiretoday.com/news/18867/ (Accessed July 20, 2007) B. Ziaie, A. Baldi, M. Lei, Y. Gu, R.A. Siegel: Hard and soft micromachining for BioMEMS: review of techniques and examples of applications in microfluidics and drug delivery, Adv. Drug Deliv. Rev. 56, 145–172 (2004) S. Beeby, G. Ensell, M. Kraft, N. White: MEMS Mechanical Sensors (Artech House, Boston 2004) H.G. Craighead: Nanoelectromechanical systems, Science 290, 1532–1536 (2000) M. Koch, A.G.R. Evans, A. Brunnschweiler: Microfluidic Technology and Applications (Research Studies, Baldock 2000) G.T.A. Kovacs: Micromachined Transducers – Sourcebook (McGraw-Hill, New York 1998) K.L. Ekinci: Electromechanical transducers at the nanoscale: Actuation and sensing of motion in nanoelectromechanical systems (NEMS), Small 1(8-9), 786–797 (2005) K.L. Ekinci, M.L. Roukes: Nanoelectromechanical systems, Rev. Sci. Instrum. 76(061101), 1–12 (2005) P. Gammel, G. Fischer, J. Bouchaud: RF MEMS and NEMS technology, devices, and applications, Bell Labs Tech. J. 10(3), 29–59 (2005) W.A. Goddard III, D.W. Brenner, S.E. Lyshevski, G.J. Iafrate (Eds.): Handbook of Nanoscience, Engineering and Technology (CRC, Boca Raton 2002) S.E. Lyshevski: Nano- and Microelectromechanical Systems: Fundamentals of Nano- and Microengineering (CRC, Boca Raton 2000) T.S. Tighe, J.M. Worlock, M.L. Roukes: Direct thermal conductance measurements on suspended

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9.28

monocrystalline nanostructures, Appl. Phys. Lett. 70(20), 2687–2689 (1997) R.G. Knobel, A.N. Cleland: Nanometre-scale displacement sensing using a single electron transistor, Nature 424(6946), 291–293 (2003) S.D. Senturia: Microsystem Design (Kluwer Academic, Boston 2001) SEM: Society for Experimental Mechanics, Bethel, 2007 SEM Annual Conference Proceedings, Springfield (2007). MEMS and Nanotechnology Clearinghouse: Material Index, http://www.memsnet.org/material/ (Accessed July 20, 2007) P. Lange, M. Kirsten, W. Riethmuller, B. Wenk, G. Zwicker, J.R. Morante, F. Ericson, J.A. Schweitz: Thisck polycrystalline silicon for surface micromechanical applications: deposition, structuring and mechanical characterization, 8th International conference on solid-state sensors and actuators (Transducers ’95), Stockholm (1995) pp. 202–205 M. Madou: Fundamentals of Microfabrication (CRC, Boca Raton 1997) B. Diem, M.T. Delaye, F. Michel, S. Renard, G. Delapierre: SOI(SIMOX) as a substrate for surface micromachining of single crystalline silicon sensors and actuators, 7th International Conference on Solid-State Sensors and Actuators (Transducers ’93), Yokohama (1993) pp. 233–236 J.M. Noworolski, E. Klaassen, J. Logan, K. Petersen, N. Maluf: Fabricationof SOI wafers with buried cavities using silicon fusion bonding ans electrochemical etchback, 8-th International conference on solid-state sensors and actuators (Transducers ’95), Stockholm (1995) pp. 71–74 T. Ikeda: Fundamentals of Piezoelectricity (Oxford Univ., New York 1990) P. Gluche, A. Floter, S. Ertl, H.J. Fecht: Commercial applications of diamond-based nano- and microtechnolgy. In: The Nano-Micro Interface, ed. by H.-J. Fecht, M. Werner (Wiley-VCH, Weinheim 2004) pp. 247–262 S. Cho, I. Chasiotis, T.A. Friedmann, J.P. Sullivan: Young’s modulus, Poisson’s ratio and failure properties of tetrahedral amorphous diamond-like carbon for MEMS devices, J. Micromech. Microeng. 15, 728–735 (2005) A.R. Krauss, O. Auciello, D.M. Gruen, A. Jayatissa, A. Sumant, J. Tucek, D.C. Mancini, N. Moldovan,

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its, device characterization, process validation, and quality control inspection. Commercial devices exist, and research in the area of MEMS/NEMS is extremely active, but many challenges remain. Advanced materials must be well characterized and

References

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Part A 9 9.29 9.30

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9.32 9.33 9.34 9.35 9.36

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229

Hybrid Metho 10. Hybrid Methods

Analyses involving real structures and components are, by their very nature, only partially specified. The central role of modern experimental analysis is to help complete, through measurement and testing, the construction of an analytical model for the given problem. This chapter recapitulates recent developments in hybrid methods for achieving this and demonstrates through examples the progress being made.

10.1 Basic Theory of Inverse Methods ............ 10.1.1 Partially Specified Problems and Experimental Mechanics ......... 10.1.2 Origin of Ill-Conditioning in Inverse Problems...................... 10.1.3 Minimizing Principle with Regularization ...................... 10.2 Parameter Identification Problems......... 10.2.1 Sensitivity Response Method (SRM) . 10.2.2 Experimental Data Study I: Measuring Dynamic Properties ....... 10.2.3 Experimental Data Study II: Measuring Effective BCs................. 10.2.4 Synthetic Data Study I: Dynamic Crack Propagation ........................ 10.3 Force Identification Problems ................ 10.3.1 Sensitivity Response Method for Static Problems ....................... 10.3.2 Generalization for Transient Loads .

231 231 232 234 235 235 237 239 240 240

10.3.3 Experimental Data Study I: Double-Exposure Holography ........ 243 10.3.4 Experimental Data Study II: One-Sided Hopkinson Bar ............. 244 10.4 Some Nonlinear Force Identification Problems ............................................. 10.4.1 Nonlinear Data Relations .............. 10.4.2 Nonlinear Structural Dynamics ....... 10.4.3 Nonlinear Space–Time Deconvolution ............................. 10.4.4 Experimental Data Study I: Stress Analysis Around a Hole ........ 10.4.5 Experimental Data Study II: Photoelastic Analysis of Cracks ....... 10.4.6 Synthetic Data Study I: Elastic–Plastic Projectile Impact ..... 10.4.7 Synthetic Data Study II: Multiple Loads on a Truss Structure 10.4.8 Experimental Data Study III: Dynamic Photoelasticity ................ 10.5 Discussion of Parameterizing the Unknowns...................................... 10.5.1 Parameterized Loadings and Subdomains .......................... 10.5.2 Unknowns Parameterized Through a Second Model ............... 10.5.3 Final Remarks ..............................

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Experimental methods do not provide a complete stress analysis solution without additional processing of the data and/or assumptions about the structural system. As eloquently stated by Kobayashi [10.1] in the earlier edition of this Handbook: “One of the frustrations of an experimental stress analyst is the lack of a universal experimental procedure that solves all problems”. Figure 10.1 shows experimental whole-field data for some sample stress analysis problems; these example prob-

lems were chosen because they represent a range of difficulties often encountered when doing experimental stress analysis using whole-field optical methods. The photoelastic data of Fig. 10.1b can directly give the stresses along a free edge; however, because of edge effects, machining effects, and loss of contrast, the quality of photoelastic data is poorest along the edge, precisely where we need good data. Furthermore, a good deal of additional data collection and processing

Part A 10

James F. Doyle

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Initial

p (x)

c)

Fig. 10.1a–c Example whole-field experimental data. (a) Moiré inplane u-displacement fringes for a point-loaded plate (the inset is the initial fringe pattern). (b) Photoelastic stress-difference fringes for a plate with a hole. (c) Double-exposure holographic out-ofplane displacement fringes for a circular plate with uniform pressure

is required if the stresses away from the free edge are of interest (this would be the case in contact and thermal problems). By contrast, the moiré methods give objective displacement information over the whole field but suffer the drawback that the fringe data must be spatially differentiated to give the strains and subsequently the stresses. It is clear from Fig. 10.1a that the fringes are too sparse to allow for differentiation; this is especially true if the stresses at the load application point are of interest. Also, the moiré methods invariably have an initial fringe pattern that must be subtracted from the loaded pattern, which leads to further deterioration of the computed strains. Double-exposure holography directly gives the deformed pattern but is so sensitive that fringe contrast is easily lost (as seen in Fig. 10.1c) and fringe localization can become a problem. The strains in this case are obtained by double spatial differentiation of the measured data on the assumption that the plate is correctly described by classical thin-plate theory – otherwise it is uncertain as to how the strains are to be obtained. Conceivably, we can overcome the limitations of each of these methods in special circumstances; in this work, however, we propose to tackle each difficulty dir-

ectly and in a consistent manner across the different experimental methods and different types of problems. That is, given a limited amount of sometimes sparse (both spatially and temporally) and sometimes (visually) poor data, determine the complete stress and strain state of the structure. The basic approach is to supplement the experimental data with analytical modeling of the problem; this is referred to as hybrid stress analysis. Experimental mechanics has a long tradition of using hybrid methods; early examples in stress analysis were the use of finite differences combined with the equilibrium and/or compatibility equations to separate the stress components in photoelasticity. Kobayashi [10.1, 2] gives an in-depth study of this early literature with an impressive array of applications and results. More recent hybrid literature is represented by Rowlands and coworkers [10.3–5] and their cited bibliography. Applications include strain gages for stress concentrations, stress pattern analysis by thermal emissions (SPATE) for thermoelastic problems, and moiré for crack analysis, to name a few. This chapter builds on this earlier research but tries to extend it beyond stress to more general applications in experimental mechanics. The ideas expressed here are elaborated in [10.6], where the thesis is advanced that the central role of experimental mechanics is to help complete construction of the analytical model using the finite element method (FEM). A single approach, called the sensitivity response method, is used across all problems and this is made feasible because of the coupling with finite element methods. The chapter begins with a general discussion of inverse problems and places the hybrid methods within the context of partially specified problems. It is shown that the two main categories of problems are parameter identification and force identification. The following sections then show solutions of these in a variety of situations involving static/dynamic and linear/nonlinear problems, and using point/whole-field data. The examples discussed are a mixture of those using experimental data to assess the practicality of the methods and those using synthetic data to assess the robustness of the algorithms. The chapter ends with a general discussion of methods of parameterizing the problem and identifying the fundamental set of unknowns.

Hybrid Methods

10.1 Basic Theory of Inverse Methods

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10.1 Basic Theory of Inverse Methods

10.1.1 Partially Specified Problems and Experimental Mechanics To begin making the connection between analysis (or model building) and experiment, consider a simple situation using the finite element method for analysis of a linear static problem; the problem is mathematically represented by [K ]{u} = {P} , where [K ] is the stiffness of the structure, {P} is the vector of applied loads, and {u} is the unknown vector of nodal displacements. (The notation is that of [10.11, 12].) The solution of these problems can be put in the generic form {u} = [K −1 ]{P} or {response} = [system]{input} . We describe forward problems as those where both the system and the input are known and only the response is unknown. A finite element analysis of a problem where the geometry and material properties (the system) are known, where the loads (the input) are known, and where we wish to determine the displacement (the response) is an example of a forward problem. An inverse problem is one where something about the system or the input is unknown but inferred by utilizing some measured response information. A common example is the measurement of load (input) and strain (response) on a uniaxial specimen to infer the Young’s modulus (system). In fact, all experimental problems can be thought of as inverse problems because we use response information to infer something about the system or the input. A word about terminology. Strictly, an inverse problem is one requiring the inverse of the system matrix.

Thus all problems involving simultaneous equations (which means virtually every problem in computational mechanics) are inverse problems. But we do not want to refer to these FEM problems as inverse problems; indeed, we want to refer to them as forward problems. Consequently, we shall call those methods, techniques, algorithms, etc., for the robust inversion of matrices inverse methods and give looser meanings to forward and inverse problems based on their cause and effect relationship. Thus a forward problem (irrespective of whether a matrix inversion is involved or not) is one for which the system is known, the inputs are known, and the responses are the only unknowns. An inverse problem is one where something about the system or the input is unknown and knowledge of something about the response is needed in order to effect a solution. Thus the key ingredient in inverse problems is the use of response data and hence its connection to experimental methods. A fully specified forward problem is one where all the materials, geometries, boundary conditions, loads, and so on are known and the displacements, stresses, and so on are required. A partially specified forward problem is one where some input information is missing. A common practice in finite element analyses is to try to make all problems fully specified by invoking reasonable modeling assumptions. Consider the following set of unknowns and the type of assumptions that could be made:

• • • •

Dimension (e.g., ice on power lines): Assume a standard deposit thickness. Report the results for a range of values about this mean. Boundary condition (e.g., loose bolt connection): Assume an elastic support/attachment. Report the results for a range of support values from very stiff to somewhat flexible. Loading (e.g., hail impacting a windshield): Assume an interaction model (hertzian, plastic). Report the results for a range of model parameters. Model (e.g., sloshing fuel tank): Assume no dynamic effects, just model the mass and stiffness. Since the unknown was not parameterized, its effect cannot be gauged.

Note that in each case because assumptions are used the results must be reported over the possible latitude of the assumptions. This adds considerably to the total cost of the analysis. Additionally, and ultimately more impor-

Part A 10.1

Before developing specific algorithms for solving problems in experimental mechanics, we will first review some of the basic theory of solving inverse problems. Inverse problems are difficult to solve since they are notoriously ill-conditioned (i.e., small changes in data result in large changes of response); a robust solution must incorporate (in addition to the measurements and our knowledge of the system) extra information either about the underlying system, or about the nature of the response functions, or both. A key idea to be introduced is that of regularization. Additional details can be found in [10.6–10].

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Solid Mechanics Topics

Part A 10.1

tantly, the use of assumptions makes the results of the analyses uncertain and even unreliable. It may appear that the most direct way of minimizing uncertainty in our assumptions is simply to measure the unknown. For an unknown dimension this may be straightforward but what about the force due to impact of hail on an aircraft wing? This is not something that can be measured directly. It is our objective here to use indirect measurements in the solution of partially specified problems to infer, rather than assume, quantities that are not directly measurable. The unknowns in a problem fall into two broad categories: system and input. Those associated with the system will be referred to as parameter identification problems, whereas those associated with the input will be referred to as force identification problems. As will be shown, by using the idea of sensitivity responses generated by the analytical model, we can formulate both sets of problems using a common underlying foundation; we will refer to the approach as the sensitivity response method (SRM). Also, quantities of interest in usual hybrid analyses (such as stress and strain distributions) are not considered basic unknowns; once the problem is fully specified these are obtained as a postprocessing operation on the model. What emerges is the central role played by models: the FEM model, the loading model, and so on. Because of the wide range of possible models needed, it would not make sense to embed these models in the inverse methods. We instead will try to develop inverse methods that use FEM and the like as external processes in a distributed computing sense. In this way, the power and versatility of stand-alone commercial packages can be harnessed.

10.1.2 Origin of Ill-Conditioning in Inverse Problems We begin with a simple illustration of the distinction between forward and inverse problems and how a system can become ill-conditioned. Reference [10.10] introduces the scenario of a zookeeper with an unknown number of animals and, based on the number of heads and feet, we try to infer the number of animals of each type. Let us continue with this scenario and suppose the zookeeper has just a number of lions (L) and tigers (T ), then the number of heads (H) and paws (P) of all his animals can be computed as      H = L +T H 1 1 L or = . P = 4L + 4T P 4 4 T

This is a well-stated forward set of equations; that is, given the populations of lions and tigers (the input), and the number of heads and paws per each animal (the system), we can compute precisely the total number of heads and paws (the output). Let us invert the problem, that is, given the total number of heads and paws, determine the populations of lions and tigers. The answer is    −1   H L 1 1 . = P T 4 4 Unfortunately, the indicated matrix inverse cannot be computed because its determinant is zero. When the determinant of a system matrix is zero, we say that the system is singular. It is important to realize that our data are precise and yet we cannot get a satisfactory solution. Also, it is incorrect to say we cannot get a solution as, in fact, we get an infinity of solutions that satisfy the given data. Thus it is not always true that N equations are sufficient to solve for N unknowns; the equations may be consistent but contain insufficient information to separate the N unknowns. Indeed, we can even overspecify our problem by including the number of eyes as additional data and still not get a satisfactory solution. The origin of the problem is that our choice of features to describe the animals (heads, paws, eyes) does not distinguish between the animals because lions and tigers have the same number of heads and paws. Let us choose the weight as a distinguishing feature and say (on average) that each lion weighs 100 and each tiger weighs 100 + Δ. The heads and weight totals are      H 1 1 L = , W 100 100 + Δ T      1 100 + Δ −1 H L . = Δ −100 1 W T As long as (on average) the animals have a different weight (Δ = 0) then we get a unique (single) inverse solution for the two populations. The heads are counted using integer arithmetic and we can assume it is precise, but assume there is some error in the weight measurement, that is, W → W ± δ. Then the estimated populations of lions and tigers are given by      1 100 + Δ −1 H L∗ = , or Δ −100 1 T∗ W ±δ δ δ L∗ = L ∓ , T ∗ = T ± , Δ Δ

Hybrid Methods

a)

I2 I1 y1

S2 d

y2

D

d S1

from our choice of weight as a purported distinguishing feature between the animals. The second thing we notice about the estimated populations is that the errors go in opposite directions. As a consequence, if the estimated populations are substituted back into the forward equations then excellent agreement with the measured data is obtained even though the populations may be grossly in error. The sad conclusion is that excellent agreement between model predictions and data is no guarantee of a good model if we suspect that the model system is ill-conditioned. As a more pertinent example of ill-conditioning, consider the measurement situation shown in Fig. 10.2 where two detectors, I1 and I2 , are used to determine the strength of two sources, S1 and S2 . Suppose the intensity obeys an inverse power law so that it diminishes as 1/r, then the detectors receive the intensities      1/r11 1/r12 S1 I1 = , I2 1/r21 1/r22 S2  rij = D2 + (yi + d j )2 , where the subscripts on r(i, j) are (sensor, source). We wish to investigate the sensitivity of this system to errors. Figure 10.2b shows the reconstructions of the sources (labeled as direct) when there is a fixed 0.001% source of error in the second sensor. The sources have strengths of 1.0 and 2.0, respectively. There is a steady increase in error as the sensors become more remote (D increases for fixed d; a similar effect occurs as d is decreased for fixed D). Least squares is popularly used to normalize a system of equations that is overdetermined (more equations than unknowns) [10.8]; let us use least squares here even though to system is determined. This leads to     S1 1/r11 1/r21 1/r11 1/r12 1/r12 1/r22 1/r21 1/r22 S2    I1 1/r11 1/r21 . = 1/r12 1/r22 I2

b) Strength 14 12 10 8 6 4 2 0 –2 –4 –6 –8 –10

Direct Least squares

–12 10

20

30

40

50 60 Distance (D/d)

Fig. 10.2a,b Two detectors measuring emissions from two sources: (a) geometry and (b) reconstructions of source strengths for fixed error in the second sensor

The inverse results are also shown in Fig. 10.2b. The least-squares solution shows very erratic behavior as D/d is increased. It is sometimes mistakenly thought that least squares will cancel errors in the data and thereby give the correct (true or smoothed) answer. Least squares will average errors in the reconstructions of the data but in all likelihood the computed solu-

233

Part A 10.1

where L and T are the exact (true) values, respectively. We notice two things about this result. First, for a given measurement error δ, the difference in weight between the animals acts as an amplifier on the errors in the population estimates; that is, the smaller Δ is, the larger is δ/Δ. When Δ is small (but not zero) we say the system is ill-conditioned – small changes in data (δ) cause large changes in results. If there were no measurement errors, then we would not notice the ill-conditioning of the problem, but the measurement errors do not cause the ill-conditioning – this is a system property arising

10.1 Basic Theory of Inverse Methods

Solid Mechanics Topics

B(u) = {u} [H]{u} with [H] being an [M × M] positive-definite matrix (the unit matrix, say), then minimizing this will lead to a unique solution for {u}. Of course, the solution will not be very accurate since none of the data were used. The key point in the inverse solution is: if we add any multiple γ times B(u) to A(u) then minimizing A(u) + γ B(u) will lead to a unique solution for {u}. Thus, the essential idea in our inverse theory is the objective to minimize     A + γ B = {d} − [A]{u} W {d} − [A]{u} 

 B∝

du dx

2 dx ∝

M−1 

({u}m − {u}m+1 )2

m=1

= ([D]{u})2

La

rg e

rm

ro r

where {d} is the data vector and [A] is the system matrix. If [A] has fewer rows (N, data) than columns (M, unknowns) then minimizing A(u), will give the correct number of equations (M); however, it will not give a unique solution for {u} because the system matrix will be rank deficient. Suppose instead we choose a nondegenerate quadratic form B(u), for example,

These equations can be solved by standard techniques such as [LU] decomposition [10.11]. As opposed to the standard least-squares form, these equations are not ill-conditioned because of the role played by γ and the regularization matrix [H] in preventing the system eigenvalues from becoming very small, and, as will be discussed shortly, can be used to add levels of smoothness to the inverse solution. The concept of regularization is a relatively recent idea in the history of mathematics; the first few papers on the subject seem to be those of [10.13, 14], although [10.9] cites earlier literature in Russian. A readable discussion of regularization is given in [10.8] and a more extensive mathematical treatment can be found in [10.7]. This latter reference has an extensive bibliography. The regularization method we will use here is generally called Tikhonov regularization [10.9, 15, 16]. Typically, the functional B involves some measure of smoothness that derives from first or higher derivatives. Consider a distribution represented by u(x) or its discretization {u} which we wish to determine from some measurements. Suppose that our a priori belief is that, locally, u(x) is not too different from a constant, then a reasonable functional to minimize is associated with the first derivative

“Best” solution od

er

Suppose that {u} is an unknown vector that we plan to solve for by some minimizing principle. Suppose, further, that our functional A(u) has the particular form of the χ 2 error function [10.8],     A(u) = {d} − [A]{u} {d} − [A]{u} ,

 [A] W[A] + γ [H] {u} = [A] W{d} .

at a

10.1.3 Minimizing Principle with Regularization

so-called trade-off curve of Fig. 10.3. The minimization problem is reduced to a linear set of normal equations by the usual method of differentiation to yield

el

rd

Part A 10.1

tion will not be smooth. Closer inspection shows that when one solution is up the other is down, and vice versa. What we are seeing is the manifestation of illconditioning. It should also be emphasized that these erratic values of S1 and S2 reconstruct the measurements, I1 and I2 , very closely (i. e., within the error bounds of the measurements). Ill-conditioning is a property of the system and not the data; errors in the data manifests the illconditioning. As it happens, the condition number of the matrix product [A ][A] (the ratio of its largest eigenvalue to its smallest) is the square of that of the matrix [A] and this is why least-squares solutions are more apt to be ill-conditioned. In this work, we alleviate this problem by introducing regularization.

er

ro r

rg e

Part A

La

234

+ γ u [H]u , where W is a general weighting array, although we will take it as diagonal. We minimize the error functional for various values of 0 < γ < ∞ along the

More regularization

Fig. 10.3 Inverse solutions involve a trade-off between achieving good data agreement and smooth modeling

Hybrid Methods

since it is nonnegative and equal to zero only when u(x) is constant. We can write the second form of B as B = {u} [D] [D]{u} ,

0 −1 · · · ⎡ ⎤ 1 −1 0 · · · ⎢ ⎥ ⎢−1 2 −1 · · ·⎥ [H] = [D] [D] = ⎢ ⎥. ⎣ 0 −1 2 · · ·⎦ ··· −1 · · · The above choice of [D] is only the simplest in an obvious sequence of derivatives. References [10.6, 8] give explicit forms for the matrices [D] and [H] for higherorder smoothing. An apparent weakness of regularization methods is that the solution procedure does not specify the amount of regularization to be used. This is not exactly true, but choosing an appropriate value for γ is an important aspect of solving inverse problems. First, we must understand that there is no correct value; the basis of the 0

choice can range from the purely subjective (the curvefit looks reasonable) to somewhat objective based on the variances of the data. A reasonable initial value of γ to try is

γ = tr [A] [A] /tr[H] , where tr is the trace of the matrix computed as the sum of diagonal components. This choice will tend to make the two parts of the minimization (model error versus data error) have comparable weights; adjustments (in factors of ten) can be made from there. To emphasize this last point, the purpose of regularization is to provide a mechanism for enforcing on the solution a priori expectations of the behavior of the solution; its determination cannot be fully objective in the sense of only using the immediate measured data. We will see in the following examples that other judgments can and should come into play. For example, suppose that we are determining a transient force history: before the event begins, we would expect the reconstructed force to show zero behavior, and if it is an impact problem we would expect the force to be of only one sign. Subjectively judging these characteristics of the reconstructions can be very valuable in assessing the amount of regularization to use.

10.2 Parameter Identification Problems In some ways the parameter identification (ID) problem is simpler than the force identification problem, so we will begin with it. It will be a good vehicle to illustrate the concepts behind the sensitivity response method. We will generalize the programming to the case of many measurements and parameters (and hence the possible need for regularization). Since the parameters can be of very different types (e.g., dimension, modulus, boundary condition), we will also include scaling of the data and parameters. It should be mentioned, however, that there is one category of parameter ID problem that is in a class of its own: the location problem (force location, damage location, and so on). What makes these so difficult is that, in addition to a search over the parameter space, they also involve a global search over the physical space of the structure. These problems are not considered here, but note is made of [10.17], which introduces some innovative ideas using the sensitivity response method.

235

10.2.1 Sensitivity Response Method (SRM) The dynamics of a general nonlinear system are described in detail in [10.12] and given by the relationship ¨ + [C]{u} ˙ = {P} − F(u) , [M]{u}

(10.1)

where [M] is the mass matrix, [C] the damping matrix, {P} the vector of applied loads, and {F} the vector of element nodal forces. The latter is a function of the deformations and would therefore be updated as part of an incremental/iterative-type solution procedure. How this is done is of no immediate concern here since we will treat the FEM as a black box that provides solutions to fully specified forward problems. Consider a situation in which we have measured a set of response histories di (t) for a known load excitation {P(t)}, and there are M unknown parameters a j that need to be identified. Let us have a set of reasonable initial guesses for the unknown parameters, then solve

Part A 10.2

where [D] is the [(M − 1) × M] first difference matrix and [H] is the [M × M] symmetric matrix given by, respectively, ⎡ ⎤ −1 1 0 · · · ⎢ ⎥ [D] = ⎣ 0 −1 1 · · ·⎦ ,

10.2 Parameter Identification Problems

Part A

Solid Mechanics Topics

Part A 10.2

StrlDent: Make guess {a}0 Iterate ∂u Compute: ψ = ∂a Solve: ~ [ΨT W Ψ + γH]{P } T = {Ψ Wd} – {ΨTWu0}

FEM: {a}, {a + Δa}

+ {ΨTW Ψ P 0} Update: a = a + Δa

Script file

SDF

GenMesh

{P}

Script file

{u}

StaDyn/NonStaD

where a¯ j is a normalizing factor, typically based on the search window for the parameter. The small change da j should also be based on the allowable search window; typically it is taken as 5%, but in some circumstances it could be chosen adaptively. Additional discussions of this point are given in [10.6, 18]. The response solution corresponding to the true parameters is different from the guessed parameter solution according to  u j − u 0  Δa j {u} = {u}0 + da j j  = {u}0 + {ψ} j Δ P˜ j , j

Fig. 10.4 Algorithm schematic of the distributed computing rela-

tionship between the inverse program and the FEM programs for implementation of the sensitivity response method (SRM) for parameter identification

the system [M]{u} ¨ 0 + [C]0 {u} ˙ 0 = {P} − {F}0 , where the subscript 0 means that the forward problem was solved with the initial guesses. In turn, change each of the parameters (one at a time) by the definite amount a j −→ a j + da j and solve ˙ j = {P} − {F} j . ¨ j + [C]0 {u} [M]{u} The sensitivity of the solution to the change of parameters is constructed from   u j − u0 a¯ j , {ψ} j ≡ da j a)

Δa j . Δ P˜ j ≡ aj It remains to determine Δa j or the equivalent nondimensional scales Δ P˜ j . Replacing Δ P˜ j with P˜ j − P˜ 0j , and using subscript i to indicate measurements di , we can write the error function as    Wi di − Qu0i + Q ψij P˜ 0j A= i j  2 −Q ψij P˜ j , j   u0i = u xi , a0j , where Q acts as a selector. This is reduced to standard form by minimizing with respect to P˜ j to give

˜ [QΨ ] W[QΨ ] + γ [H] { P}  = [QΨ ] W{d − Qu0 + Qψ P˜ 0 } . b)

b L = 254 mm b = 25.4 mm h = 6.3 mm 40 elements 2024 aluminum

h

L

#1

L

#2

L

P (t)

Acceleration (g)

236

υ1

100 0

–100

2

υ2

P (t) 0

2

4

6

Fig. 10.5a,b An impacted aluminum beam: (a) geometry and properties and (b) recorded accelerations

8 10 Time (ms)

Hybrid Methods

#1

Acceleration (g)

Iterated

100 0 –100

100 0 –100 Nominal

Nominal

1000

Part A 10.2

Acceleration (g)

237

#2

Forward Recorded

Iterated

0

10.2 Parameter Identification Problems

2000

3000

4000

5000 Time (μs)

0

1000

2000

3000

4000

5000 Time (μs)

Fig. 10.6 Comparison of the recorded responses and the forward solution responses

The system of equations is solved iteratively with the new parameters used to compute updated responses and sensitivities. An implementation of this approach is shown in Fig. 10.4: StrIDent is the inverse solver, GenMesh, the FEM modeler program, produces the changed structure data files (SDF) that are then used in the FEM (StaDyn/NonStaD) programs. Note that there is a nice separation of the inverse problem and the FEM analysis. The significant advantage of the algorithm is that it is independent of the underlying problem, be it static/dynamic, linear/nonlinear, and so on.

10.2.2 Experimental Data Study I: Measuring Dynamic Properties This example shows how the dynamic properties of an impacted beam were obtained. Note that wave propagation characteristics are dominated by the ratio √ c0 = E/ρ and therefore wave responses are not the most suitable for obtaining separated values of E and ρ. Since mass density can be easily and accurately determined by weighing, we will illustrate the determination of the Young’s modulus and damping. Fig. 10.8a–c Pressure-loaded circular plate: (a) geometry and gage positions, (b) center mesh, and (c) the elastic

boundary condition, modeled with an overhang of unknown modulus 

Pressure gauge

Ring and bolts

Pressurized cylinder

Plate and gages p

ε1

ε2

ε3

ε4

ε5

80

778

292

590

140

140

Fig. 10.7 Photograph of pressurized plate and maximum recorded

strains

a)

b) 5 3

1

2

4 Diameter: 107mm (4,214 in.) Thickness: 3.3 (0.131in.) 2024 T-3 aluminum Gage: EA-13-062AQ-350

c)

p (x,y) = specified

E2 = ?

238

Part A

Solid Mechanics Topics

a) Strain (µε)

b) Strain (µε)

1200

–4 FEM Theory Experiment

1000

1200 FEM Experiment

1000

Part A 10.2

800

1

800

600

3

600

400

400 2

200

5

0

200 0

0

20

40

60 80 Nominal pressure (psi)

0

20

40

60 80 Nominal pressure (psi)

Fig. 10.9a,b Pressurized plate comparisons: (a) plate with fixed boundary and (b) plate with identified flexible boundary

Data frame

P(l ) P(t)

a0

Symmetry

a0: 6.35 mm L = 63.5 mm 2h = 12.7 mm b = 25.4 mm Aluminum

Fig. 10.10 Double cantilevered beam specimen: geometry, properties and mesh

Figure 10.5 shows the geometry and the acceleration data recorded for the impact of the beam. The force transducer described in [10.6] was used to record

the force history and PCB 303 A accelerometers (made by Piezotronics, Inc.) were used for the response. The DASH-18 system recorded the data. The impact was on the wide side of the beam, which increased the amount of damping due to movement through the air. The damping is implemented in the FEM code as [C] = α[M] = (η/ρ)[M], which is a form of proportional damping. The parameter α has units of s−1 and this will be identified along with the modulus. The search window for this type of problem can be set quite narrow since we generally have good nominal values. Consequently, convergence was quite rapid (three or c)

a) u (x, y) δ = 64 nm σ0 = 241MPa

υ (x, y)

δ

b)

σ0

0

5

10

15 Iterations

Fig. 10.11a–c Double cantilever beam synthetic data: (a) moiré u(x, y) and v(x, y) fringes at 10.6 μs, (b) deformed shape (exaggerated ×10), and (c) convergence behavior using a single frame of data with 0.2% noise

Hybrid Methods

a)

10.2 Parameter Identification Problems

239

b)

P21 P19

a (t)

(60×103 in./s) a· (t) P3 P1 0

5

10

15 Time (μs)

0

5

10

15

20

25 30 Time (μs)

Fig. 10.12a,b Reconstruction of the DCB specimen behavior. (a) Cohesive force history. (b) Crack tip position and speed

four iterations) and the values obtained were Nominal:

c0 = 5080.00 m/s (200 000 in./s) ,

Iterated:

α = 0.0 s−1 , c0 = 4996.18 m/s (196 700 in./s) , α = 113.37 s−1 .

Figure 10.6 shows the response reconstructions using the nominal values and the iterated values compared to the measured responses. It is apparent that taking the damping into account can improve the quality of the comparisons considerably. To improve the correspondence over a longer period of time would require iterating on the effective properties of the clamped condition. Physically, the beam was clamped in a steel vise and not fixed as in the model. For wave propagation problems, some energy is transmitted into the vise. Determining effective boundary conditions is discussed next.

10.2.3 Experimental Data Study II: Measuring Effective BCs This second experiment is that of a plate with a uniform pressure. In this case, the pressure loading is actually known with a good deal of confidence but, because the plate is formed as the end of a pressurized cylinder with bolted clamps, the true boundary conditions are not rigidly clamped as was originally intended. The ob-

jective will be to get a more realistic modeling of the boundary condition. A photograph of the plate forming the end of a cylinder is shown in Fig. 10.7. Sixteen bolts along with an aluminum ring are used to clamp the plate to the walls of the cylinder. Note that the cylinder has an increased wall thickness to facilitate this. Figure 10.8 shows the geometry of the plate and the gage locations. Gages 4 and 5 are not exactly on the edge of the plate, but the gage backing was trimmed so that they could be put as close as possible to the edge. Each gage ended up being approximately 1.8 mm (0.07 in.) from the edge; the meshes, such as that in Fig. 10.8b, were adjusted so as to have a node at the gage positions. Figure 10.9a shows the recorded strains. There is very good linearity with very little offset; consequently, the data was used directly as part of the inverse solutions. The strains at the maximum pressure are shown in Fig. 10.7. Figure 10.9a shows that modeling the boundary condition as fixed does not give a good comparison with experiment. Figure 10.8b models the problem as an overhanging plate with specified thickness but unknown modulus; the pressure load will be taken as specified. With E 2 as the single unknown, the converged result gave E 2 = 0.26E 1 and the reconstructions are shown in Fig. 10.9b; the standard deviation in the strain is 18 μe. The comparison between the model and ex-

Part A 10.2

1500m/s

240

Part A

Solid Mechanics Topics

~

10

1 × P1 ~

5

1 × P2 +

Part A 10.3

~

{ψ}1 × P1

+

=

~

{ψ}2 × P2

=

{d}

Fig. 10.13 The data {d} are conceived as synthesized from the superpositions of unit-load solutions {ψ}1 and {ψ}2 modified by scalings P˜1 and P˜2

periment is considerably improved. The results hardly changed when both moduli were assumed simultaneous unknowns.

10.2.4 Synthetic Data Study I: Dynamic Crack Propagation A theme of this work is the central role played by the model in stress analysis. We illustrate here how it can be used in the design of a complicated experiment. That is, it is generally too expensive in both time and money to design experiments by trial and error and a wellconstructed model can be used to narrow the design parameters. To make the scenario explicit, consider a double cantilever beam (DCB) specimen made from two aluminum segments bonded together as shown in Fig. 10.10. The crack (or delamination) is constrained to grow along the interface, and the interface properties of the bond line are the basic unknowns of the experiment. This delamination is modeled as a cohesive layer described by σ (v) = σ0 [v/δ] e(−v/δ) , where v is the separation distance and (σ0 , δ) are the parameters to be identified. Reference [10.19] contains details about this type of crack modeling that is implemented in the nonlinear FEM code NonStaD. To get accurate estimates of the interface properties, it is necessary to record the data as the crack passes

through the frame of view. Typical frames are shown in Fig. 10.11a along with the exaggerated deformed shape in Fig. 10.11b. The applied load is a rectangular pulse of long duration with a rise time of 3 μs and, for identification purposes, is taken as known. This is a problem with a wide range of scales. As seen from Fig. 10.11b, the tip deflection is of order h/10 but h is of order δ × 105 . At rupture, the deflections along the line of symmetry are 1δ ∼ 10δ. To increase the accuracy of the data, the frames came from just the 12 mm × 12 mm window indicated in Fig. 10.10a. The noise added to the synthetic data had a specified standard deviation of 0.2 × δ; with a moiré sensitivity of 100 line/mm, this works out to about 0.2% of a fringe. The convergence behavior is shown in Fig. 10.11c where a single v(x, y) frame of data at 10.6 μs was used; both parameters have converged by the tenth iteration. The results are robust with reasonable initial guesses for the parameters. In the present tests, the allowable search windows spanned 10–200% of the exact values, with the initial guesses being 50%. Keep in mind that, during the iteration process, at each iteration a complete crack propagation problem is solved three times (once for the nominal parameters, and once each for the variation of the parameters); thus the computational cost is quite significant. Once the parameters are determined, reconstructions of any quantity can be obtained, which is the utility of the hybrid methods. Figure 10.12a shows, for example, the interface force histories at every other node. These histories are almost identical to the exact histories, which is not surprising because of the goodness of the parameter estimates. The quality of these reconstructions is greatly superior to those achieved in [10.20], which determined the interface forces directly; this underlines the value of a reparameterization of unknowns of a problem. The position of the crack tip is quite apparent in Fig. 10.12a; using the peak value of the force as the indicator, Fig. 10.12b shows the crack tip position and its speed. The speed is not constant because of the decrease in beam stiffness as the crack propagates.

10.3 Force Identification Problems In the inverse problems of interest here, the applied loads and displacements are unknown but we know some information about the responses (either in the

form of displacements or strains). In particular, assume that we have a vector of measurements d(t) which are related to the structural degree of freedom (DoF) ac-

Hybrid Methods

cording to {d(t)} ⇔ [Q]{u(t)} ,

10.3.1 Sensitivity Response Method for Static Problems Since there is a relation between {u} and {P}, then (for minimization purposes) we can take either as the basic set of unknowns; here it is more convenient to use the loads because for most problems the size of {P} is substantially smaller than the size of {u}. Let the unknown forces be represented as the collection ˜ = [Φ]{ P} ˜ , (10.2) {P} = [{φ}1 {φ}2 · · · {φ} Mp ]{ P} where each {φ}m is a known distribution of forces and P˜m the corresponding unknown scale. As a particular case, {φ}m could be a single unit force, but it can also represent a uniform or a Gaussian pressure distribution; for traction distribution problems it has a triangular distribution [10.20]. We will refer to each {φ}m distribution as a unit or perturbation load; there are Mp such loads. Solve the series of forward FEM problems [K ]{ψ}1 = {φ}1 , [K ]{ψ}2 = {φ}2 , [K ]{ψ} Mp = {φ} Mp .

... ,

The {ψ}m responses will have imbedded in them the effects of the complexity of the structure; consequently, usual complicating factors (for deriving analytical solutions used in hybrid methods) such as geometry, material, and boundary conditions are all included in {ψ}m , which in turn is handled by the finite element method. Furthermore, note that the responses are not necessarily just displacement; they could be strain or principal stress difference as required by the experimental method, and consequently the inverse program does not need to perform differentiation to get strains. In other words, anything particular to the modeling or mechanics of the structure is handled by the FEM program. We can therefore write the actual forward solution as ˜ , {u} = [Ψ ]{ P} [Ψ ] = [{ψ}1 {ψ}2 · · · {ψ} Mp ] , (10.3)

˜ is not known. In this way, the coefficients although { P} ˜ act as scale factors on the solutions in [Ψ ]. of { P} This idea is shown schematically in Fig. 10.13 where the two-load problem is conceived as the scaled superposition of two separate one-load problems. This superposition idea leads to simultaneous equations to ˜ solve for the unknown scales { P}. The measured data at a collection of locations are related to the computed responses as ˜ . {d} ⇐⇒ [Q]{u} = [Q][Ψ ]{ P} If supplementary information is available (e.g., additional data or a relationship among the unknowns), it is incorporated simply by appending to this set of equations before the least-squares procedure is used. The error functional can be written as a function of P˜ only and minimized to get ˜ = [QΨ ] W{d} . [Ψ  Q WQΨ + γ H]{ P} (10.4)

This reduced system of equations is of size [Mp × Mp ] and is relatively inexpensive to solve since it is symmetric and small. Once the loads are known, the actual forward solution is then obtained from (10.3), which involves a simple matrix product. The key to understanding our approach to the computer implementation is the series of solutions in the matrix [Ψ ] – it is these that can be performed externally (in a distributed computing sense) by a commercial FEM code. The basic relationship is shown in Fig. 10.14 and is laid out in a similar style to Fig. 10.4. If the number of unknowns are computationally too large, then a parameterized loading scheme can be used, where the loading is assumed to be synthesized from speci-

StrlDent: Form: – – – [A ] = [Q ][Ψ ] –

φm (t)

FEM: GenMesh StaDyn

–

[B ] = [A ]T{d} Solve: – – – – – [A TA + γH ]{P } = {B }

Script file

{ψ}m (t)

Response file

Fig. 10.14 Algorithm schematic of the distributed computing relationship between the inverse program StrIDent and the FEM programs. For static problems, the FEM program is run once for each load position; for dynamic problems, a {ψ}m (t) history at all data locations is computed for each force location

241

Part A 10.3

where [Q] is a selection matrix that plays the role of selecting the subset of responses participating in forming the data. We want to find the forces {P(t)} that make the system best match the measurements. We will introduce the ideas behind the method by first considering static problems, and then generalize them to the dynamic case.

10.3 Force Identification Problems

242

Part A

Solid Mechanics Topics

fied forms [10.21]. The experimental study to follow is a simple example.

10.3.2 Generalization for Transient Loads

of the structure. Because of the linearity of the system, the responses are also similar to each other but shifted an amount mΔT . This basic idea is illustrated in Fig. 10.15. The error functional is

Let the force history be represented as

Part A 10.3

P(t) =

Mp 

P˜m φm (t) ,

˜ , {P} = [Φ]{ P}

(10.5)

m=1

where φm (t) are specified functions of time; they are similar to each other but shifted an amount mΔT . We will consider specific forms later. The array [Φ] has as columns each φm (t) discretized in time. The vector {P} has the force history P(t) discretized in time. This force causes the response at a particular point in the model u(x, t) =

Mp 

P˜m ψm (x, t) ,

A = {d − Qu} W{d − Qu} ˜ . ˜  W{d − QΨ P} = {d − QΨ P} Force reconstructions show the manifestations of illconditioning as ΔT is changed because, with small ΔT , the data are incapable of distinguishing between neighbor force values Pn + Δ, Pn+1 − Δ. Time regularization is therefore important and connects components within ˜ that is, { P}, ˜ = { P} ˜  [H]{ P} ˜ . ˜  [D] [D]{ P} B = { P} Minimizing with respect to P˜m then gives

˜ = [QΨ ] W{d} . [QΨ ] W[QΨ ] + γ [H] { P}

˜ , {u(x)} = [Ψ ]{ P}

m=1

(10.6)

(10.7)

where ψm (x, t) is the response to force φm (t) at the particular point, and [Ψ ] are all the discretized responses arranged as columns. As in the static case, we refer to φm (t) as unit or perturbation forces and ψm (t) as sensitivity responses. These responses are obtained by the finite element method, and therefore will have imbedded in them the effects of the complexity

The core array is symmetric of size [Mp × Mp ]. There are a variety of functions that could be used for φm (t); the main requirement is that they have compact support in both the time/space and frequency domains. Reference [10.22] used a Gram–Schmidt reduction scheme to generate a series of smooth orthogonal functions from the basic triangular pulse; a set

a) P (t)

b) Actual force

φ (t)

Pertubation force

ψ (t)

Sensitivity response

Time P (t) P3 P2 P1

P4

P5

P6

Measured response

P7 Time

0

200

400

600

800

1000 1200 1400 Time (μs)

Fig. 10.15a,b The sensitivity response concept for transient load identification. (a) A discretized history is viewed as a series of scaled triangular perturbations. (b) How the scaled perturbations superpose to give the actual histories

Hybrid Methods

¯ = {[Φ]1 { P} ˜ 1 , [Φ]2 { P} ˜ 2 , · · · , [Φ] Np { P} ˜ N p } , { P} ˜ n is the history Pn (t) arranged as a vector where each { P} of Mp time components. This collection of force vectors causes the collection of responses ⎫ ⎡ ⎧ ⎤ ⎪ [Ψ ]11 [Ψ ]12 · · · [Ψ ]1Np {u}1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎬ ⎢ ⎨ {u}2 ⎪ ⎢ [Ψ ]21 [Ψ ]22 · · · [Ψ ]2Np ⎥ ⎢ ⎥ .. ⎪ = ⎢ .. .. .. .. ⎥ ⎪ ⎪ ⎪ . ⎣ . ⎦ . . . ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ {u} Nd [Ψ ] Nd 1 [Ψ ] Nd 1 · · · [Ψ ] Nd Np ⎫ ⎧ ˜ 1 ⎪ ⎪ { P} ⎪ ⎪ ⎪ ⎪ ⎪ ˜ ⎪ ⎨ { P}2 ⎬ (10.8) × . ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˜ Np ,⎭ { P} where {u}i is a history of a displacement (DoF) at the i measurement location. We also write this as  ˜ j or {u} ¯ . ¯ P} [Ψ ]ij { P} {u}i = ¯ = [ψ]{ j

The [Mp × Mp ] array [Ψ ]ij relates the i location response history to the j applied perturbation force. It contains the responses ψm (t) arranged as a vector. Let the Nd sensor histories of data be organized as ¯ = {{d}1 , {d}2 , · · · , {d} Nd } , {d}

243

where {d}n is a vector of Md components (it is simplest to have Md = Mp but this is not required). The measured data DoF are a subset of the computed responses, that is, ¯ u} ¯ i ⇐⇒ [Q]{ {d} ¯ or

¯ ψ]{ ¯ ⇐⇒ [Q ¯ . ¯ P} {d}

(10.9)

Define a data error functional as ¯ ψ¯ P} ¯ ψ¯ P} ¯ ¯ = {d¯ − Q ¯  W{d − Q E( P) 



¯ [Ht ]{ P} ¯ + γs { P} ¯ [Hs ]{ P} ¯ , + γt { P} which has incorporated regularization in both time [Ht ] and space [Hs ]. Time regularization connects components within each P˜ j only, while space regularization connects components in P˜ j across near locations. The sparsity of the matrices [Ht ] and [Hs ] are quite different and therefore their effect on the solution is also quite different, even for same order regularizations. Minimiz¯ gives ing with respect to { P}

¯ ψ] ¯ ψ] ¯ ¯ + γt [Ht ] + γs [Hs ] { P} ¯  W[Q [Q ¯ ψ] ¯ . ¯  W{d} = [Q

(10.10)

The core matrix is symmetric, of size [(Mp × Np ) × (Mp × Np )], and in general is fully populated. Only when Np is large (there are a large number of force histories) does the computational cost of solving (10.10) become significant. The computer implementation is shown in Fig. 10.14. The sensitivity response is computed only once for a given force location irrespective of the number of sensors, hence the FEM cost is simply that of a forward solution (times the number of forces).

10.3.3 Experimental Data Study I: Double-Exposure Holography Figure 10.1c shows a double-exposure holographic fringe pattern for the out-of-plane displacement of a pressure-loaded plate. The plate forms one end of a pressurized cylinder and the inner white dots in the figure show the flexing part of the plate. The small change of pressure was applied accurately using a manometer. As sometimes happens with holography, the fringe pattern is not of uniform quality. Furthermore, there is a slight asymmetry in the fringe pattern which could be due to a slight overall rigid-body motion of the cylinder. Figure 10.16b shows the limited number of discretized data points used. Because of the expected symmetries in the results the points were distributed somewhat uniformly so as to provide averaging. Figure 10.16c shows

Part A 10.3

of these is illustrated in Fig. 10.15b. Here we will use the triangular pulse directly; this is equivalent to using a linear interpolation between the discretized points, as shown in Fig. 10.15a. The advantage of the triangle is that the scales P˜m have the direct meaning of being the discretized force history and hence there is no need to actually perform the summation in (10.5). In fact, without losing the advantage of the straight triangular pulse, it is also possible to use a smoothed triangular pulse; this is the scheme implemented in StrIDent. There are a number of features we now wish to add to the above developments, primarily the ability to determine multiple unknown force histories. There are two situations of interest here: the first is that of multiple isolated forces and the second is traction distributions. The main difference is whether the forces are related to each other or not; isolated forces are a finite number of forces which are not related, but traction distributions contain very many forces which are continuously related to their neighbors and therefore space regularization can be used. Let the Np unknown force histories be organized as

10.3 Force Identification Problems

244

Part A

Solid Mechanics Topics

a)

b)

c)

Diameter: 107 mm (4.215 in) Thickness: 3.12 mm (0.123 in) 6061 aluminum

Part A 10.3

α: 17 deg, β: 0 deg λ: 632nm (24.9 μin.)

Fig. 10.16a–c Holographic experiment of a pressure-loaded plate. (a) Cylindrical pressure vessel properties and holographic parameters. (b) Discretized data positions. (c) FEM mesh

a)

b)

Nominal: 7.63 kPa (1.09psi) Computed: 7.77 kPa (1.11psi)

σxx

σxy

Fig. 10.17a,b Reconstructions. (a) Displacement fringe pattern: top recorded, bottom reconstructed. (b) Stress contours

the FEM mesh used. The data were relocated onto the nearest nodes of this mesh by the scheme discussed in [10.6]. The edge of the plate was modeled as being rigidly clamped. The pressure was taken as the only unknown in formulating the inverse problem. Since the number of

Strain gages P (t)

Steel

Rs (t)

Ra (t)

Accelerometers

data far exceed the number of unknowns, regularization is not required. The nominal measured and computed pressures are given in Fig. 10.17. The comparison is very good. Figure 10.17a shows a comparison of the reconstructed fringe pattern (bottom) with that measured (top). Since the pressure comparison is good, it is not surprising that this comparison is also good. A theme running through the hybrid solutions in this work is that, once the unknown forces (or whatever parameterization of the unknowns is used) have been determined, the analytical apparatus is then already available for the complete stress analysis. Figure 10.17b shows, for example, the σxx and σxy stress distributions. What is significant about these reconstructions is that (in comparison to traditional ways of manipulating such holographic data) the process of directly differentiating experimental data is avoided. Furthermore, this would still be achievable even if the plate was more complicated by having, say, reinforcing ribs (in which case the strains are not related to the second derivatives of the displacement). Note also that, because the experimental data were not directly differentiated, only a limited number of data points needed to be recorded.

aluminum

51

Accelerometers #1 #2 L

L = 305 mm D = 102 mm h = 6.3 mm 6061 aluminum

Fig. 10.18 Square plate with a central hole. The Hopkinson bar is

attached to the left side and its two strain gages are sufficient to determine both the input load and the transmitted load

10.3.4 Experimental Data Study II: One-Sided Hopkinson Bar It often occurs that we need to apply a known force history to a structure. In some instances, where the frequency of excitation is low, we can use an instrumented hammer as is common in modal analysis [10.23]. In other instances, we need a high-energy high-frequency input, and common force transducers do not have the required specifications. The one-sided Hopkinson bar

Hybrid Methods

b)

Strain (ue)

0

0.1

Part A 10.3

100

0 –0.1

–100

0

1000

2000

3000

4000 Time (μs)

0

1000

2000

3000

4000 Time (μs)

Fig. 10.19a,b Experimentally recorded data. (a) Strains from the OSHB. (b) Velocities of the plate computed from measured accelerations

force input to the plate. These force reconstructions are shown labeled as P and Rs , respectively, in Fig. 10.20. The impact force, as expected, is essentially that of a single pulse input. The reconstructions of the force at the connection point, on the other hand, are quite complicated, persisting with significant amplitude for the full duration of the recording. Note that the connection force is larger than the impact force because of the high impedance of the plate. The Rs force from the bar was used as input to the forward problem for the plate and the computed

P (t) Accelerometers Hopkinson bar

R s (t), R 2a (t)

#2 1

P/P0

(OSHB) was designed for this purpose. More details and examples can be found in [10.24, 25]. Figure 10.18 shows a schematic of a problem used to test the ability of accelerometers to reconstruct forces on a twodimensional plate. This force will be compared to that determined by the strain gage responses from the onesided Hopkinson bar. Figure 10.18 shows the specimen dimensions and placement of the accelerometers. The specimen was machined from a 6.3 mm (1/4 in.)-thick sheet of aluminum to be a 305 mm (12 in.) square with a 76 mm (4 in.)-diameter hole at its center. A small hole was drilled and tapped at the midpoint of the left edge for the connection of the OSHB. Accelerometers were placed on the same edge as the force input and on the opposite edge, at 25.4 mm (2 in.) off the force input line. The data were collected at a 4 μs rate over 4000 μs. The force input to the OSHB was from the impact of a steel ball attached to a pendulum. The acceleration data were converted to velocities and detrended, the strain gage data were modified to account for the slight nonlinearity of the Wheatstone bridge (since the resistance change of semiconductor gages is relatively large). These data are shown in Fig. 10.19. The inverse problem was divided into two separate problems corresponding to the bar and the plate. The data from the two strain gages on the bar were used simultaneously to reconstruct the impact force and the

245

Measured Forward

Velocity (m/s)

a)

10.3 Force Identification Problems

R s (t), R 1a (t)

0 #1

0

1000

2000

3000

4000 Time (μs)

Fig. 10.20 Force reconstructions for the plate with a hole. The reference load is P¯0 = 2224 N (500 lb)

246

Part A

Solid Mechanics Topics

Part A 10.4

velocities are compared to the measured in Fig. 10.19. The comparison is quite good considering the extended time period of the comparison. The data from the accelerometers were then used as separate single sensor inputs to reconstruct the force at the connection point. These force reconstructions are shown labeled as R1a and R2a in Fig. 10.20. The reconstructed forces at the connection point agree quite closely both in character and magnitude. This is significant because the force transmitted across the boundary

is quite complex and has been reconstructed from different sensor types placed on different structural types. Reference [10.6] discusses a collection of possible variations of the one-sided Hopkinson bar. The basic idea expressed is that, because a general finite element modeling underlies the force identification method, and because we can determine multiple unknown forces simultaneously, we are given great flexibility in the design of the rod and its attachment to the structure.

10.4 Some Nonlinear Force Identification Problems Nonlinearities can arise in a number of forms: material nonlinearity such as plasticity and rubber elasticity, geometric nonlinearity as occurs with large displacements and rotations, and nonlinearities associated with the data relation. We discuss examples of each.

10.4.1 Nonlinear Data Relations In photoelasticity there is a nonlinear relationship between observed fringes and applied loads. To see this, recall that fringes are related to stresses by [10.26]

fσ N h

2 2 = (σxx − σ yy )2 + 4σxy .

From this, while it is true that a doubling of the load doubles the stress and therefore doubles the fringe orders, this is not true for nonproportional loading. That is, the fringe pattern caused by two loads acting simultaneously is not equal to the sum of the fringe patterns of the separate loads even though the stresses would be equal. Since our inverse method is based on superposition, we must modify it to account for this nonlinear relation. Based on this discussion, it may seem that photoelasticity is not as attractive as other methods. This is perhaps true, but the main reason it is included here is as an example of data with a nonlinear relation to mechanical variables. It may well be that sensors of the future will also have a nonlinear relationship and the algorithms developed here will be available to take advantage of them. For example, there is currently a good deal of activity in developing pressure-sensitive paints, fluorescent paints, and the like; see [10.27, 28] for a discussion of their properties and some of the literature. These inexpensive methods, most likely, will have

a nonlinear relation between the data and the mechanical variables. We need to generate a linear relation between the fringe data and the applied loads. As usual for nonlinear problems, we begin with a linearization about a known state. That is, suppose we have a reasonable estimate of the stresses as [σxx , σ yy , σxy ]0 ; then the true stresses are only a small increment away. A Taylor series expansion about this known (guessed) state provides

2

fσ 2 N ≈ (σxx − σ yy )2 + 4σxy 0 h + 2 (σxx − σ yy )0

× (Δσxx − Δσ yy ) + (4σxy )0 Δσxy . Let the stress increments (Δσxx , Δσ yy , Δσxy ) be computed from the scaled unit loads so that for all components of force we have  {ψxx , ψ yy , ψxy } j Δ P˜ j . {Δσxx , Δσ yy , Δσxy } = j

The stress sensitivities {ψ} j must be computed for each component of force. Noting that Δ P˜ = P˜ − P˜0 , we write the approximate fringe relation as

2   fσ G¯ j P˜o j + G¯ j P˜o j (10.11) N ≈ σ02 − h j

j

with

2 σ02 ≡ (σxx − σ yy )2 + 4σxy , 0 G¯ j ≡ [2(σxx − σ yy )0 (ψxx − ψ yy ) j + 8(σxy )0 ψxy j ] . Equation (10.11) is our linear relation between fringes and applied loads. Let the data be expressed in the form

Hybrid Methods

N¯ ≡ N f σ /h so that the minimization principle leads to

[QG] W[QG] + γ [H] P˜ = [QG] W{ N¯ 2 − Qσ02 + QG P˜0 } , (10.12)

10.4.2 Nonlinear Structural Dynamics The dynamics of a general nonlinear system are described by (10.1) and rewritten here as ˜ − {F} . [M]{u} ¨ + [C]{u} ˙ = {P} − {F} = [Φ]{ P} There are two sets of loads: {P}, which is the set of unknown applied loads to be identified, and the element nodal forces {F}, which depend on the deformations and are also unknown. Both {P} and {F} must be identified as part of the solution. We begin the solution by approximating the element nodal forces by a Taylor series expansion about a known (guessed) configuration ! ∂F {Δu} = {F}0 + [K T ]0 {u − u0 } {F} ≈ {F}0 + ∂u = {F}0 − [K T ]0 {u}0 + [K T ]0 {u} , where the subscript 0 means known state and {F0 } − [K T ]0 {u}0 is a residual force vector due to the mismatch between the correctly reconstructed nodal forces {F} and its linear approximation [K T ]0 {u}0 ; at convergence, it is not zero. The square matrix [K T ]0 is the tangent stiffness matrix. We have thus represented the nonlinear problem as a combination of known forces {F0 } − [K T ]0 {u}0 (known from the previous iteration)

StrlDent: ~

Make guess: {P }

[Φ]

Script file

[Ψ]

StaDyn

{P}

Script file

247

Iterate Update: G, σ0 Solve: ~ [GT WG + γH]{P }

[σxx, σyy, σxy ]

–2

= {GTWN } – {GTWσ02} ~ + {GTWP0}

StaDyn FEM:

Fig. 10.21 Algorithm schematic of the distributed computing relationship between the inverse program and the FEM programs when nonlinear updating is needed

and a force linearly dependent on the current displacements, [K T ]0 {u}. The governing equation is rearranged in the form [M]{u} ¨ + [C]{u} ˙ + [K T ]0 {u} ˜ − {F}0 + [K T ]0 {u}0 . = [Φ]{ P} This is a linear force identification problem with un˜ and known {F}0 and [K T ]0 {u}0 . Solve the known { P} three linear dynamics problems ¨ + [C][ψ] ˙ + [K T ]0 [Ψ ] = [Φ] , [M][ψ] [M]{ψ¨ F } + [C]{ψ˙ F } + [K T ]0 {ψF } = −{F}0 , [M]{ψ¨ u } + [C]{ψ˙ u } + [K T ]0 {ψu } = [K T ]0 {u}0 . Note that the left-hand operator is the same in each case. The total response can be represented as ˜ + {ψF } + {ψu } , {u} = [Ψ ]{ P} and the data relation becomes [Q]{u} − {d} = 0

⇒

˜ = {d} − {QψF } − {Qψu } = {d ∗ } . [QΨ ]{ P} The minimum principle leads to

˜ [QΨ ] W[QΨ ] + γ [H] { P} = [QΨ ] {d − QψF − Qψu } ,

(10.13)

which is identical in form to the linear dynamic case except for the carryover terms {ψF } and {ψu }. The core array is symmetric, of size [Mp × Mp ]; note that it does not change during the iterations and hence the decomposition need be done only once. The right-hand side

Part A 10.4

where, as before, [Q] plays the role of a selector. This ˜ to is solved repeatedly using the newly computed { P} ˜ 0. replace { P} In terms of the computer implementation shown in Fig. 10.21, the sensitivity solution {ψ} is determined only once for each load component, but the updating of the stress field {σ0 } requires an FEM call on each iteration. Both the left- and right-hand side of the system of equations must be reformed (and hence decomposed) on each iteration. In all iterative schemes, it is important to have a method of obtaining good initial guesses. There are no hard and fast rules for the present type of problem; however, the rate of convergence depends on the number of unknown forces, this suggests using minimal forces initially to obtain an idea of the unknowns to use as starter values. Furthermore, the effect of regularization is sensitive to the number of unknowns and therefore it is wise to obtain starter values without regularization.

10.4 Some Nonlinear Force Identification Problems

248

Part A

Solid Mechanics Topics

StrlDent: Make guess: P Form: [ΨTΨ + γH ] Iterate

Part A 10.4

[Φ]

Script file

[Ψ]

StaDyn

{P}

Script file

[F ], [u]

NonStaD

[F ], [u]

Script file

FEM:

Update: P, u, F Solve: ~ [ΨTΨ + γH ] {P } = {ΨTd} +{ΨTψu} +{ΨTψF}

{ψu}, {ψF}

StaDyn

Fig. 10.22 A possible algorithm schematic for the distributed computing relationship between the nonlinear dynamic inverse program and the FEM programs

10

∑ ⇒ {d}

9

3

8

Δt

7

Data frames

6

∑ ⇒ {d}

5

2

4

Whole-field {ψ} snapshots Perturbation φm forces

3 2 1 φm 0

∑ ⇒ {d} 1

1

2

3

4

5

6

7

8

9

10

∑ ⇒ P (t)

Fig. 10.23 Relationship between frames due to the perturbation

forces and the measured frames

a)

b)

P

Fig. 10.24a,b Photoelastic fringe pattern due to distributed load: (a) experimentally recorded and (b) reconstructed

changes during the iterations due to the updating associated with {ψF } and {ψu }.

The basic algorithm is: starting with an initial guess for the unknown force histories, the effective data in (10.13) is updated, then P˜ is obtained from solving (10.13), from which new effective data are obtained and the process is repeated until convergence. The implementation details, however, are crucial to its success. A very important issue that needs to be addressed is the formation and assemblage of the matrices associated with the above formulation and, as much as possible, to have the FEM portion of the inverse solution shifted external to the inverse programming; this is discussed in great detail in [10.6, 21]. The main conclusion is that, since the Newton–Raphson iterations scheme does not require the exact tangent stiffness matrix for convergence, then, perhaps at the expense of the number of iterations and the radius of convergence, the sensitivity responses can be obtained from the external FEM program by using an approximate tangent stiffness matrix. An obvious choice would be to use the linear elastic stiffness matrix; another could be a linear elastic stiffness matrix that is based on an updated or mean geometry. The implementation actually uses two structural models: the true model and a separate model for estimating the tangent stiffness matrix. While not a proof as such, the following examples make the key point that it is the Newton–Raphson iterations on the equilibrium, and not the quality of the [K T ] estimate, that produces the converged result. The schematic for the algorithm as implemented in the program StrIDent is shown in Fig. 10.22. Note that [u] and [F] are histories of every DoF at every node, and therefore can require a very large storage space. Also note that the linear FEM program is expected to give the dynamic responses to the loads {F}0 and [K T ]0 {u}0 ; both are loadings where every DoF at every node is an individual history. Forming the product [K T ]0 {u}0 , where each component of {u}0 is an individual history, is not something usually available in commercial FEM packages. Requirements such as this need to be implemented as special services in the FEM programs available to the inverse program. It seems that for the nonlinear problem there is not a definite separation (of what must be done internally and what can be done externally) as there is for linear problems.

10.4.3 Nonlinear Space–Time Deconvolution Consider an experiment that uses high-speed photography to record moiré or photoelastic fringes. The scenario we envision here is that the force history is

Hybrid Methods

φ1 (t)

⇒

amount mΔT , {ψ}4 is a snapshot of all the displacements at 50 μs caused by φ2 . Similarly, {ψ}3 , {ψ}2 , and {ψ}1 are caused by φ3 , φ4 , and φ5 at 50 μs, respectively. Therefore, for data frame {d}2 , P˜5 contributes to {u}1 , P˜4 contributes to {u}2 , all the way to P˜1 contributing to {u}5 ; any component greater than 5 does not contribute. It is this reversal in time that allows the scheme to be computationally efficient. Thus, a snapshot of the displacements (say) at time step i is {u}i = {ψ}i P˜1 + {ψ}i−1 P˜2 + {ψ}i−2 P˜3 + · · · + {ψ}1 P˜i + 0 P˜i+1 + · · · + 0 P˜Mp , where {u} is a snapshot of all the displacements (DoF) at this instant of time. Note that force components occurring at time steps greater than i do not contribute. We can write this relation as {u}i =

a)

{ψ}(i+1−m) P˜m +

m=1

Mp 

0 P˜m

or

m=i+1

˜ . {u} = [Ψ ]{ P}

{ψ}1 , {ψ}2 , {ψ}3 , . . . {ψ} Mp .

A frame of data at time mΔt has contributions only up to that time but the sequence of snapshots are reversed in time. For example, let the time step be Δt = 10 μs and consider data frame {d}2 , the frame of all the displacements (DoF) at 50 μs caused by P(t). In this, {ψ}5 is a snapshot of all the displacements at 50 μs caused by φ1 . Since φm are similar to each other but shifted an

i 

(10.14)

The [Mu × Mp ] array [Ψ ] (Mu is the total number of DoFs) relates the snapshot of displacements to the applied force. Let the Nd frames of data be organized as   ¯ = {d}1 , {d}2 , · · · , {d} Nd  , {d} b) Traction

0.6

0.08 P1 + P2

0.4

0.06

0.2 0.04 0 0.02 –0.2 P1 – P2 0

–0.4 –0.6

0

5

10

15 20 Node position

– 0.02

0

5

Fig. 10.25a,b Iterative results: (a) estimation of resultant load and (b) traction distribution

10

15 Node position

249

Part A 10.4

unknown (it is due to blast loading or ballistic impact, say) and we are especially interested in local (such as at the notch root) stress behavior. The mechanics of the underlying event or the data relation can be nonlinear. One thing that is clear is, since the total number of frames (photographs) is severely limited, the frames cannot be used directly to obtain history behavior. Conceivably, we can overcome this limitation by simply paying the price of having multiple cameras [10.29]. Here we propose to tackle the problem directly; that is, given a limited number of frames of sparse data, determine the complete history of the event including stresses and strains. Figure 10.23 attempts to show the relationship between the force and the observed frames; while both axes are time axes, it tries to show how a single frame of data has both early time and late time force information. Let a single unknown force history be represented by (10.5). Each perturbation force causes a snapshot at each time step; the sequence is the same for each force just shifted one time step. Thus it is sufficient to consider just the first perturbation force and its snapshots

10.4 Some Nonlinear Force Identification Problems

250

Part A

Solid Mechanics Topics

a)

b)

P1

P2

P3

full details, nonlinear versions of the problems represented by (10.12, 10.13) are covered by replacing the right-hand side by [QG] [W]{N 2 − Qσ02 + QG P˜0 } or

Part A 10.4

Width: 60 mm (2.35 in.) Thick: 2.03 mm (0.08 in.) Crack: 11.7 mm (0.46 in.) Load: 6.67 N (1.50 lb)

[QΨ ] {d − QψF − Qψu }

as is appropriate for nonlinear data relations and nonlinear dynamics, respectively.

10.4.4 Experimental Data Study I: Stress Analysis Around a Hole Fig. 10.26a,b Crack under predominantly mode I loading: (a) experimental light-field photoelastic fringe data and (b) modeling and reconstruction

¯ n is a vector of Md components. The minimizwhere {d} ing principle leads to

¯ ψ] ¯ ψ] ˜ ¯  W[Q ¯ + γt [ H¯ t ] + γs [ H¯ s ] { P} [Q ¯ ψ] ¯ . ¯  W{d} = [Q

(10.15)

This is the core of the algorithm and is quite similar in form to that of (10.10). Thus, without going into the a)

b)

Fig. 10.27a,b Crack tip fringe patterns. (a) Experimental dark-field fringe pattern for green separation. (b) Reconstructions with nearfield experimental white fringes indicated as data points υ = 50 m/s .

Line of symmetry

Fig. 10.28 Blunt projectile impacting a stop

Stop a = 63.5 mm b = 12.7 mm h = 2.5 mm Aluminum

Figure 10.24a shows the recorded photoelastic fringe pattern where the load is distributed over an area. It is not the intention here to determine the precise distribution of this load per se – to do this would require collecting data very close to the distribution. Rather, in the spirit of hybrid analysis, we will do a stress analysis in the region close to the hole precisely where the data is of poor quality. The distributed load was achieved by placing a soft rubber layer between the model and the applied load; consequently, there was no possibility of a priori knowing the distribution, only the resultant. More details on this example can be found in [10.30]. The results to be reported used data taken from both the top and bottom regions away from the edges and from the hole. Two studies were performed. In the first, the load distribution is represented by two forces a distance of three nodes apart. Figure 10.25a shows the sum and difference of these forces as the position is varied. The results clearly indicate the center of the resultant load at P1 − P2 = 0. The second study took into account the distributed nature of the applied load. The initial guess for this study was taken as the resultant load with all others being zero. Figure 10.25b shows the reconstructed distribution. The precise distribution depends on the amount of regularization used. A key point, however, is that the solution in the vicinity of the hole is not sensitive to the precise nature of the distribution of applied load. That is, all reconstructed distributions regardless of regularization gave nearly the same solutions in the vicinity of the hole. Figure 10.24b shows the reconstructed fringe pattern corresponding to the distribution of Fig. 10.25b. The patterns close to the hole are almost identical. The patterns near the distribution show a slight difference which might be due to the presence of a shearing traction. Needless to say, reconstructing the fringe patterns is just one example of the entire results now available

Hybrid Methods

from postprocessing the stress analysis of the fully specified model.

10.4.5 Experimental Data Study II: Photoelastic Analysis of Cracks

a)

+50 m/s

Remote u· (t)

–50 m/s

Contact

An epoxy specimen was cut to the dimensions shown in Fig. 10.26 and an artificial crack was machined using a very fine blade. The specimen was then loaded as described in [10.31] and sent through a stress freezing cycle. Figure 10.26a shows a zoomed out image of the stress frozen pattern; this is a light-field image of the green separation. The crack, as machined, is at a small angle to the edge and therefore there is a small amount of mixed-mode behavior. Figure 10.27a shows the green separation with the image oriented with respect to the crack. The mixed-mode behavior is evident in the inner fringes. A very versatile method of crack parameter extraction is that of the virtual crack closure technique (VCCT) first introduced in [10.32]. The way we will use the FEM modeling is first to use the photoelastic data to determine the effective loading and then use this loading with the VCCT to determine the stress intensities. Figure 10.26b shows the modeling: the clamp/pins are replaced with point support boundary conditions at the bottom and three applied loads on the top. The elevated (stress-freezing) temperature properties were used; the Young’s modulus was estimated as 16.75 MPa (2430 psi) and f¯σ of 200 N/m (1.1 lb/in.). Data remote from the crack tip but not close to the load points were used. First-order regularization was used with a relatively large value of regularization – this forced the distribution to be linear. The results were {P} = {0.351, 0.330, 0.318} P0 , P0 = 6.46 N (1.45 lb) .

0

b)

40

80

120

160 200 Time (μs)

σxx

εxx

–0.5

0

0.5

1

1.5 2 Strain (%)

Fig. 10.29a,b Projectile behavior: (a) velocity responses and (b) nonlinear σxx /¯xx stress–strain response at different locations

The expected resultant load value is P0 = 6.67 N (1.50 lb). It is interesting that the load distribution picks up a slight bending action. Figure 10.26b shows a reconstruction of the fringe pattern using these load values; the comparison with Fig. 10.26a is quite reasonable. The loads were then used to give the stress intensity factors. K 1 /K 0 = 1.18 (1.28) , K 2 /K 0 = 0.03 (0.03) , √ where K 0 = σ∞ πa is the stress intensity factor for an infinite sheet. The parenthetical values are results using the expected load distributed uniformly and computed using the VCCT. Reference [10.6] studied the same problem using the extrapolation method for the three color separations, as well as the Levenberg–Marquardt method for nonlinear parameter identification. The red, green, blue separation results for K 1 /K 0 were 1.37,

251

Part A 10.4

The objective of this study is to determine stress intensity factors using photoelastic data; however, unknown forces will be used as an intermediate parameterization. To elaborate further on this point, stress intensity √ factors are usually presented in graphs as K i = σ∞ πa f (a/W), where W is a geometric factor (such as the width) and f (a/W) shows the influence of the geometry on the crack behavior. This formula gives us a clue as to how to approach crack problems (and singularity-type problems in general): the FEM is used to characterize the geometric effects, and the role of the experiment is to characterize the effective remote loading. This is essentially what we will do in this problem.

10.4 Some Nonlinear Force Identification Problems

252

Part A

Solid Mechanics Topics

a)

b) Reconstructed Exact

Part A 10.4

Initial

5

1

10

2

15

3

30

0

20

40

60

80

100 120 Time (μs)

0

20

40

60

80

100 120 Time (μs)

Fig. 10.30a,b Convergence behavior with 1% noise. (a) First three iterations. (b) Samples over last 20 iterations. Numbers to the left are the iteration counter

1.34, and 1.19, respectively. Note that this required the use of data very close to the crack tip.

10.4.6 Synthetic Data Study I: Elastic–Plastic Projectile Impact One challenge of ballistic experiments is to be able to determine the force due to impact and penetration. References [10.33, 34] discuss two possible experimental methods; what they have in common is that they record data at the back end of the projectile from which they try to infer the applied loading at the front end. Two more recent examples [10.35, 36] used measurements off the side of the projectile. Figure 10.28 shows a blunt projectile impacting a stop – the stop, as implied, does not deform. This a)

P1 H

b)

L = 254 mm H = 127 mm b = 25.4 mm h = 2.5 mm Aluminum

σ

σY = E /500

P2 4

8

4

2 ε

8

2 L

0

0.25

0.5

0.75

1 1.25 Strain (%)

Fig. 10.31a,b A truss with two uncorrelated load histories: (a) geometry and (b) stress–strain behavior of members that yielded

example will be used to explore some of the issues involved in trying to use remote measurements to infer contact loads when the contact region undergoes elastic–plastic deformations. The program NonStaD can model dynamic contact problems [10.19] and was used to generate the synthetic data for the moving projectile contacting the rigid stop. A blunt projectile was chosen because it gives a sharper rise time in the contact force history than the corresponding rounded projectile. Although the six contacting nodes have individual histories only the resultant force history will be identified here. That is, StrIDent has the ability to identify individual forces or resultants on groups of nodes; in the present case, just the resultant will be identified. Figure 10.29a shows the velocity responses along the length of the projectile and Fig. 10.29b shows the stress–strain excursions experienced. As can be seen, there is a substantial amount of plasticity with multiple unloadings and reloadings; however, it is interesting to note that, although the head of the projectile experiences a good deal of plasticity, the rebound velocity shows only a slight loss of momentum. Figure 10.30 shows the resultant force history as the full heavy line. Note that the contact time depends primarily on the length of the projectile and not on the contact conditions. Figure 10.30 shows the convergence behavior using data just from the tail end. The

Hybrid Methods

a)

P2 (t)

Initial

Initial

1

1

5

5

9

9

13

13 1000

1500 Time (μs)

0

Part A 10.4

P1 (t)

500

253

b)

Reconstructed Exact

0

10.4 Some Nonlinear Force Identification Problems

500

1000

1500 Time (μs)

Fig. 10.32 Convergence behavior with 1% noise. Numbers to the left are the iteration counter

initial guess is based on the linear behavior and it looks quite good. This is not surprising because essentially the signal propagated to the monitor is dominated by the elastic response, and the measured response is insensitive to the particulars of the behavior in the vicinity of the contact. This accounts for the success of force identification methods even in the presence of localized plasticity as occurs in the impact of, say, a steel ball on aluminum. There are small but significant differences between the linear reconstructed force and the true force shown as the initial results in Fig. 10.30a. The phase delay is due to the change of modulus above yielding. The first few iterations then attempt to correct for this phase shift. By the fifth iteration this initial behavior has been corrected but, in the process, the maximum force has become large. Bear in mind that this maximum force will primarily affect the long-time response seen in the trailing edge. The next 20 or so iterations of Fig. 10.30b are concerned with reducing this peak force. Convergence is slow but, overall, the final comparison is quite good. That the reconstructed force could have the large excursions and still return to the correct answer attests to the basic robustness of the algorithm.

10.4.7 Synthetic Data Study II: Multiple Loads on a Truss Structure As a second elastic–plastic example, consider the multimember truss shown in Fig. 10.31. This is subject to

two uncorrelated loads and their histories are shown in Fig. 10.32 as the solid lines; note that the two forces are not identical.

Strain gages Photoelastic model

Impactor

Impactor: V = 4.3 m/s D = 19 mm Steel

Cordin 180 camera

Model: L = 508 mm W = 254 mm h = 6.35 mm Epoxy

Fig. 10.33 Experimental setup for recording dynamic photoelastic

photographs

a)

b)

Line of symmetry

P N = 0.5 N=1 N = 0.75

R .

Fig. 10.34a,b Photoelastic experiment: (a) recorded image and (b) discretized data superimposed on (partial view) mesh

254

Part A

Solid Mechanics Topics

Part A 10.4

Again, NonStaD was used to generate synthetic velocity data at each of the joints. With this particular loading, the members can experience many load–unload cycles. Three of the members yield and their stress– strain excursions are shown in Fig. 10.31b. Member 2, which is closest to P1 , shows a significant amount of plasticity. The convergence results are shown in Fig. 10.32 where the vertical velocity of the three center joints were used as data inputs. The initial guess for P1 is poor while that for P2 is quite good. This can be explained by noting that P1 on its own will cause the three members to yield whereas P2 on its own will not cause any yielding. The forces are reconstructed exceptionally well by the tenth or so iteration. Since there is 1% noise, the fluctuations never settle down. Initial attempts to reconstruct the forces over a longer period of time were not especially successful. The longer period encompasses the stage when the members, especially member 2, reverses stress sign. It seems that, when the members are in the unloading/reloading stage, the responses are sensitive to the history of stress up to that point; small variations in initial history can cause large variations in this behavior. This led, in [10.21], to the development of a time sliding algorithm. The basic solution strategy is to reconstruct the forces with a time base that expands (or slides) with the iterations. If the new force estimates are weighted in favor of retaining the already estimated early-time behavior then this gives a robust approach to getting to the unload/reloading stage. Such a strategy also helped considerably to give better initial guesses. Many examples are given in [10.21].

wave (indicated by R) propagating downward along the surface. It is the objective here to determine the contact force history from this single frame of data. One of the difficulties in using photoelasticity for dynamic problems is in identifying the fringe orders. In static problems a variety of techniques can be used including the use of white light where the zero fringe order can be identified as the black fringe. For dynamic crack propagation problems [10.38], the pattern close to the crack tip has a definite monotonic ordering easily counted from the remote state. For general dynamic problems with multiple reflections and data similar to the 300 μs frame in Fig. 10.34, there is just the grey scale image with no obvious ordering. It is worth mentioning, however, that all fringe orders need not be identified since only a subset of the data need be used for the inverse method to succeed. Thus a)

Initial

1 4 7 10 13 0

100

200

300 400 Time (μs)

b)

10.4.8 Experimental Data Study III: Dynamic Photoelasticity The experimental setup is shown in Fig. 10.33, the Cordin high-speed camera can record a sequence 27 photographs with a frame rate up to every 8 μs. The impact is generated by a steel ball. If the ball contacted the epoxy directly, then the contact time would be relatively long and wave propagation effects would be hardly noticeable. The epoxy plate is attached to the one-sided Hopkinson bar so that the impact is actually steel on steel and the strain gages on the bar allow the force at the plate to be computed as discussed earlier. A frame of data is shown in Fig. 10.34a. Reference [10.37] has complete sets of such fringe patterns for different specimen arrangements and different layered materials. The photograph shows the Rayleigh

Frame Force 100

Photoelastic

Bar 0 0

100

200

300

400

500

600 700 Time (μs)

Fig. 10.35a,b Convergence behavior of the force reconstructions. (a) Force reconstructions during iterations. (b) Comparison with force measured from the one-sided Hopkinson bar

Hybrid Methods

a)

b)

Fig. 10.36a,b Data reconstructions: (a) isochromatic fringe pattern and (b) contours of the fringe pattern at every 0.5 fringe order

results are shown in Fig. 10.36a; it has all the characteristics of Fig. 10.34a. What is very interesting is the actual contours of intensity as shown in Fig. 10.34b. These contours are plotted at every 0.1 of a fringe with the whole and half orders being the thick lines, and they show the very complex nature of the fringe pattern, especially in the vicinity of the Rayleigh wave. In particular, they show that the center of the white and black fringes cannot be relied upon to be whole- and halforder fringes, respectively. This reinforces the point that fringe identification can be a significant challenge in dynamic photoelasticity.

10.5 Discussion of Parameterizing the Unknowns The goal of the experimental work is to complete the specification of the given problem. As discussed in the Introduction, there are a variety of unknowns in the problem and consequently there is a choice as to how to parameterize these unknowns. This discussion, in light of the examples covered, attempts to frame the issues involved in choosing the appropriate parameterization. Consider the typical static FEM problem stated as [K (a j )]{u} = {P} . The stiffness, as indicated, depends on a number of parameters a j . From a partially specified problem point of view, the only allowable unknowns are the parameters a j and the applied loads {P}. Thus quantities of interest such as stresses or deformed shapes are not considered basic unknowns – once the problem is fully specified these are obtained as a postprocessing operation. However, force identification and parameter identification are often interchangeable problems; which approach is preferred will depend on the specifics of any particular problem. Some general ideas are discussed next.

255

10.5.1 Parameterized Loadings and Subdomains When the loading is distributed it may require a number of parameters to characterize it and there are a number of situations that may prevail depending on our a priori information and the number of sensors available. If there are a good number of sensors available, then the loading on each node can be considered unknown. If there are fewer sensors but the distribution is expected to be somewhat smooth, then again the loading on each node can be considered unknown and regularization used to maintain a well-behaved solution. If the number of unknowns is computationally too large, then a parameterized loading scheme can be used. A challenging situation arises where there are not many sensors and the distribution is not simple. Consider the situation of an airfoil in an unsteady stream or a building under wind loading. Both will have a complicated loading (in space and time) but we are interested somehow in characterizing (and therefore estimating) the stresses. In these

Part A 10.5

in Fig. 10.34b only those fringes that are obvious or clearly defined were identified and discretized. However, the data used must be distributed over the entire model otherwise the time information would be distorted. Figure 10.35a shows the convergence behavior; in spite of the very poor initial guess, the solution eventually converges to the nearly correct history. The frame of data was taken at time step t = 300 μs but the solution was determined over a time period of 400 μs; the horizontal trailing edge is indicative of the first-order time regularization used when no real information is in the data. A detailed plot of the force history is shown in Fig. 10.35b; the comparison is with the force determined from the one-sided Hopkinson bar. Overall, the comparison is quite good. It is expected that the results could be improved if more frames of data were used and if the parameters of the model (modulus, density, fringe constant, and so on) were tweaked as described earlier in the parameter identification section. The latter was not done here because the limited number frames did not warrant it. As additional processing, the determined force history was used to reconstruct the fringe pattern, and the

10.5 Discussion of Parameterizing the Unknowns

256

Part A

Solid Mechanics Topics

a) Unknowns: • Loads • Properties • BCs • Flaws

Region of interest Unknown event

Part A 10.5

b)

Free body cut

Region of interest

ty tx

Tractions

Fig. 10.37a,b Schematic of the subdomain concept. (a) The complete system is comprised of the unknowns of interest plus remote unknowns of no immediate interest. (b) The analyzed system is comprised of the unknowns of interest plus the unknown boundary tractions

cases it is necessary to develop a model (preferably, a physics- or mechanics-based model) for the loading. It usually happens that the distribution is in terms of parameters that appear nonlinearly in the relation. What the holographic analysis of the plate (Fig. 10.1c), the interferometric analysis of the beam (Fig. 10.1a), the photoelastic analysis of the hole (Fig. 10.24), all have in common is that they use the experimental data indirectly to infer some other information about the problem. That is, the data do not directly give the quantities of interest (the bending strains in the plate, for example) and additional processing is required to complete the stress analysis. Loads were introduced as a convenient intermediate parameterization of the unknowns. The loads as intermediate parameters may have physical significance, as for the Output Stresses

Input

Equilibrium

Loads

Hooke's law

Analysis Heat flow Input

Conduction

Temp Output

Fig. 10.38 Concatenated modeling: the conduction model is used to generate temperature sensitivities that are inputs to Hooke’s law, which in turn produces load sensitivities to the equilibrium modeling

applied pressure in the holographic case, but the traction distribution of Fig. 10.25 is not intended to be the actual traction distribution. The inferred loads are understood to be a set of loads that are consistent with the given experimental data and are capable of accurately reconstructing the stress state in the vicinity of the data. There are additional reasons for choosing forces as the set of unknowns. Often in experimental situations, it is impossible, inefficient, or impractical to model the entire structure. Indeed, in many cases, data can only be collected from a limited region. Furthermore, the probability of achieving a good inverse solution is enhanced if the number of unknowns can be reduced. All these situations would be taken care of if we could just isolate the subregion of interest from the rest of the structure: the aircraft wing detached from the fuselage, the loose joint separate from the beams, the center span independent of the rest of the bridge, and so on. The basic idea of our approach to this concern is shown schematically in Fig. 10.37. The original problem has, in addition to the primary unknowns of interest, additional unknowns associated with the rest of the structure. Through a free-body cut, these are removed from the problem so that the subdomain has the primary unknowns of interest plus a set of additional unknown tractions that are equivalent to the remainder of the structure. While these unknown tractions appear as applied loads on the subdomain, they are, essentially, a parametric representation of the effect the rest of the structure has on the subregion of interest.

10.5.2 Unknowns Parameterized Through a Second Model To help fix the ideas expressed here, consider the thermoelastic stress analysis of a nonuniform cylinder with a hot fluid slowly moving through its core. Further, assume we are using integrated photoelasticity [10.39, 40]; then the fringe distribution observed on any crosssection is   fσ (σzz − σ yy )2 + 4σz2y dx , N(y, z) = h where the integration is along the light path. If the problem is parameterized with the stresses as the unknowns, there is insufficient data to determine the stress distributions even in the simple case of axisymmetric stresses [10.40]. Even if the problem is parameterized with the temperatures as the unknowns then there still would never be a sufficient number of independent light paths to determine the great many unknowns.

Hybrid Methods

concatenating many models generalizes to problems involving residual stresses, radiation, and electrostatics, to name a few. It can also be extended to complex interacting systems such as loads on turbine blades in an engine.

10.5.3 Final Remarks What these examples and discussions show is that a crucial step in inverse analysis is parameterizing the unknowns of a problem; the specific choices made depend on the problem type, the number of sensors available, and the sophistication of the modeling capability. The examples also show that we have tremendous flexibility as to how this can be done; this flexibility is afforded by incorporating FEM-type modeling as an integral part of the experimental analysis.

References 10.1

10.2 10.3

10.4

10.5

10.6

10.7

10.8

10.9 10.10

10.11 10.12

A.S. Kobayashi: Hybrid experimental-numerical stress analysis. In: Handbook on Experimental Mechanics, ed. by A.S. Kobayashi (VCH, Weinheim 1993) pp. 751–783 A.S. Kobayashi: Hybrid experimental-numerical stress analysis, Exp. Mech. 23, 338–347 (1983) T.H. Baek, R.E. Rowlands: Hybrid stress analysis of perforated composites using strain gages, Exp. Mech. 41(2), 195–203 (2001) B.J. Rauch, R.E. Rowlands: Stress separation of thermoelastically measured isopachics, Exp. Mech. 41(4), 358–367 (2001) J. Rhee, R.E. Rowlands: Moiré-numerical hybrid analysis of cracks in orthotropic media, Exp. Mech. 42(3), 311–317 (2002) J.F. Doyle: Modern Experimental Stress Analysis: Completing the Solution of Partially Specified Problems (Wiley, Chichester 2004) A. Neumaier: Solving ill-conditioned and singular linear systems: a tutorial on regularization, SIAM J. Appl. Math. 40(3), 636–666 (1998) W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling: Numerical Recipes, 2nd edn. (Cambridge Univ. Press, Cambridge 1992) A.N. Tikhonov, V.Y. Arsenin: Solutions of Ill-Posed Problems (Wiley, New York 1977) S. Twomey: Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam 1977) J.F. Doyle: Static and Dynamic Analysis of Structures (Kluwer, Dordredt 1991) J.F. Doyle: Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability (Springer, New York 2001)

257

10.13

10.14

10.15 10.16

10.17

10.18 10.19

10.20

10.21

10.22

10.23 10.24

D.L. Phillips: A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9(1), 84–97 (1962) S. Twomey: On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadratures, J. Assoc. Comput. Mach. 10, 97–101 (1963) O.M. Alifanov: Methods of solving ill-posed inverse problems, J. Eng. Phys. 45(5), 1237–1245 (1983) Y. Martinez, A. Dinth: A generalization of Tikhonov’s regularizations of zero and first order, Comput. Math. Appl. 12B(5/6), 1203–1208 (1986) A.J. Quintana: A Global Search Method for Damage Detection in General Structures. M.Sc. Thesis (Purdue University, West Lafayette 2004) M. Kleiber: Parameter Sensitivity in Nonlinear Mechanics (Wiley, Chichester 1997) S.-W. Choi: Impact Damage of Layered Material Systems. Ph.D. Thesis (Purdue University, West Lafayette 2002) S.-M. Cho: A Sub-Domain Inverse Method for Dynamic Crack Propagation Problems. M.Sc. Thesis (Purdue University, West Lafayette 2000) S.-M. Cho: Algorithms for Identification of the Nonlinear Behavior of Structures. Ph.D. Thesis (Purdue University, West Lafayette 2004) J.F. Doyle: A wavelet deconvolution method for impact force identification, Exp. Mech. 37, 404–408 (1997) D.J. Ewins: Modal Testing: Theory and Practice (Wiley, New York 1984) R.A. Adams, J.F. Doyle: Multiple force identification for complex structures, Exp. Mech. 42(1), 25–36 (2002)

Part A 10

What we can do instead is use a second modeling (in this case a conduction model) to parameterize the temperature distribution in terms of a limited number of boundary temperatures or heat sources. The idea is shown in Fig. 10.38 as a concatenation of modelings: the stresses are related to the applied loads through the equilibrium model, the applied loads are related to the temperature distribution through Hooke’s law, and the temperature distribution is related to the parameterized heat sources through the conduction model. Working in reverse, the conduction model uses the boundary temperature fields to compute the temperature sensitivity fields. These in turn are used in the equilibrium modeling to compute the sensitivity responses to find the stresses. Thus the stress parameterization is ultimately related to the heat parameterization. This idea of using a second model or

References

258

Part A

Solid Mechanics Topics

10.25

10.26 10.27

Part A 10

10.28

10.29

10.30

10.31

10.32

10.33

R.A. Adams: Force Identification in Complex Structures. M.Sc. Thesis (Purdue University, West Lafayette 1999) J.W. Dally, W.F. Riley: Experimental Stress Analysis, 3rd edn. (McGraw-Hill, New York 1991) T. Liu, M. Guille, J.P. Sullivan: Accuracy of pressure sensitive paint, AIAA J. 39(1), 103–112 (2001) H. Sakaue, J.P. Sullivan: Time response of anodized aluminum pressure sensitive paint, AIAA J. 39(10), 1944–1949 (2001) J.W. Dally, R.J. Sanford: Multiple ruby laser system for high speed photography, Opt. Eng. 21, 704–708 (1982) U.-T. Kang: Inverse Method for Static Problems Using Optical Data. Ph.D. Thesis (Purdue University, West Lafayette 2002) J.F. Doyle, S. Kamle, J. Takezaki: Error analysis of photoelasticity in fracture mechanics, Exp. Mech. 17, 429–435 (1981) E.F. Rybicki, M.F. Kanninen: A finite element calculation of stress intensity factors by a modified crack-closure integral, Eng. Fract. Mech. 9, 931– 938 (1977) A.L. Chang, A.M. Rajendran: Novel in-situ ballistic measurements for validation of ceramic constitutive models, 14th US Army Symp. Solid Mech., ed.

10.34

10.35

10.36

10.37

10.38 10.39 10.40

by K.R. Iyer, S.-C. Chou (Batelle, Columbus 1996) pp. 99–110 H.D. Espinosa, Y. Xu, H.-C. Lu: A novel technique for penetrator velocity measurements and damage identification in ballistic penetration experiments. In: 14th US Army Symposium on Solid Mechanics, ed. by K.R. Iyer, S.-C. Chou (Batelle, Columbus 1996) pp. 111–120 H.A. Bruck, D. Casem, R.L. Williamson, J.S. Epstein: Characterization of short duration stress pulses generated by impacting laminated carbonfiber/epoxy composites with magnetic flyer plates, Exp. Mech. 42(3), 279–287 (2002) J. Degrieck, P. Verleysen: Determination of impact parameters by optical measurement of the impactor displacement, Exp. Mech. 42(3), 298–302 (2002) S.A. Rizzi: A Spectral Analysis Approach to Wave Propagation in Layered Solids. Ph.D. Thesis (Purdue University, West Lafayette 1989) J.W. Dally: Dynamic photoelastic studies of dynamic fracture, Exp. Mech. 19, 349–361 (1979) H.K. Aben: Integrated Photoelasticity (McGrawHill, New York 1980) J.F. Doyle, H.T. Danyluk: Integrated photoelasticity for axisymmetric problems, Exp. Mech. 18(6), 215– 220 (1978)

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Statistical An

11. Statistical Analysis of Experimental Data

James W. Dally

characterized statistically. Regression analysis provides a method to fit a straight line or a curve through a series of scattered data points on a graph. The adequacy of the regression analysis can be evaluated by determining a correlation coefficient. Methods for extending regression analysis to multivariate functions exist. In principle, these methods are identical to linear regression analysis; however, the analysis becomes much more complex. The increase in complexity is not a concern, because computer subroutines are available that solve the tedious equations and provide the results in a convenient format. Many probability functions are used in statistical analyses to represent data and predict population properties. Once a probability function has been selected to represent a population, any series of measurements can be subjected to a chi-squared (χ 2 ) test to check the validity of the assumed function. Accurate predictions can be made only if the proper probability function has been selected. Finally, statistical methods for accessing error propagation are discussed. These methods provide a means for determining error in a quantity of interest y based on measurements of related quantities x1 , x2 , ..., xn and the functional relationship y = f (x1 ‚x2 ‚ ..., xn ).

11.1

Characterizing Statistical Distributions ... 11.1.1 Graphical Representations of the Distribution........................ 11.1.2 Measures of Central Tendency ........ 11.1.3 Measures of Dispersion .................

260 260 261 262

11.2 Statistical Distribution Functions ........... 263 11.2.1 Gaussian Distribution ................... 263 11.2.2 Weibull Distribution ..................... 265 11.3 Confidence Intervals for Predictions ....... 267 11.4 Comparison of Means............................ 270

Part A 11

Statistical methods are extremely important in engineering, because they provide a means for representing large amounts of data in a concise form that is easily interpreted and understood. Usually, the data are represented with a statistical distribution function that can be characterized by a measure of central tendency (the mean x¯ ) and a measure of dispersion (the standard deviation Sx ). A normal or Gaussian probability distribution is by far the most commonly employed; however, in some cases, other distribution functions may have to be employed to adequately represent the data. The most significant advantage resulting from the use of a probability distribution function in engineering applications is the ability to predict the occurrence of an event based on a relatively small sample. The effects of sampling error are accounted for by placing confidence limits on the predictions and establishing the associated confidence levels. Sampling error can be controlled if the sample size is adequate. Use of Student’s t distribution function, which characterizes sampling error, provides a basis for determining sample size consistent with specified levels of confidence. Student’s t distribution also permits a comparison to be made of two means to determine whether the observed difference is significant or whether it is due to random variation. Statistical methods can also be employed to condition data and to eliminate an erroneous data point (one) from a series of measurements. This is a useful technique that improves the data base by providing strong evidence when something unanticipated is affecting an experiment. Regression analysis can be used effectively to interpret data when the behavior of one quantity y depends upon variations in one or more independent quantities x1 , x2 , ..., xn . Even though the functional relationship between quantities exhibiting variation remains unknown, it can be

260

Part A

Solid Mechanics Topics

11.5 Statistical Safety Factor ......................... 271 11.6 Statistical Conditioning of Data ............. 272 11.7 Regression Analysis............................... 272 11.7.1 Linear Regression Analysis............. 272 11.7.2 Multivariate Regression................. 274

Part A 11.1

Experimental measurements of quantities such as pressure, temperature, length, force, stress or strain will always exhibit some variation if the measurements are repeated a number of times with precise instruments. This variability, which is fundamental to all measuring systems, is due to two different causes. First, the quantity being measured may exhibit significant variation. For example, in a materials study to determine fatigue life at a specified stress level, large differences in the number of cycles to failure are noted when a number of specimens are tested. This variation is inherent to the fatigue process and is observed in all fatigue life measurements. Second, the measuring system, which includes the transducer, signal conditioning equipment, analogue-to-digital (A/D) converter, recording instrument, and an operator may introduce error in the measurement. This error may be systematic or random, depending upon its source. An instrument operated out of calibration produces a systematic error, whereas reading errors due to interpolation on a chart are random. The accumulation of random errors in a measuring system produces a variation that must be examined in relation to the magnitude of the quantity being measured. The data obtained from repeated measurements represent an array of readings not an exact result. Maximum information can be extracted from such an array of

11.7.3 Field Applications of Least-Square Methods .............. 275 11.8 Chi-Square Testing ............................... 277 11.9 Error Propagation ................................. 278 References .................................................. 279

readings by employing statistical methods. The first step in the statistical treatment of data is to establish the distribution. A graphical representation of the distribution is usually the most useful form for initial evaluation. Next, the statistical distribution is characterized with a measure of its central value, such as the mean, the median, or the mode. Finally, the spread or dispersion of the distribution is determined in terms of the variance or the standard deviation. With elementary statistical methods, the experimentalist can reduce a large amount of data to a very compact and useful form by defining the type of distribution, establishing the single value that best represents the central value of the distribution (mean), and determining the variation from the mean value (standard deviation). Summarizing data in this manner is the most meaningful form of presentation for application to design problems or for communication to others who need the results of the experiments. The treatment of statistical methods presented in this chapter is relatively brief; therefore, only the most commonly employed techniques for representing and interpreting data are presented. A formal course in statistics, which covers these techniques in much greater detail as well as many other useful techniques, should be included in the program of study of all engineering students.

11.1 Characterizing Statistical Distributions For the purposes of this discussion, consider the data obtained from an experiment conducted n times to measure the ultimate tensile strength of a fully tempered beryllium copper alloy. The data obtained represent a sample of size n from an infinite population of all possible measurements that could have been made. The simplest way to present these data is to list the strength measurements in order of increasing magnitude, as shown in Table 11.1. These data can be arranged into seven groups to give a frequency distribution as shown in Table 11.2. The ad-

vantage of representing data in a frequency distribution is that the central tendency is more clearly illustrated.

11.1.1 Graphical Representations of the Distribution The shape of the distribution function representing the ultimate tensile strength of beryllium copper is indicated by the data groupings of Table 11.2. A graphical presentation of this group data, known as a histogram, is shown in Fig. 11.1. The histogram method of presen-

Statistical Analysis of Experimental Data

Table 11.1 The ultimate tensile strength of beryllium cop-

per, listed in order of increasing magnitude Strength ksi (MPa)

Sample number

Strength ksi (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

170.5 (1175) 171.9 (1185) 172.6 (1190) 173.0 (1193) 173.4 (1196) 173.7 (1198) 174.2 (1201) 174.4 (1203) 174.5 (1203) 174.8 (1206) 174.9 (1206) 175.0 (1207) 175.4 (1210) 175.5 (1210) 175.6 (1211) 175.6 (1211) 175.8 (1212) 175.9 (1213) 176.0 (1214) 176.1 (1215)

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

176.2 (1215) 176.2 (1215) 176.4 (1217) 176.6 (1218) 176.7 (1219) 176.9 (1220) 176.9 (1220) 177.2 (1222) 177.3 (1223) 177.4 (1223) 177.7 (1226) 177.8 (1226) 178.0 (1228) 178.1 (1228) 178.3 (1230) 178.4 (1230) 179.0 (1236) 179.7 (1239) 180.1 (1242) 181.6 (1252)

tation shows the central tendency and variability of the distribution much more clearly than the tabular method of presentation of Table 11.2. Superimposed on the histogram is a curve showing the relative frequency of the occurrence of a group of measurements. Note that the points for the relative frequency are plotted at the midpoint of the group interval. A cumulative frequency diagram, shown in Fig. 11.2, is another way of representing the ultimate strength data from the experiments. The cumulative frequency is the number of readings having a value less

Number of observations 20

261

Relative frequency 0.5

18 0.4

16 14

0.3

12 10

0.2

8 6

0.1

4 2

0 0 167 169 171 173 175 177 179 181 183 185 Ultimate tensile strength Su (ksi)

Fig. 11.1 Histogram with superimposed relative frequency

diagram

than a specified value of the quantity being measured (ultimate strength) divided by the total number of measurements. As indicated in Table 11.2, the cumulative frequency is the running sum of the relative frequencies. When the graph of cumulative frequency versus the quantity being measured is prepared, the end value for the group intervals is used to position the point along the abscissa.

11.1.2 Measures of Central Tendency While histograms or frequency distributions are used to provide a visual representation of a distribution, numerical measures are used to define the characteristics of the distribution. One basic characteristic is the central tendency of the data. The most commonly employed measure of the central tendency of a distribution of data

Table 11.2 Frequency distribution of ultimate tensile strength Intervals group ksi (MPa)

Observations in the group

Relative frequency

Cumulative frequency

169.0 − 170.9 (1166 − 1178) 171.0 − 172.9 (1179 − 1192) 173.0 − 174.9 (1193 − 1206) 175.0 − 176.9 (1207 − 1220) 177.0 − 178.9 (1221 − 1234) 179.0 − 180.9 (1235 − 1248) 181.0 − 182.9 (1249 − 1261) Total

1 2 8 16 9 3 1 40

0.025 0.050 0.200 0.400 0.225 0.075 0.025

0.025 0.075 0.275 0.675 0.900 0.975 1.000

Part A 11.1

Sample number

11.1 Characterizing Statistical Distributions

262

Part A

Solid Mechanics Topics

It is evident that a typical set of data may give different values for the three measures of central tendency. There are two reasons for this difference. First, the population from which the samples were drawn may not be Gaussian (for which the three measures are expected to coincide). Second, even if the population is Gaussian, the number of measurements n is usually small and deviations due to a small sample size are to be expected.

Cumulative frequency 1 0.8 0.6 0.4

11.1.3 Measures of Dispersion 0.2

Part A 11.1

0 167 169 171 173 175 177 179 181 183 185 Ultimate tensile strength Su (ksi)

Fig. 11.2 Cumulative frequency diagram

is the sample mean x, ¯ which is defined as x¯ =

n  xi , n

(11.1)

i=1

where xi is the i-th value of the quantity being measured and n is the total number of measurements. Because of time and costs involved in conducting tests, the number of measurements is usually limited; therefore, the sample mean x¯ is only an estimate of the true arithmetic mean μ of the population. It is shown later that x¯ approaches μ as the number of measurements increases. The mean value of the ultimate strength data presented in Table 11.1 is x¯ = 176.1 ksi (1215 MPa). The median and mode are also measures of central tendency. The median is the central value in a group of ordered data. For example, in an ordered set of 41 readings, the 21-st reading represents the median value with 20 readings lower than the median and 20 readings higher than the median. In instances when an even number of readings are taken, the median is obtained by averaging the two middle values. For example, in an ordered set of 40 readings, the median is the average of the 20-th and 21-st readings. Thus, for the ultimate tensile strength data presented in Table 11.1, the median is 1/2(176.1 + 176.2) = 176.15 ksi (1215 MPa). The mode is the most frequent value of the data; therefore, it is located at the peak of the relative frequency curve. In Fig. 11.1, the peak of the relative probability curve occurs at an ultimate tensile strength Su = 176.0 ksi (1214 MPa); therefore, this value is the mode of the data set presented in Table 11.1.

It is possible for two different distributions of data to have the same mean but different dispersions, as shown in the relative frequency diagrams of Fig. 11.3. Different measures of dispersion are the range, the mean deviation, the variance, and the standard deviation. The standard deviation Sx is the most popular and is defined as:  1/2 n (xi − x) ¯2 . (11.2) Sx = n −1 i=1

Because the sample size n is small, the standard deviation Sx of the sample represents an estimate of the true standard deviation σ of the population. Computation of Sx and x¯ from a data sample is easily performed with most scientific-type calculators. Expressions for the other measures of dispersion, namely, the range R, mean deviation dx , and variance Sx2 are given by R = xL − xs ,  n   xi − x¯  dx = , n

(11.3) (11.4)

i=1

Relative frequency Small dispersion

Large dispersion

μ

Fig. 11.3 Relative frequency diagrams with large and small dispersions

Statistical Analysis of Experimental Data

Sx2 =

2 n   xi − x¯ , n −1

(11.5)

i=1

estimate of the mean x. ¯ As the sample size n is increased the estimates of x, ¯ Sx , and Sx2 improve, as shown in the discussion of Sect. 11.4. Variance is an important measure of dispersion because it is used in defining the normal distribution function. Finally, a measure known as the coefficient of variation Cv is used to express the standard deviation Sx as a percentage of the mean x. ¯ Thus, Cv =

Sx (100) . x¯

(11.6)

The coefficient of variation represents a normalized parameter that indicates the variability of the data in relation to its mean.

11.2 Statistical Distribution Functions As the sample size is increased, it is possible in tabulating the data to increase the number of group intervals and to decrease their width. The corresponding relative frequency diagram, similar to the one illustrated in Fig. 11.1, will approach a smooth curve (a theoretical distribution curve) known as a distribution function. A number of different distribution functions are used in statistical analyses. The best-known and most widely used distribution in experimental mechanics is the Gaussian or normal distribution. This distribution is extremely important because it describes random errors in measurements and variations observed in strength determinations. Other useful distributions include the binomial, exponential, hypergeometric, chi-square χ 2 , Relative frequency 0.5

11.2.1 Gaussian Distribution The Gaussian or normal distribution function, as represented by a normalized relative frequency diagram, is shown in Fig. 11.4. The Gaussian distribution is completely defined by two parameters: the mean μ and the standard deviation σ. The equation for the relative frequency f in terms of these two parameters is given by: (11.7)

where

0.3

x¯ − μ (11.8) . σ Experimental data (with finite sample sizes) can be analyzed to obtain x¯ as an estimate of μ and Sx as an estimate of σ. This procedure permits the experimentalist to use data drawn from small samples to represent the entire population. The method for predicting population properties from a Gaussian (normal) distribution function utilizes the normalized relative frequency diagram shown z=

0.2 0.1 0

F, Gumbel, Poisson, Student’s t, and Weibull distributions; the reader is referred to [11.1–5] for a complete description of these distributions. Emphasis here will be on the Gaussian and Weibull distribution functions because of their wide range of application in experimental mechanics [11.6–13].

1 2 f (z) = √ e−(z /2) , 2π

0.4

μ–3σ μ–2σ –3 –2

μ–σ –1

μ 0

μ+σ 1

μ+2σ μ+3σ x 2 3 z

Fig. 11.4 The normal or Gaussian distribution function

263

Part A 11.2

where xL is the largest value of the quantity in the distribution and xs is the smallest value. Equation (11.4) indicates that the deviation of each reading from the mean is determined and summed. The average of the n deviations is the mean deviation. The absolute value of the difference (xi − x) ¯ must be used in the summing process to avoid cancellation of positive and negative deviations. The variance of the population σ 2 is estimated by Sx2 , where the denominator (n − 1) in (11.2) and (11.5) serves to reduce error introduced by approximating the true mean μ with the

11.2 Statistical Distribution Functions

264

Part A

Solid Mechanics Topics

A(0, +1) leads to the following determinations:

f (z)

Area =

Part A 11.2

–2

–1

0

1

f (z) dz

z2

z1 –3

z2

∫z

1

A(−1, +1) = p(−1, +1) = 0.3413 + 0.3413 = 0.6826 , A(−2, +2) = p(−2, +2) = 0.4772 + 0.4772 = 0.9544 , A(−3, +3) = p(−3, +3) = 0.49865 + 0.49865 = 0.9973 , A(−1, +2) = p(−1, +2) = 0.3413 + 0.4772 = 0.8185 . z

2

3

Fig. 11.5 Probability of a measurement of x between limits of z 1 and z 2 . The total area under the curve f (z) is 1

in Fig. 11.4. The area A under the entire curve is given by (11.7) as 1 A= √ 2π

∞

e−(z

2 /2)

dz = 1 .

(11.9)

−∞

Equation (11.9) implies that the population has a value z between −∞ and +∞ and that the probability of making a single observation from the population with a value −∞ ≤ z ≤ +∞ is 100%. While the previous statement may appear trivial and obvious, it serves to illustrate the concept of using the area under the normalized relative frequency curve to determine the probability p of observing a measurement within a specific interval. Figure 11.5 shows graphically, with the shaded area under the curve, the probability that a measurement will occur within the interval between z 1 and z 2 . Thus, from (11.7) it is evident that: z 2 p(z 1 , z 2 ) = z1

1 f (z) dz = √ 2π

z 2

e−(z

2 /2)

dz .

z1

(11.10)

Evaluation of (11.10) is most easily accomplished by using tables that list the areas under the normalized relative frequency curve as a function of z. Table 11.3 lists one-side areas between limits of z 1 = 0 and z 2 for the normal distribution function. Since the distribution function is symmetric about z = 0, this one-sided table is sufficient for all evaluations of the probability. For example, A(−1, 0) =

Because the normal distribution function has been well characterized, predictions can be made regarding the probability of a specific strength value or measurement error. For example, one may anticipate that 68.3% of the data will fall between limits of x¯ ± 1.0 Sx , 95.4% between limits of x¯ ± 2.0 Sx , and 99.7% between limits of x¯ ± 3.0 Sx . Also, 81.9% of the data should fall between limits of x¯ − 1.0 Sx and x¯ + 2.0 Sx . In many problems, the probability of a single sample exceeding a specified value z 2 must be determined. It is possible to determine this probability by using Table 11.3 together with the fact that the area under the entire curve is unity (A = 1); however, Table 11.4, which lists one-sided areas between limits of z 1 = z and z 2 ⇒ ∞, yields the results more directly. The use of Tables 11.3 and 11.4 can be illustrated by considering the ultimate tensile strength data presented in Table 11.1. By using (11.1) and (11.2), it is easy to establish estimates for the mean x¯ and standard deviation Sx as x¯ = 176.1 ksi (1215 MPa) and Sx = 2.25 ksi (15.5 MPa). The values of x¯ and Sx characterize the population from which the data of Table 11.1 were drawn. It is possible to establish the probability that the ultimate tensile strength of a single specimen drawn randomly from the population will be between specified limits (by using Table 11.3), or that the ultimate tensile strength of a single sample will not be above or below a specified value (by using Table 11.4). For example, one determines the probability that a single sample will exhibit an ultimate tensile strength between 175 ksi (1207 MPa) and 178 ksi (1234 MPa) by computing z 1 and z 2 and using Table 11.3. Thus, 175 − 176.1 = −0.489 , 2.25 178 − 176.1 z2 = = 0.844 , 2.25 p( − 0.489, 0.844) = A(−0.489, 0) + A(0, 0.844) = 0.1875 + 0.3006 = 0.4981 . z1 =

Statistical Analysis of Experimental Data

11.2 Statistical Distribution Functions

265

Table 11.3 Areas under the normal distribution curve from z 1 = 0 to z 2 (one side) 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.0000

0.0040

0.0080

0.0120

0.0160

0.0199

0.0239

0.0279

0.0319

0.0359

0.1

0.0398

0.0438

0.0478

0.0517

0.0557

0.0596

0.0636

0.0675

0.0714

0.0753

0.2

0.0793

0.0832

0.0871

0.0910

0.0948

0.0987

0.1026

0.1064

0.1103

0.1141

0.3

0.1179

0.1217

0.1255

0.1293

0.1331

0.1368

0.1406

0.1443

0.1480

0.1517

0.4

0.1554

0.1591

0.1628

0.1664

0.1700

0.1736

0.1772

0.1808

0.1844

0.1879

0.5

0.1915

0.1950

0.1985

0.2019

0.2054

0.2088

0.2123

0.2157

0.2190

0.2224

0.6

0.2257

0.2291

0.2324

0.2357

0.2389

0.2422

0.2454

0.2486

0.2517

0.2549

0.7

0.2580

0.2611

0.2642

0.2673

0.2704

0.2734

0.2764

0.2794

0.2823

0.2852

0.8

0.2881

0.2910

0.2939

0.2967

0.2995

0.3023

0.3051

0.3078

0.3106

0.3233

0.9

0.3159

0.3186

0.3212

0.3238

0.3264

0.3289

0.3315

0.3340

0.3365

0.3389

1.0

0.3413

0.3438

0.3461

0.3485

0.3508

0.3531

0.3554

0.3577

0.3599

0.3621

1.1

0.3643

0.3665

0.3686

0.3708

0.3729

0.3749

0.3770

0.3790

0.3810

0.3830

1.2

0.3849

0.3869

0.3888

0.3907

0.3925

0.3944

0.3962

0.3980

0.3997

0.4015

1.3

0.4032

0.4049

0.4066

0.4082

0.4099

0.4115

0.4131

0.4147

0.4162

0.4177

1.4

0.4192

0.4207

0.4222

0.4236

0.4251

0.4265

0.4279

0.4292

0.4306

0.4319

1.5

0.4332

0.4345

0.4357

0.4370

0.4382

0.4394

0.4406

0.4418

0.4429

0.4441

1.6

0.4452

0.4463

0.4474

0.4484

0.4495

0.4505

0.4515

0.4525

0.4535

0.4545

1.7

0.4554

0.4564

0.4573

0.4582

0.4591

0.4599

0.4608

0.4616

0.4625

0.4633

1.8

0.4641

0.4649

0.4656

0.4664

0.4671

0.4678

0.4686

0.4693

0.4699

0.4706

1.9

0.4713

0.4719

0.4726

0.4732

0.4738

0.4744

0.4750

0.4758

0.4761

0.4767

2.0

0.4772

0.4778

0.4783

0.4788

0.4793

0.4799

0.4803

0.4808

0.4812

0.4817

2.1

0.4821

0.4826

0.4830

0.4834

0.4838

0.4842

0.4846

0.4850

0.4854

0.4857

2.2

0.4861

0.4864

0.4868

0.4871

0.4875

0.4878

0.4881

0.4884

0.4887

0.4890

2.3

.4893

0.4896

0.4898

0.4901

0.4904

0.4906

0.4909

0.4911

0.4913

0.4916

2.4

0.4918

0.4920

0.4922

0.4925

0.4927

0.4929

0.4931

0.4932

0.4934

0.4936

2.5

0.4938

0.4940

0.4941

0.4943

0.4945

0.4946

0.4948

0.4949

0.4951

0.4952

2.6

0.4953

0.4955

0.4956

0.4957

0.4959

0.4960

0.4961

0.4962

0.4963

0.4964

2.7

0.4965

0.4966

0.4967

0.4968

0.4969

0.4970

0.4971

0.4972

0.4973

0.4974

2.8

0.4974

0.4975

0.4976

0.4977

0.4977

0.4978

1.4979

0.4979

0.4980

0.4981

2.9

0.4981

0.4982

0.4982

0.4983

0.4984

0.4984

0.4985

0.4985

0.4986

0.4986

3.0

0.49865

0.4987

0.4987

0.4988

0.4988

0.4988

0.4989

0.4989

0.4989

0.4990

z2

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

This simple calculation shows that the probability of obtaining an ultimate tensile strength between 175 ksi (1207 MPa) and 178 ksi (1234 MPa) from a single specimen is 49.8%. The probability of the ultimate tensile strength of a single specimen being less than 173 ksi (1193 MPa) is determined by computing z 1 and using Table 11.4. Thus, 173 − 176.1 = −1.37 , 2.25 p(−∞, −1.37) = A(−∞, −1.37) = A(1.37, ∞) = 0.0853 . z1 =

Thus, the probability of drawing a single sample with an ultimate tensile strength less than 173 ksi (1193 MPa) is 8.5%.

11.2.2 Weibull Distribution In investigations of the strength of materials due to brittle fracture, of crack initiation toughness, or of fatigue life, researchers often find that the Weibull distribution provides a more suitable approach to the statistical analysis of the available data. The Weibull distribution function p(x) is defined

Part A 11.2

z2

266

Part A

Solid Mechanics Topics

Table 11.4 Areas under the normal distribution curve from z 1 to z 2 ⇒ ∞ (one side)

Part A 11.2

z1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

0.5000 0.4602 0.4207 0.3821 0.3446 0.3085 0.2743 0.2430 0.2119 0.1841 0.1587 0.1357 0.1151 0.0968 0.0808 0.0668 0.0548 0.0446 0.0359 0.0287 0.0228 0.0179 0.0139 0.0107 0.00820 0.00621 0.00466 0.00347 0.00256 0.00187

0.4960 .4562 0.4168 0.3783 0.3409 0.3050 0.2709 0.2389 0.2090 0.1814 0.1562 0.1335 0.1131 0.0951 0.0793 0.0655 0.0537 0.0436 0.0351 0.0281 0.0222 0.0174 0.0136 0.0104 0.00798 0.00604 0.00453 0.00336 0.00248 0.00181

0.4920 0.4522 0.4129 0.3745 0.3372 0.3015 0.2676 0.2358 0.2061 0.1788 0.1539 0.1314 0.1112 0.0934 0.0778 0.0643 0.0526 0.0427 0.0344 0.0274 0.0217 0.0170 0.0132 0.0102 0.00776 0.00587 0.00440 0.00326 0.00240 0.00175

0.4880 0.4483 0.4090 0.3707 0.3336 0.2981 0.2643 0.2327 0.2033 0.1762 0.1515 0.1292 0.1093 0.0918 0.0764 0.0630 0.0516 0.0418 0.0336 0.0268 0.0212 0.0166 0.0129 0.00990 0.00755 0.00570 0.00427 0.00317 0.00233 0.00169

0.4840 0.4443 0.4052 0.3669 0.3300 0.2946 0.2611 0.2296 0.2005 0.1736 0.1492 0.1271 0.1075 0.0901 0.0749 0.0618 0.0505 0.0409 0.0329 0.0262 0.0207 0.0162 0.0125 0.00964 0.00734 0.00554 0.00415 0.00307 0.00226 0.00164

0.4801 0.4404 0.4013 0.3632 0.3264 0.2912 0.2578 0.2266 0.1977 0.1711 0.1469 0.1251 0.1056 0.0885 0.0735 0.0606 0.0495 0.0401 0.0322 0.0256 0.0202 0.0158 0.0122 0.00939 0.00714 0.00539 0.00402 0.00298 0.00219 0.00159

0.4761 0.4364 0.3974 0.3594 0.3228 0.2877 0.2546 0.2236 0.1949 0.1685 0.1446 0.1230 0.1038 0.0869 0.0721 0.0594 0.0485 0.0392 0.0314 0.0250 0.0197 0.0154 0.0119 0.00914 0.00695 0.00523 0.00391 0.00288 0.00212 0.00154

0.4721 0.4325 0.3936 0.3557 0.3192 0.2843 0.2514 0.2206 0.1922 0.1660 0.1423 0.1210 0.1020 0.0853 0.0708 0.0582 0.0475 0.0384 0.0307 0.0244 0.0192 0.0150 0.0116 0.00889 0.00676 0.00508 0.00379 0.00280 0.00205 0.00149

0.4681 0.4286 0.3897 0.3520 0.3156 0.2810 0.2483 0.2177 0.1894 0.1635 0.1401 0.1190 0.1003 0.0838 0.0694 0.0571 0.0465 0.0375 0.0301 0.0239 0.0188 0.0146 0.0113 0.00866 0.00657 0.00494 0.00368 0.00272 0.00199 0.00144

0.4641 0.4247 0.3859 0.3483 0.3121 0.2776 0.2451 0.2148 0.1867 0.1611 0.1379 0.1170 0.0985 0.0823 0.0681 0.0559 0.0455 0.0367 0.0294 0.0233 0.0183 0.0143 0.0110 0.00840 0.00639 0.00480 0.00357 0.00264 0.00193 0.00139

as: −[(x−x 0 )/b]2

p(x) = 1 − e p(x) = 0

for x > x0 , for x < x0 ,

(11.11)

where x0 , b, and m are the three parameters which define this distribution function. In studies of strength, p(x) is taken as the probability of failure when a stress x is placed on the specimen. The parameter x0 is the zero strength since p(x) = 0 for x < x0 . The constants b and m are known as the scale parameter and the Weibull slope parameter (modulus), respectively. Four Weibull distribution curves are presented in Fig. 11.6 for the case where x0 = 3, b = 10, and m = 2, 5, 10, and 20. These curves illustrate two important features of the Weibull distribution. First, there

is a threshold strength x0 and if the applied stress is less than x0 , the probability of failure is zero. Second, the Weibull distribution curves are not symmetric, and the distortion in the S-shaped curves is controlled by the Weibull slope parameter m. Application of the Weibull distribution to predict failure rates of 1% or less of the population is particularly important in engineering projects where reliabilities of 99% or greater are required. To utilize the Weibull distribution requires knowledge of the Weibull parameters. In experimental investigations, it is necessary to conduct experiments and obtain a relatively large data set to accurately determine x0 , b, and m. Consider as an illustration, Weibull’s own work in statistically characterizing the fiber strength of Indian cotton. In this example, an unusually large sam-

Statistical Analysis of Experimental Data

Probability of failure (%) 100 90

11.3 Confidence Intervals for Predictions

Reduced variate y = log10 loge (1–P)–1 1

Probability of failure (%) 5

m = 20

2

10

80

P = m /(n +1) n = 3000 S0 = 0.46 g

99.9 99 90

70 60 x0 = 3 b = 10 m P (x) = 1 – e– [(x –3)/10]

50 40 30 20

0.5

70 50

0

30

–0.5

10

–1

10

–1.5 0

x0

5

10

15

20

25 Stress x

Fig. 11.6 The Weibull distribution function

(11.12)

where k is the order number of the sequenced data and n is the total sample size. At this stage it is possible to prepare a graph of probability of failure P(x) as a function of strength x to obtain a curve similar to that shown in Fig. 11.6. However, to determine the Weibull parameters x0 , b, and m requires additional conditioning of the data. From (11.11), it is evident that: e[(x−x0 )/b] = [1 − p(x)]−1 . m

(x − x0 ) b

m

= ln[1 − p(x)]−1 .

0

0.25

0.5

0.75

1 1.25 log (S – S0)

Taking log10 of both sides of (11.14) gives a relation for the slope parameter m. Thus: m=

(11.14)

log10 ln[1 − p(x)]−1 . log10 (x − x0 ) − log10 b

(11.15)

The numerator of (11.15) is the reduced variate y = log10 ln[1 − p(x)]−1 used for the ordinate in preparing a graph of the conditioned data as indicated in Fig. 11.7. Note that y is a function of p alone and for this reason both the p and y scales can be displayed on the ordinates (Fig. 11.7). The lead term in the denominator of (11.15) is the reduced variate x = log10 (x − x0 ) used for the abscissa in Fig. 11.7. In the Weibull example, the threshold strength x0 was adjusted to 0.46 g so that the data would fall on a straight line when plotted against the reduced x and y variates. The constant b is determined from the condition that: log10 b = log10 (x − x0 )

(11.13)

Taking the natural log of both sides of (11.13) yields: 

–0.25

Fig. 11.7 Fiber strength of Indian cotton shown in graphical format with Weibull’s reduced variate (from data by Weibull [11.10])

ple (n = 3000) was studied by measuring the load to fracture (in grams) for each fiber. The strength data obtained was placed in sequential order with the lowest value corresponding to k = 1 first and the largest value corresponding to k = 3000 last. The probability of failure p(x) at a load x is then determined from: k , p= n +1

2 – 0.5

when

y=0.

(11.16)

Note from Fig. 11.7 that y = 0 when log10 (x − x0 ) = 0.54, which gives b = 0.54. Finally m is given by the slope of the straight line when the data is plotted in terms of the reduced variates x and y. In this example problem m = 1.48.

11.3 Confidence Intervals for Predictions When experimental data are represented with a normal distribution by using estimates of the mean x¯ and

standard deviation Sx and predictions are made about the occurrence of certain measurements, questions arise

Part A 11.3

0

267

268

Part A

Solid Mechanics Topics

Part A 11.3

concerning the confidence that can be placed on either the estimates or the predictions. One cannot be totally confident in the predictions or estimates because of the effects of sampling error. Sampling error can be illustrated by drawing a series of samples (each containing n measurements) from the same population and determining several estimates of the mean x¯1 , x¯2 , x¯3 , . . .. A variation in x¯ will occur, but fortunately this variation can also be characterized by a normal distribution function, as shown in Fig. 11.8. The mean of the x and x¯ distributions is the same; however, the standard deviation of the x¯ distribution Sx¯ (sometimes referred to as the standard error) is less than Sx because Sx (11.17) Sx¯ = √ . n When the standard deviation of the population of the x¯ is known, it is possible to place confidence limits on the determination of the true population mean μ from a sample of size n, provided n is large (n > 25). The confidence interval within which the true population mean μ is located is given by the expression (x¯ − zSx¯ ) < μ < [x¯ + zSx¯ ] ,

(11.18)

where x¯ − zSx¯ is the lower confidence limit and x¯ + zSx¯ is the upper confidence limit. The width of the confidence interval depends upon the confidence level required. For instance, if z = 3 in (11.18), a relatively wide confidence interval exists; therefore, the probability that the population mean μ will be located within the confidence interval is high (99.7%). As the width of the confidence interval decreases, the probability that the population mean μ will _

Table 11.5 Confidence interval variation with confidence level interval = x¯ + zSx¯ Confidence level (%)

z

Confidence level (%)

z

99.9 99.7 99.0 95.0

3.30 3.00 2.57 1.96

90.0 80.0 68.3 60.0

1.65 1.28 1.00 0.84

fall within the interval decreases. Commonly used confidence levels and their associated intervals are shown in Table 11.5. When the sample size is very small (n < 20), the standard deviation Sx does not provide a reliable estimate of the standard deviation μ of the population and (11.18) should not be employed. The bias introduced by small sample size can be removed by modifying (11.18) to read (x¯ − t(α)Sx¯ ) < μ < [x¯ + t(α)Sx¯ ] ,

(11.19)

where t(α) is the statistic known as Student’s t. α is the level of significance (the probability of exceeding a given value of t). Student’s t distribution is defined by a relative frequency equation f (t), which can be expressed as   t 2 (ν+1)/2 , (11.20) f (t) = F0 1 + ν where F0 is the relative frequency at t = 0 required to make the total area under the f (t) curve equal to unity and ν is the number of degrees of freedom. f (t ) 0.5

f (x) or f (x )

υ = ∞ (normal distribution)

0.4

υ = 10 υ=5 υ=1

Distribution of x 0.3

_

Distribution of x 0.2 s sx = x √n _

sx_

0.1

sx

μ

_

x or x

Fig. 11.8 Normal distribution of individual measurements of the quantity x and of the mean x¯ from samples of size n

0 –5

–4

–3

–2

–1

0

1

2

3

4

5 t

Fig. 11.9 Student’s t distribution for several degrees of freedom ν

Statistical Analysis of Experimental Data

11.3 Confidence Intervals for Predictions

269

Table 11.6 Student’s t distribution for ν degrees of freedom showing t(α) as a function of area A (one side) 0.995

0.99

0.975

0.95

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

63.66 9.92 5.84 4.60 4.03 3.71 3.50 3.36 3.25 3.17 3.11 3.06 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.84 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.76 2.76 2.75 2.70 2.66 2.62 2.58

31.82 6.96 4.54 3.75 3.36 3.14 3.00 2.90 2.82 2.76 2.72 2.68 2.65 2.62 2.60 2.58 2.57 2.55 2.54 2.53 2.52 2.51 2.50 2.49 2.48 2.48 2.47 2.47 2.46 2.46 2.42 2.39 2.36 2.33

12.71 4.30 3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23 2.20 2.18 2.16 2.14 2.13 2.12 2.11 2.10 2.09 2.09 2.08 2.07 2.07 2.06 2.06 2.06 2.05 2.05 2.04 2.04 2.02 2.00 1.98 1.96

6.31 2.92 2.35 2.13 2.02 1.94 1.90 1.86 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.75 1.74 1.73 1.73 1.72 1.72 1.72 1.71 1.71 1.71 1.71 1.70 1.70 1.70 1.70 1.68 1.67 1.66 1.65

Confidence level α 0.90 0.80 3.08 1.89 1.64 1.53 1.48 1.44 1.42 1.40 1.38 1.37 1.36 1.36 1.35 1.34 1.34 1.34 1.33 1.33 1.33 1.32 1.32 1.32 1.32 1.32 1.32 1.32 1.31 1.31 1.31 1.31 1.30 1.30 1.29 1.28

The distribution function f (t) is shown in Fig. 11.9 for several different degrees of freedom ν. The degrees of freedom equal the number of independent measurements employed in the determination. It is evident that, as ν becomes large, Student’s t distribution approaches the normal distribution. One-side areas for the t distribution are listed in Table 11.6 and illustrated in Fig. 11.10.

1.376 1.061 0.978 0.941 0.920 0.906 0.896 0.889 0.883 0.879 0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860 0.859 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854 0.851 0.848 0.845 0.842

0.75

0.70

0.60

0.55

1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.681 0.679 0.677 0.674

0.727 0.617 0.584 0.569 0.559 0.553 0.549 0.546 0.543 0.542 0.540 0.539 0.538 0.537 0.536 0.535 0.534 0.534 0.533 0.533 0.532 0.532 0.532 0.531 0.531 0.531 0.531 0.530 0.530 0.530 0.529 0.527 0.526 0.524

0.325 0.289 0.277 0.271 0.267 0.265 0.263 0.262 0.261 0.260 0.260 0.259 0.259 0.258 0.258 0.258 0.257 0.257 0.257 0.257 0.257 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.255 0.254 0.254 0.253

0.158 0.142 0.137 0.134 0.132 0.131 0.130 0.130 0.129 0.129 0.129 0.128 0.128 0.128 0.128 0.128 0.128 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.126 0.126 0.126 0.126

The term t(α)Sx¯ in (11.19) represents the measure from the estimated mean x¯ to one or the other of the confidence limits. This term may be used to estimate the sample size required to produce an estimate of the mean x¯ with a specified reliability. Noting that one-half of the bandwidth of the confidence interval is δ = t(α)Sx¯ and using (11.17), it is apparent that the sample size is given

Part A 11.3

ν

270

Part A

Solid Mechanics Topics

Part A 11.4

Student's t 10 9 8 7 α = 0.001 6 5 0.01 4 0.02 3 0.05 2 0.1 1 0 2 4 6 8 10

Since t(α) depends on n, a trial-and-error solution is needed to establish the sample size n needed to satisfy the specifications. For the data of Table 11.1, n = 40: therefore ν = 39 and t(α) = t(0.995) = 2.71 from Table 11.6. The value t(α) = t(0.995) is used since 0.5% of the distribution must be excluded on each end of the curve to give a two-sided area corresponding to a reliability of 99%. Substituting into (11.21) yields   2.71(2.25) 2 = 12.00 . n= 1.76 12

14 16 18 20 Degrees of freedom ν

Fig. 11.10 Student’s t statistic as a function of the degrees of freedom ν with α as the probability of exceeding t as a parameter

by  n=

t(α)Sx¯ δ

2 .

(11.21)

The use of (11.21) can be illustrated by considering the data in Table 11.1, where Sx = 15.51 MPa and x¯ = 1214.2 MPa. If this estimate of μ is to be accurate to ±1% with a reliability of 99%, then δ = (0.01)(176.1) = 12.14 MPa .

This result indicates that a much smaller sample than 40 will be sufficient. Next try n = 12, ν = 11, and t(α) = 3.11; then   3.11(2.25) 2 = 15.80 . n= 1.76 Finally, with n = 15, ν = 14, and t(α) = 2.98; then   2.98(2.25) 2 n= = 14.50 . 1.76 Thus, a sample size of 15 would be sufficient to ensure an accuracy of ±1% with a confidence level of 99%. The sample size of 40 listed in Table 11.1 is too large for the degree of accuracy and confidence level specified. This simple example illustrates how sample size can be reduced and cost savings effected by using statistical methods.

11.4 Comparison of Means Because Student’s t distribution compensates for the effect of small sample bias and converges to the normal distribution for large samples, it is a very useful statistic in engineering applications. A second important application utilizes the t distribution as the basis for a test to determine if the difference between two means is significant or due to random variation. For example, consider the yield strength of a steel determined with a sample size of n 1 = 20 which gives x¯1 = 540.6 MPa, and Sx1 ¯ = 41.65 MPa. Suppose now that a second sample from another supplier is tested to determine the yield strength and the results are n 2 = 25, x¯1 = 562.6 MPa, and Sx2 ¯ = 38.34 MPa. Is the steel from the second supplier superior in terms of yield strength? The standard deviation of can be expressed the difference in means S(x2− ¯ x1) ¯

as

 2 S(2x2− ¯ x1) ¯ = Sp

1 1 + n1 n2

 = Sp2

n1 + n2 , n1n2

(11.22)

where Sp2 is the pooled variance, which can be expressed as Sp2 =

2 + (n − 1)S2 (n 1 − 1)Sx1 2 x2 . n1 + n2 − 2

(11.23)

The statistic t can be computed from the expression t=

|x¯2 − x¯1 | . S(x2− ¯ x1) ¯

(11.24)

A comparison of the value of t determined from (11.24) with a value of t(α) obtained from Table 11.6 provides

Statistical Analysis of Experimental Data

a statistical basis for deciding whether the difference in means is real or due to random variations. The value of t(α) to be used depends upon the degrees of freedom ν = n 1 + n 2 − 2 and the level of significance required. Levels of significance commonly employed are 5% and 1%. The 5% level of significance means that the probability of a random variation being taken for a real difference is only 5%. Comparisons at the 1% level of significance are 99% certain; however, in such a strong test, real differences can often be attributed to random error.

11.5 Statistical Safety Factor

In the example being considered, (11.23) yields = 20.68 MPa, Sp2 = 230.1 MPa, (11.22) yields S(2x2− ¯ x1) ¯ and (11.24) yields t = 1.848. For a 5% level of significance test with ν = 43 and α = 0.05 (the comparison is one-sided, since the t test is for superiority), Table 11.6 indicates that t(α) = 1.68. Because t > t(α), it can be concluded with a 95% level of confidence that the yield strength of the steel from the second supplier was higher than the yield strength of steel from the first supplier.

zR =

x¯S − x¯σ , SS−σ

where SS−σ =



SS2 − Sσ2

R (%)

zR

R (%)

zR

50 90 95 99

0 1.288 1.645 2.326

99.9 99.99 99.999 99.9999

3.091 3.719 4.265 4.753

Stress

Strength

(11.25)

Region of failure

(11.26)

and the subscripts ‘S’ and ‘σ’ refer to strength and stress, respectively. The reliability associated with the value of z R determined from (11.25) may be determined from a table showing the area A(z R ) under a standard normal distribution curve by using: R = 0.5 + A(z R ) .

Table 11.7 Reliability R as a function of the statistic z R

(11.27)

Typical values of R as a function of the statistic z R are given in Table 11.7. The reliability determined in this manner incorporates a safety factor of 1. If a safety factor of N is to be specified together with a reliability, then (11.25)

Fig. 11.11 Superimposed normal distribution curves for strength and stress showing the region of failure

and (11.26) are rewritten to give a modified relation for z R as x¯S − N x¯ σ zR = , SS2 + Sσ2

(11.28)

which can be rearranged to give the safety factor N as   1 x¯S − z R SS2 + Sσ2 . N= (11.29) x¯σ

Part A 11.5

11.5 Statistical Safety Factor In experimental mechanics, it is often necessary to determine the stresses acting on a component and its strength in order to predict whether failure will occur or if the component is safe. The prediction can be a difficult if both the stress σij , and the strength S y , are variables since failure will occur only in the region of overlap of the two distribution functions, as shown in Fig. 11.11. To determine the probability of failure, or conversely, the reliability, the statistic z R is computed by using the equation

271

272

Part A

Solid Mechanics Topics

11.6 Statistical Conditioning of Data

Part A 11.7

Previously it was indicated that measurement error can be characterized by a normal distribution function and that the standard deviation of the estimated mean Sx¯ can be reduced by increasing the number of measurements. In most situations, sampling cost places an upper limit on the number of measurements to be made. Also, it must be remembered that systematic error is not a random variable; therefore, statistical procedures cannot serve as a substitute for precise accurately calibrated and properly zeroed measuring instruments. One area where statistical procedures can be used very effectively to condition experimental data is with the erroneous data point resulting from a measuring or recording mistake. Often, this data point appears questionable when compared with the other data collected, and the experimentalist must decide whether the deviTable 11.8 Deviation ratio DR0 used for statistical conditioning of data Number of measurements n

Deviation ratio DR0

Number of measurements n

Deviation ratio DR0

4 5 7 10 15

1.54 1.65 1.80 1.96 2.13

25 50 100 300 500

2.33 2.57 2.81 3.14 3.29

ation of the data point is due to a mistake (and hence should be rejected) or due to some unusual but real condition (and hence should be retained). A statistical procedure known as Chauvenet’s criterion provides a consistent basis for making the decision to reject or retain such a point from a sample containing several readings. Application of Chauvenet’s criterion requires computation of a deviation ratio (DR) for each data point, followed by comparison with a standard deviation ratio DR0 . The standard deviation ratio DR0 is a statistic that depends on the number of measurements, while the deviation ratio DR for a point is defined as xi − x¯ . (11.30) DR = Sx The data point is rejected when DR > DR0 and retained when DR ≤ DR0 . Values for the standard deviation ratio DR0 are listed in Table 11.8. If the statistical test of (11.30) indicates that a single data point in a sequence of n data points should be rejected, then the data point should be removed from the sequence and the mean x¯ and the standard deviation Sx should be recalculated. Chauvenet’s method can be applied only once to reject a data point that is questionable from a sequence of points. If several data points indicate that DR > DR0 , then it is likely that the instrumentation system is inadequate or that the process being investigated is extremely variable.

11.7 Regression Analysis Many experiments involve the measurement of one dependent variable, say y, which may depend upon one or more independent variables, x1 , x2 , . . ., xk . Regression analysis provides a statistical approach for conditioning the data obtained from experiments where two or more related quantities are measured.

11.7.1 Linear Regression Analysis Suppose measurements are made of two quantities that describe the behavior of a process exhibiting variation. Let y be the dependent variable and x the independent variable. Because the process exhibits variation, there is not a unique relationship between x and y, and the data, when plotted, exhibit scatter, as illustrated

in Fig. 11.12. Frequently, the relation between x and y that most closely represents the data, even with the scatter, is a linear function. Thus Yi = mxi + b ,

(11.31)

where Yi is the predicted value of the dependent variable yi for a given value of the independent variable xi . A statistical procedure used to fit a straight line through scattered data points is called the least-squares method. With the least-squares method, the slope m and the intercept b in (11.31) are selected to minimize the sum of the squared deviations of the data points from the straight line shown in Fig. 11.12. In utilizing the leastsquares method, it is assumed that the independent

Statistical Analysis of Experimental Data

variable x is free of measurement error and the quantity  Δ2 = (yi − Yi )2 (11.32)

y = mx + b

(a)

0

0

x1

(b)

distribution of y is a measure of the correlation. When the dispersion is small, the correlation is good and the regression analysis is effective in describing the variation in y. If the dispersion is large, the correlation is poor and the regression analysis may not be adequate to describe the variation in y. The adequacy of regression analysis can be evaluated by determining a correlation coefficient R2 that is given by the expression   n − 1 {y2 } − m{xy} 2 , (11.34) R = 1− n −2 {y2 }



where {y2 } = y2 − ( y)2 /n and {xy} = xy −



( x) ( y)/n. When the value of the correlation coefficient R2 = 1, perfect correlation exists between y and x. If R2 equals zero, no correlation exists and the variations observed in y are due to random fluctuations and not Table 11.9 Probability of obtaining a correlation coefficient R2 due to random variations in y

y = mx + b

x

Fig. 11.12 Linear regression analysis is used to fit a leastsquares line through scattered data points

Probability 0.02

n

0.10

0.05

5 6 7 8 10 15 20 30 40 60 80 100

0.805 0.729 0.669 0.621 0.549 0.441 0.378 0.307 0.264 0.219 0.188 0.168

0.878 0.811 0.754 0.707 0.632 0.514 0.444 0.362 0.312 0.259 0.223 0.199

0.934 0.882 0.833 0.789 0.716 0.592 0.516 0.423 0.367 0.306 0.263 0.235

0.01 0.959 0.917 0.874 0.834 0.765 0.641 0.561 0.464 0.403 0.337 0.291 0.259

Part A 11.7

posed on the linear regression graph

y

0

x

Fig. 11.13 Distribution of y at a fixed value of x superim-

Solving (b) for m and b yields



x y − n xy and m =  2

x − n x2



y−m x b= (11.33) , n where n is the number of data points. The slope m and intercept b define a straight line through the scattered data points such that Δ2 is minimized. In any regression analysis it is important to establish the correlation between x and y. Equation (11.31) does not predict the exact values that were measured, because of the variation in the process. To illustrate, assume that the independent quantity x is fixed at a value x1 and that a sequence of measurements is made of the dependent quantity y. The data obtained would give a distribution of y, as illustrated in Fig. 11.13. The dispersion of the

0

273

y

is minimized at fixed values of x. After substituting (11.31) into (11.32), the minimization process of Δ2 implies that ∂Δ2 ∂  = (yi − mx − b)2 = 0 , ∂b ∂b ∂Δ2 ∂  = (yi − mx − b)2 = 0 . ∂m ∂m Differentiating yields  2 (yi − mx − b) (−x) = 0 ,  2 (yi − mx − b) (−1) = 0 .

11.7 Regression Analysis

274

Part A

Solid Mechanics Topics

changes in x. Because random variations in y exist, a value of R2 = 1 is not obtained even if y(x) is linear. To interpret correlation coefficients 0 < R2 < 1, the data in Table 11.9 is used to establish the probability of obtaining a given R2 due to random variations in y. As an example, consider a regression analysis with n = 15, which gives R2 = 0.65 as determined by (11.34). Reference to Table 11.9 indicates that the probability of obtaining R2 = 0.65 due to random variations is slightly less than 1%. Thus one can be 99% certain that the regression analysis represents a true correlation between y and x.

Part A 11.7

11.7.2 Multivariate Regression Many experiments involve measurements of a dependent variable y that depends upon several independent variables x1 , x2 , x3 , . . . , etc. It is possible to represent y as a function of x1 , x2 , x3 , . . . , by employing the multivariate regression equation Yi = a + b1 x1 + b2 x2 + . . . + bk xk ,

(11.35)

where a, b1 , b2 , . . . , bk are regression coefficients. The regression coefficients a, b1 , b2 , . . . , bk are determined by using the method of least squares in a manner similar to that employed for

linear regression analysis where the quantity Δ2 = (yi − Yi )2 is minimized. Substituting (11.32) into (11.35) yields Δ2 =



(yi − a − b1 x1 − b2 x2 − . . . − bk xk )2 . (11.36)

 ∂Δ2 =2 (yi − a − b1 x1 − b2 x2 ∂bk − . . . − bk xk )(−xk ) = 0 .

(11.37)

Equation (11.37) leads to the following set of k + 1 equations, which can be solved for the unknown regression coefficients a, b1 , b2 , . . . , bk ,    an + b1 x 1 + b2 x 2 + . . . bk xk  = yi ,     a x 1 + b1 x12 + b2 x 1 x 2 + . . . bk x1 xk  = yi x1 ,     x 1 x 2 + b2 x22 + . . . bk x2 xk a x 2 + b1  = yi x2 , .. .



x k + b1  = yi xk .

a



x 1 x k + b2



x 2 x k + . . . bk



xk2

(11.38)

The correlation coefficient R2 is again used to determine the degree of association between the dependent and independent variables. For multiple regression equations, the correlation coefficient R2 is given as n −1 R2 = 1 − n −k   2 {y } − b1 {yx 1 } − {yx 2 } − . . . − {yx k } , × {y2 } (11.39)

Differentiating yields  ∂Δ2 =2 (yi − a − b1 x1 − b2 x2 ∂a − . . . − bk xk )(−1) = 0 ,  ∂Δ2 =2 (yi − a − b1 x1 − b2 x2 ∂b1 − . . . − bk xk )(−x1 ) = 0 ,  ∂Δ2 (yi − a − b1 x1 − b2 x2 =2 ∂b2 − . . . − bk xk )(−x2 ) = 0 , .. .

where

   y xk yx k − {yx k } = and n  2  y y2 − . {y2 } = n This analysis is for linear, noninteracting, independent variables; however, the analysis can be extended to include cases where the regression equations would have higher-order and cross-product terms. The nonlinear terms can enter the regression equation in an additive manner and are treated as extra variables. With well-established computer routines for regression analysis, the set of (k + 1) simultaneous equations given by (11.38) can be solved quickly and inexpensively. No significant difficulties are encountered in adding extra terms to account for nonlinearities and interactions. 

Statistical Analysis of Experimental Data

where

11.7.3 Field Applications of Least-Square Methods

Linear Least-Squares Method Consider a calibration model in photoelasticity and write the equation for the fringe order N(x, y) as

N(x, y) =

h G(x, y) + E(x, y) , fσ

E(x, y) = Ax + By + C .

(b)

h G(x, y) + Axk + Byk + C . fσ

(11.40)

Note that (11.40) is linear in terms of the unknowns (h/ f σ ), A, B, and C. For m selected data points, with m > 4, an overdeterministic system of linear equations results from (11.40). This system of equations can be expressed in matrix form as N = aw ,

⎛

The solution of the set of m equations for the unknowns h/ f σ , A, B, and C can be achieved in a least-squares sense through the use of matrix methods. Note that a N = cw , where c = a a , and that w = c−1 a N , where c−1 is the inverse of c. Solution of the matrix w gives the column elements, which are the unknowns. This form of solution is easy to accomplish on a small computer which can be programmed to perform the matrix manipulations. The matrix algebra outlined above is equivalent to minimizing the cumulative error E squared which is E=

For any selected point (xk , yk ) in the field, Nk is determined by N(x, y) =

⎛ ⎞ ⎞ ⎛ ⎞ G 1 x1 y1 1 N1 h/ f σ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ G 2 x2 y2 1⎟ ⎜ N2 ⎟ A ⎟ ⎜ ⎟ ⎟w = ⎜ a = N=⎜ ⎜ ⎟. ⎜ .. ⎟ ⎜ .. ⎟ ⎝ B ⎠ ⎝ . ⎠ ⎝ . ⎠ C G m xm ym 1 Nm

(a)

where G(x, y) is the analytical representation of the difference of the principal stresses (σ1 − σ2 ) in the calibration model, h is the model thickness, f σ is the material fringe value, and E(x, y) is the residual birefringence. Assume a linear distribution for E(x, y) which can be expressed as

275

Part A 11.7

The least-squares method is an important mathematical process used in regression analysis to obtain regression coefficients. Sanford [11.12] showed that the leastsquares method could be extended to field analysis of data obtained with optical techniques (photoelasticity, moiré, holography, etc.). With these optical methods, a fringe order N, related to a field quantity such as stress, strain, or displacement, can be measured at a large number of points over a field (x, y). The applications require an analytical representation of the field quantities as a function of position (x, y) over the field. Several important problems including calibration, fracture mechanics, and contact stresses have analytical solutions where coefficients in the governing equations require experimental data for complete evaluation. Two examples will be described which introduce both the linear and nonlinear least-squares method applied over a field (x, y).

11.7 Regression Analysis

2 m   h G(xk , yk ) + Ax k + Byk + C − Nk . fσ k=1

(11.41)

The matrix operations apply the least-squares criteria, which require that ∂E ∂E ∂E ∂E = = = =0. ∂(h/ f σ ) ∂A ∂B ∂C

(11.42)

The advantage of this statistical approach to calibration of model materials in optical arrangements is the use of full field data to reduce errors due to discrepancies in either the model materials or the optical systems. Nonlinear Least-Squares Method In the preceding section a linear least-squares method provided a direct approach to improving the accuracy of calibration with a single-step computation of an overdeterministic set of linear equations. In other experiments involving either the determination of unknowns arising in stresses near a crack tip or contact stresses near a concentrated load, the governing equations are nonlinear in terms of the unknown quantities. In these cases, the procedure to be followed involves linearizing the governing

276

Part A

Solid Mechanics Topics

Part A 11.7

equations, applying the least-squares criteria to the linearized equations, and finally iterating to converge to an accurate solution for the unknowns. To illustrate this statistical approach, consider a photoelastic experiment that yields an isochromatic fringe pattern near the tip of a crack in a specimen subjected to mixed-mode loading. In this example, there are three unknowns K I , K II , and σ0x , which are related to the experimentally determined fringe orders Nk at positions (rk , θk ). The governing equation for this mixed-mode fracture problem is   1  N fσ 2 = (K I sin θ + 2K II cos θ)2 h 2πr  + (K II sin θ)2 θ 2σ0x sin K I sin θ(1 + 2 cos θ) +√ 2 2πr  2 . + K II (1 + 2 cos2 θ + cos θ) + σ0x

In matrix form the set of m equations represented by (11.46) can be written as f = aΔK , where the matrices are defined as ⎛ ⎞ f1 ⎜ ⎟ ⎜ f2 ⎟ ⎟ f =⎜ ⎜ .. ⎟ , ⎝ . ⎠ fm ⎛ ⎞ ∂ f 1 /∂K I ∂ f 1 /∂K II ∂ f 1 /∂σ0x ⎜ ⎟ ⎜ ∂ f 2 /∂K I ∂ f 2 /∂K II ∂ f 2 /∂σ0x ⎟ ⎟, a=⎜ .. ⎜ ⎟ ⎝ ⎠ . ∂ f m /∂K I ∂ f m /∂K II ∂ f m /∂σ0x ⎛ ⎞ ΔK I ⎜ ⎟ ΔK = ⎝ΔK II ⎠ . Δσ0x

(11.43)

Equation (11.43) can be solved in an overdeterministic sense, by forming the function f (K I , K II , σ0x ) as 1  (K I sin θk + 2K II cos θk )2 f k (K I , K II , σ0x ) = 2πrk  2σ0x θk + (K II sin θk )2 + √ sin 2 2πrk  · K I sin θk (1 + 2 cos θk )  + K II (1 + 2 cos2 θk + cos θk )   Nk f σ 2 2 − =0, + σ0x h (11.44)

where k = 1, 2, 3, . . . m and (rk , θk ) are coordinates defining a point on an isochromatic fringe of order Nk . A Taylor series expansion of (11.44) yields:     ∂ fk ∂ fk ΔK I + ΔK II ( f k )i+1 = ( f k )i + ∂K I i ∂K II i   ∂ fk + Δσ0x , (11.45) ∂σ0x i where i refers to the i-th iteration step and ΔK I , ΔK II , and Δσ0x are corrections to the previous estimate of ΔK I , ΔK II , and Δσ0x . It is evident from (11.45) that corrections should be made to drive f (K I , K II , σ0x ) toward zero. This fact leads to the iterative equation       ∂ fk ∂ fk ∂ fk ΔK I + ΔK II + Δσ0x ∂K I i ∂K II i ∂σ0x i = −( f k )i . (11.46)

(11.47)

The least-squares minimization process is accomplished by multiplying, from the left, both sides of (11.47) by the transpose of matrix a, to give a f = a aΔK or d = cΔK , where d = a f c = a a . Finally, the correction terms are given by: ΔK = c−1 d .

(11.48)

The solution of (11.48) gives ΔK I , ΔK II , and Δσ0x , which are used to correct initial estimates of K I , K II , and σ0x and obtain a better fit of the function f k (K I , K II , σ0x ) to m data points. The procedure is executed on a small computer programmed using MATLAB. One begins by assuming initial values for K I , K II , and σ0x . Then, the elements of the matrices f and a are computed for each of the m data points. The correction matrix ΔK is then computed from (11.48) and finally the estimates of the unknowns are corrected by noting that (K I )i+1 = (K I )i + ΔK I , (K II )i+1 = (K II )i + ΔK II , (σ0x )i+1 = (σ0x )i + Δσ0x .

(11.49)

Statistical Analysis of Experimental Data

The procedure is repeated until each element in the correction matrix ΔK becomes acceptably small. As convergence is quite rapid, the number of iterations re-

11.8 Chi-Square Testing

277

quired for accurate estimates of the unknowns is usually small.

11.8 Chi-Square Testing The chi-square (χ 2 ) test is used in statistics to verify the use of a specific distribution function to represent the population from which a set of data has been obtained. The chi-square statistic χ 2 is defined as  k   (n o − n e )2 ne

i=1

,

(11.50)

where n o is the actual number of observations in the i-th group interval, n e is the expected number of observations in the i-th group interval based on the specified distribution, and k is the total number of group intervals. The value of χ 2 is computed to determine how closely the data fits the assumed statistical distribution. If χ 2 = 0, the match is perfect. Values of χ 2 > 0 indicate the possibility that the data are not represented by the specified distribution. The probability p that the value of χ 2 is due to random variation is illustrated in Fig. 11.14. The degree of freedom is defined as ν = n −k ,

(11.51)

κ2 30 or

re c

t

Hypothesis not correct

tc

25

no

P = 0.1

pr ob

ab ly

P = 0.01

No reason to reject hypothesis

Table 11.10 Chi-squared χ 2 computation for grouped ulti-

yp

ot

he

sis

is

20

mate tensile strength data

H

15

P = 0.9

10 ta Da

5

sus

pec

t

to as

og

o

od

P = 0.99

Data may be falsified

0

0

5

10 15 20 25 Number of degrees of freedom ν

Fig. 11.14 Probability of χ 2 values exceeding those shown,

as a function of the number of degrees of freedom

Group interval 0 − 170.9 171 − 172.9 173 − 174.9 175 − 176.9 177 − 178.9 179 − 180.9 180 − ∞

Number observed 1 2 8 16 9 3 1

Number expected 0.468 2.904 9.096 13.748 9.836 3.360 0.588

(no − ne )2 /ne 0.604 0.281 0.132 0.369 0.071 0.039 0.289 1.785 = χ 2

Part A 11.8

χ2 =

where n is the number of observations and k is the number of conditions imposed on the distribution. As an example of the χ 2 test, consider the ultimate tensile strength data presented in Table 11.1 and judge the adequacy of representing the ultimate tensile strength with a normal probability distribution described with x¯ = 1214.2 MPa and Sx = 15.51 MPa. By using the properties of a normal distribution function, the number of specimens expected to fall in any strength group can be computed. The observed number of specimens in Table 11.1 exhibiting ultimate tensile strengths within each of seven group intervals, together with the computed number of specimens in a normal distribution in the same group intervals, are listed in Table 11.10. The computation of the χ 2 value (χ 2 = 1.785) is also illustrated in the table. The number of groups is n = 7. Because the two distribution parameters x¯ and Sx were determined by using these data, k = 2; therefore, the number of degrees of freedom is ν = n − k = 7 − 2 = 5. Plotting these results (ν = 5 and χ 2 = 1.785) in Fig. 11.14 shows that the point falls in the region where there is no reason to expect that the hypothesis in not correct. The hypothesis is to represent the tensile strength with a normal probability function. The χ 2 test does not prove the validity of this hypothesis, but instead fails to disprove it. The lines dividing the χ 2 − ν graph of Fig. 11.14 into five different regions are based on probabilities

278

Part A

Solid Mechanics Topics

Part A 11.9

of obtaining χ 2 values greater than the values shown by the curves. For example, the line dividing the regions marked “no reason to reject hypothesis” and “hypothesis probably is not correct” has been selected at a probability level of 10%. Thus, there is only one chance in 10 that data drawn from a population correctly represented by the hypothesis would give a χ 2 value exceeding that specified by the p > 0.10 curve. The “hypothesis rejected region” is defined with the p > 0.01 curve indicating only one chance in 100 of obtaining a χ 2 value exceeding those shown by this curve. The χ 2 function can also be used to question if the data have been adjusted. Probability levels of 0.90 and 0.99 have been used to define regions where data is suspected as too good and data may be falsified. For the latter classification there are 99 chances out of 100 that the χ 2 value will exceed that determined by a χ 2 analysis of the data. The χ 2 statistic can also be used in contingency testing where the sample is classified under one of two categories – pass or fail. Consider, for example, an inspection procedure with a particular type of strain gage where 10% of the gages are rejected due to etching imperfections in the grid. In an effort to reduce this rejection rate, the manufacturer has introduced new

Table 11.11 Observed and expected inspection results Group interval

Number observed

Number expected

(no − ne )2 /ne

Passed Failed

1840 160

1800 200

0.89 8.00 8.89 = χ 2

clean-room techniques that are expected to improve the quality of the grids. On the first lot of 2000 gages, the failure rate was reduced to 8%. Is this reduced failure rate due to chance variation, or have the new clean-room techniques improve the manufacturing process? A χ 2 test can establish the probability of the improvement being the result of random variation. The computation of χ 2 for this example is illustrated in Table 11.11. Plotting the results from Table 11.11 on Fig. 11.14 after noting that ν = 1 shows that χ 2 = 8.89 falls into the region “hypothesis is not correct”. In this case the hypothesis was that there has been no improvement. The χ 2 test has shown that there is less than one chance in a 100 of the improvement in rejection rate (8% instead of 10%) being due to random variables. Thus, one can conclude with confidence that the new clean-room techniques were effective in improving yield.

11.9 Error Propagation Previous discussions of error have been limited to error arising in the measurement of a single quantity: however, in many engineering applications, several quantities are measured (each with its associated error) and another quantity is predicted on the basis of these measurements. For example, the volume V of a cylinder could be predicted on the basis of measurements of two quantities (diameter D and length L). Thus, errors in the measurements of diameter and length will propagate through the governing mathematical formula V = π DL/4 to the quantity (volume, in this case) being predicted. Because the propagation of error depends upon the form of the mathematical expression being used to predict the reported quantity, standard deviations for several different mathematical operations are listed below. For addition and/or subtraction of quantities (y = x1 ± x2 ± . . . ± xn ), the standard deviation S y¯ of the mean y¯ of the projected quantity y is given by 2 + S2 + . . . + S2 . (11.52) S y¯ = Sx1 x2 xn ¯ ¯ ¯

For multiplication of quantities (y = x1 x2 . . . xn ), the standard deviation S y¯ is given by   2 S S2¯ S2¯ ¯ + x2 + . . . + xn . S y¯ = (x¯1 x¯2 . . . x¯n ) x1 2 2 x¯n2 x¯1 x¯2 (11.53)

For division of quantities (y = x1 /x2 ), the standard deviation S y¯ is given by   2 2 Sx2 S¯ x¯1  ¯ S y¯ =  x1 + . x¯2 x¯12 x¯22

(11.54)

For calculations of the form (y = x1k ), the standard deviation S y¯ is given by S y¯ = k x¯1k−1 Sx1 ¯ .

(11.55)

Statistical Analysis of Experimental Data

For calculations of the form (y = x 1/k ), the standard deviation S y¯ is given by 1/k

x¯ S y¯ = 1 Sx1 . k x¯ 1 ¯

(11.56)

Consider for example a rectangular rod where independent measurements of its width, thickness, and length have yielded x¯1 = 40 with Sx1 ¯ = 0.10, x¯2 = 10 with Sx2 ¯ = 0.04, and x¯3 = 330 with Sx3 ¯ = 0.8, where all dimensions are in millimeters. The volume of the bar is V = x¯1 x¯2 x¯3 .

= (40)(10)(330)  (0.10)2 (0.04)2 (0.8)2 × + + (40)2 (10)2 (330)2 = (132)(5.303) = 700 mm3 This determination of S y¯ for the volume of the bar can be used together with the properties of a normal probability distribution to predict the number of bars with volumes within specific limits. The method of computing the standard error of a quantity S y¯ as given by (11.52)–(11.56), which are based on the properties of the normal probability distribution function, should be used where possible. However, in many engineering applications, the number of measurements that can be made is small; therefore, the data x¯1 , x¯2 , . . . , x¯n and Sx1 ¯ , Sx2 ¯ , . . . , Sxn ¯ needed for statistical based estimates of the error are not available. In these instances, error estimates can still be made but the results are less reliable. A second method of estimating error when data are limited is based on the chain rule of differential calculus. For example, consider a quantity y that is a function

of several variables y = f (x1 , x2 , . . ., xn ) .

(11.57)

Differentiating yields ∂y ∂y ∂y dx 1 + dx 2 + . . . + dx n . (11.58) dy = ∂x1 ∂x2 ∂xn In (11.58), dy is the error in y and dx1 , dx2 , . . ., dxn are errors involved in the measurements of x1 , x2 , . . ., xn . The partial derivatives ∂y/∂x1 , ∂y/∂x2 , . . ., ∂y/∂xn can be determined exactly from (11.57). Frequently, the errors dx1 , dx2 , . . ., dxn are estimates based on the experience and judgment of the test engineer. An estimate of the maximum possible error can be obtained by summing the individual error terms in (11.58). Thus        ∂y   ∂y   ∂y         dx 1  +  dx 2  + . . . +  dx n  . dy max =  ∂x1 ∂x2 ∂xn (11.59)

Equation (11.59) gives a worst-case estimate of error, because the maximum errors dx1 , dx2 , . . . , dxn are assumed to occur simultaneously and with the same sign. A more realistic equation for estimating error is obtained by squaring both sides of (11.58) to give  n   ∂y 2 2 (dxi )2 (dy) = ∂xi i=1    n  ∂y ∂y + dx i dx j , (11.60) ∂xi ∂x j i=1, j=1

where i = j. If the errors dxi are independent and symmetrical with regard to positive and negative values then the cross-product terms will tend to cancel and (11.60) reduces to dy =    2  2 2 ∂y ∂y ∂y dx 1 + dx 2 + . . . + dx n . ∂x1 ∂x2 ∂xn (11.61)

References 11.1 11.2 11.3

R.M. Bethea, R.R. Rhinehart: Applied Engineering Statistics (Dekker, New York 1991) J.T. McClave, T. Sincich: Statistics, 10th edn. (Prentice Hall, Upper Saddle River 2006) A. Agresti, C. Franklin: Statistics: The Art and Science of Learning from Data (Prentice Hall, Upper Saddle River 2006)

279

11.4

11.5 11.6

R. Hogg, A. Craig, J. McKean: Introduction to Mathematical Statistics, 6th edn. (Prentice Hall, Upper Saddle River 2005) P.S. Mann: Introductory Statistics, 5th edn. (Wiley, New York 2005) D.L. Harnett, J.F. Horrell: Data, Statistics and Decision Models with EXCEL (Wiley, New York 1998)

Part A 11

The standard error of the volume can be determined by using (11.53). Thus,   2 2 2 S Sx2 Sxn ¯ ¯ ¯ S y¯ = (x¯1 x¯2 . . . x¯ n ) x1 + + . . . + x¯n2 x¯12 x¯22

References

280

Part A

Solid Mechanics Topics

11.7

11.8 11.9 11.10

G.W. Snedecor, W.G. Cochran: Statistical Methods, 8th edn. (Iowa State Univ. Press, Ames 1989) W.C. Navidi: Statistics for Engineers and Scientists (McGraw-Hill, New York 2006) H. Pham (Ed.): Springer Handbook of Engineering Statistics (Springer, Berlin, Heidelberg 2006) W. Weibull: Fatigue Testing and Analysis of Results (Pergamon, New York 1961)

11.11 11.12

11.13

J.S. Milton, J.C. Arnold: Introduction to Probability and Statistics (McGraw-Hill, New York 2006) R.J. Sanford: Application of the least squares methods to photoelastic analysis, Exp. Mech. 20, 192–197 (1980) J.R. Sanford, J.W. Dally: A general methods for determining mixed-mode stress intensity factors from isochromatic fringe patterns, Eng. Fract. Mech. 11, 621–633 (1979 )

Part A 11

281

Part B

Contact M Part B Contact Methods

12 Bonded Electrical Resistance Strain Gages Robert B. Watson, Raleigh, USA 13 Extensometers Ian McEnteggart, Buckinghamshire, UK 14 Optical Fiber Strain Gages Chris S. Baldwin, Lanham, USA 15 Residual Stress Yuri F. Kudryavtsev, Markham, Canada 16 Nanoindentation: Localized Probes of Mechanical Behavior of Materials David F. Bahr, Pullman, USA Dylan J. Morris, Gaithersburg, USA 17 Atomic Force Microscopy in Solid Mechanics Ioannis Chasiotis, Urbana, USA

283

Robert B. Watson

The bonded resistance strain gage is an analog electrical sensor ideally suited to the task of measuring surface stains on solid materials. The two most common devices are the popular etched-foil sensor, and the somewhat more exotic semiconductor gage. Etched foil strain gages are produced with thin foil (typically 5 µm) bonded to a thin insulating carrier (typically 25 µm). Semiconductor gages are produced from a semiconductor material (typically 0.1 mm-thick silicon), commonly with an insulating carrier, but often bonded directly to a specimen. Strain gages work on the same principle as engineering strain; i. e., a starting gage length is deformed to a final gage length with a corresponding change in electrical resistance proportional to the deformation. Foundational characteristics, including sensitivity, thermal response, and limiting properties, and use of these sensors in experimental stress analysis applications, including shear measurement, thermal expansion measurement, and principal strain measurement, are reviewed. Electrical circuits used in the interrogation of strain gages are discussed, along with special considerations and exploitation of specific circuit arrangements for solving unique problems. Potential error sources and correction techniques for both the sensors and circuits are covered.

The present-day incarnation of the ubiquitous electrical resistance strain gage, the ancestral wire version of which was invented independently in the late 1930s by Ed Simmons at the California Institute of Technology and Arthur Ruge at the Massachusetts Institute of Technology, from principles first discovered by Lord Kelvin in 1856, has become the de facto standard for the study of material deformations, be it a quick check, or an ex-

12.1 Standardized Strain-Gage Test Methods . 284 12.2 Strain and Its Measurement .................. 284 12.3 Strain-Gage Circuits .............................. 12.3.1 Elementary Circuits ..................... 12.3.2 The Potentiometer Circuit ............ 12.3.3 The Wheatstone Bridge Circuit .....

285 285 286 287

12.4 The Bonded Foil Strain Gage.................. 12.4.1 Manufacturing ........................... 12.4.2 Gage Factor – A Practical Theory... 12.4.3 Strains from Every Direction......... 12.4.4 Gage Factor – The Manufacturer’s Value ............ 12.4.5 Transverse Sensitivity Error – Numerical Examples ................... 12.4.6 The Influence of Temperature Changes.............. 12.4.7 Control of Foil Strain Gage Thermal Output 12.4.8 Foil Strain Gages and the Wheatstone Bridge ......... 12.4.9 Performance Characteristics ......... 12.4.10 Gage Selection Procedure ............ 12.4.11 Specific Applications ...................

291 291 292 294

303 306 310 312

12.5 Semiconductor Strain Gages .................. 12.5.1 Manufacturing ........................... 12.5.2 Strain Sensitivity ........................ 12.5.3 Temperature Sensitivity .............. 12.5.4 Special Circuit Considerations ...... 12.5.5 Installation Techniques...............

325 325 327 328 331 332

295 296 297 302

References .................................................. 332

haustive research. Prior to the invention of the bonded wire strain gage, Charles Kearns of Hamilton Standard produced a crude version of the modern-day strain gage in 1936 by cementing flattened carbon composition resistor elements to a propeller blade [12.1]. These devices, though suited only for dynamic measurements due to their inherent lack of electrical resistance stability, were instrumental in the solving of aircraft propeller

Part B 12

Bonded Elect

12. Bonded Electrical Resistance Strain Gages

284

Part B

Contact Methods

Part B 12.2

blade failures. Kearns notes that in-flight propeller blade failures went from as many as 41 in the early 1930s, to zero in 1940. Development in 1952 of today’s familiar etched-foil strain gage is attributed to the Saunders-Roe Company in the United Kingdom [12.2]. During its 70-plus year history, many technical articles, textbooks, and manufacturer’s white papers have been dedicated to the examination and use of these simple looking sensors. Even a cursory review of, say, the Society for Experimental Mechanics (SEM) archives will provide clear conformation of the intense scrutiny

and study to which these little devices have been subjected, particularly during the formative years of both the society and the sensor. This chapter attempts to briefly convey information on foundational characteristics and use, in specific experimental stress analysis applications, of electrical resistance strain gages, specifically the foil type and its cousin, the semiconductor strain gage, which is discussed last. An idiomatic distinction between these two sensor types could be that the former is electric in nature, and the latter electronic.

12.1 Standardized Strain-Gage Test Methods Fig. 12.1 Strain-gage layout and nomenclature

Bars: 45° axes alignment marks

End loops Gage length

Triangles: Grid center alignment marks

End loops Transition

Outer grid lines Inner grid lines

Overall pattern length

Solder tab length

Solder tab width Tab spacing Grid width

Outer line width Inner line width Space width

Engineering data supplied with each package of strain gages is critical to the successful use of the gages in accurate determination of specimen strains. A few standards organizations around the globe have produced documents detailing a format for uniform data collection and/or presentation. Among others, these include the American Society for Testing and Materials (ASTM E 251) [12.3], the Organisation Internationale de Metrologie Legale (OIML International Recommendation no. 62) [12.4], and the German Standards (VDI/VDE 2635) [12.5]. The following sections will draw heavily on the data supplied by manufacturers. To provide for a common terminology within this chapter, typical strain-gage structure and nomenclature are given in Fig. 12.1.

12.2 Strain and Its Measurement Strain gages provide for the determination of surface strains beneath the active measuring area of the gage (gage length × grid width). Thus, it is best to begin with a definition of exactly what is being measured. There are several accepted variations on the definition of strain, each having a particular applicability to a given study. The most commonly accepted definition used in the practice of experimental stress analysis is called engineering strain and is defined as the change in length of a prescribed initial length ε=

(L 1 − L 0 ) ΔL = , L0 L

(12.1)

where ε is the engineering strain (m/m, or some other dimensionless length unit), L 1 the final length (m) and L 0 is the initial length = L(m). Strain gages work in precisely the manner defined by engineering strain. The gage has an initial length (L 0 ) and when stretched (e.g., with the specimen to which it is attached), a final length (L 1 ). Fortuitously, as the conductor of the gage changes length, there is a corresponding change in resistance. It is this resistance change that is measurable and relatable to engineering strain.

Bonded Electrical Resistance Strain Gages

12.3 Strain-Gage Circuits

Common to both types of strain gages that will be discussed (etched-foil and semiconductor) are the electrical circuits used to interrogate them. These circuits become a significant part of the measurement system and will be discussed before gage characteristics proper. Stretching an electrical conductor causes its resistance to change. A full explanation and derivation of this behavior will be presented in later sections specific to the foil strain gage and the semiconductor gage, but to study circuits used with strain gages it is helpful to keep in mind that applied strain causes the gage to change resistance. It is the function of the circuit to ascertain these resistance changes, which provides for the measurement of strain. Any electrical circuit which can determine resistance of a conductor is a candidate for use with strain gages, but the minute resistance changes involved require sensitive and stable systems. For typical bonded foil gages used in stress analysis applications, there is approximately a 2 μΩ/Ω change in resistance per 1 μm/m (με) of applied strain (note that μm/m and με are used interchangeably). For typical semiconductor gages, this sensitivity can increase to over 100 μΩ/Ω /με, which somewhat simplifies the resistance measurement task. Most commonly, either the simple potentiometer circuit, or the Wheatstone bridge circuit are chosen to interrogate strain gages, but even more elementary circuits will suffice.

12.3.1 Elementary Circuits Electrical resistance is related to voltage (V ) and current (I ) by Ohm’s law, V = IR, indicating that resistance measurement is accomplished using either a reference voltage or reference current. Each method is illustrated in Fig. 12.2. As suggested by the figures, applying either a constant voltage or constant current across a strain gage a)

E =Vc

b)

Rg+ΔRg I0 -ΔI

E 0 +ΔE Rg+ΔRg

Ic Vc

Ic

Fig. 12.2a,b Elementary circuit arrangements: (a) constant voltage, (b) constant current

and ascertaining changes to these quantities as the gage resistance changes with applied strained, the resistance change can be related to the strain which causes it. The simple ohm-meter is the most basic form of this circuit; and, an ohm-meter can serve as a strain-gage instrument, but the resolution and accuracy of strains calculated from resistance data aquired this way are questionable. However, an inexpensive strain gage instrument can be built using a simple personal-computer-based ohm-meter board, whereby the computer can serve as both the data acquisition and data analysis system. Constant-Voltage Excitation Referring to Fig. 12.2a, if a constant-voltage power supply Vc is connected across a single strain gage Rg , according to Ohm’s law there will be an initial current flow I0 through the unstrained gage prescribed by

I0 =

E0 . Rg

(12.2)

For an increase in gage resistance, the current through the gage will drop from I0 to I0 − ΔI , given by I0 − ΔI =

E0 . Rg + ΔRg

(12.3)

Substituting E 0 = I0 Rg and solving for the relative change in current yields   ΔRg Rg 1 ΔI . = R (12.4) = ΔR g I0 +1 1+ g ΔRg

Rg

Equation (12.4) shows that the relative change in current for the constant-voltage circuit is a nonlinear function of the relative change in gage resistance. Later it will be shown that, in the elastic-stress region of the gage, resistance change is a linear function of strain. Therefore, in the constant-voltage circuit, the relative current change is a nonlinear function of strain, which is an obvious disadvantage to using this circuit. Use of constant voltage across the strain gage produces a current-sensitive circuit. As such, there are limitations imposed on resolution by the self-heating effect of electrical current passing through the gage. While using a high voltage is beneficial to extracting the best resolution, the allowable safe power density in the gage restricts this available benefit. Thus, the circuit is most useful with gages of lower resistance and higher

Part B 12.3

12.3 Strain-Gage Circuits

285

Contact Methods

Part B 12.3

strain sensitivity. As is discussed in Sect. 12.5.4, even though semiconductor gages satisfy the sensitivity requirement, use of constant-voltage excitation with these sensors causes other problems. Constant-Current Excitation In like manner to the analysis of the constant-voltage circuit, the same procedure is applied to constantcurrent excitation (Fig. 12.2b). In this circuit ΔI = 0 at all times, and there is a change in voltage drop across the gage as its resistance changes with applied strain. By measuring this voltage drop, the change in resistance, and hence the applied strain, can be determined. From Ohm’s law

E 0 + ΔE = Ic (Rg + ΔRg ) .

(12.5)

Noting that Ic = I0 , and substituting (12.2) into (12.5) and simplifying yields ΔRg ΔE = . E0 Rg

(12.6)

From this expression it should be clear that the change in potential across a gage is linear with the unit resistance change of the gage, and hence with applied strain. Equation (12.6) also demonstrates that the constantcurrent circuit maximizes efficiency, since the unit voltage drop is precisely equivalent to the unit change in gage resistance, which suggests, with everything else equal, that gages of highest resistance and highest strain sensitivity will produce the highest measurement resolution. However, since the applied current must remain small ( 0.1 A) to avoid destructive self-heating in the gage, the output signal of the constant-current circuit is low, typically, on the order of μV, and this is on top of the source voltage, which is normally several volts.

12.3.2 The Potentiometer Circuit The potentiometer circuit shown schematically in Fig. 12.3 essentially combines both of the elementary circuits described above, or at least provides for similar operation to both. The primary benefits from studying this circuit come from the fact that it is simple to construct, versatile in use, and a second strain gage can be used in place of the ballast resistor Rb , which under certain measurement conditions allows for improved signal level and/or temperature compensation. In general, however, the potentiometer circuit suffers from the same low-level signal which plagues the elementary circuits. Since semiconductor gages can have over 50 times the strain sensitivity of foil gages,

I Rb C Vc or Ic E 0 +ΔE

Part B

Rg +ΔRg

286

Rm

E

Fig. 12.3 Potentiometer circuit arrangement showing capacitor-coupled output and display meter resistance

and since they are also prone to very high temperature sensitivity, this circuit is an attractive choice. Like the elementary circuits, perhaps the greatest limitation of the potentiometer circuit comes from the large common-mode voltage inherent in its output, consigning its use to dynamic applications. As depicted in Fig. 12.3, the potentiometer circuit has only three major components 1. the power source (either constant voltage or constant current) 2. the ballast resistor 3. the strain gage The circuit can be operated in either constantvoltage or constant-current mode and one benefit is that, regardless of mode, the output signal is a change in voltage, which somewhat simplifies its measurement. In addition to the three major components in the potentiometer circuit, adding an output circuit with capacitive coupling (also shown in the figure) separates the high-level direct-current (DC) voltage component present due to the voltage source from the low-level voltage changes produced by changes in gage resistance. Of course, this additional RC circuit relegates its use to dynamic strains above the low-frequency cutoff value, f c = 1/(2π Rm C). A device for capturing and displaying the output voltage from the circuit is also necessary, which can have a significant effect on the resolution, speed, and accuracy of the readings. With either mode of excitation, to avoid resistively loading the measurement circuit, it is beneficial to have a capture/display device with high impedance (Rm ) so as not to load the measurement circuit.

Bonded Electrical Resistance Strain Gages

E0 =

Rg Vc . Rg + Rb

(12.7)

Letting r = Rb /Rg , the change in output voltage caused by resistance changes in both the ballast resistor and strain gage can be written as   ΔRg ΔRb r ΔE = − (12.8) (1 − η) , Vc Rb (1 + r)2 Rg where η = 1−

1+



1 1+r



1 ΔRg Rg

b − r ΔR Rb

.

(12.9)

The term η contains the nonlinear component of the output. If the ballast resistor is a matching strain gage to Rg , mounted on an unstrained sample of the same material and experiencing the same temperature profile as Rg , then it can be shown that this circuit achieves temperature compensation of the strain-gage temperature response. Also, if the ballast gage sees equal and opposite strain levels, the circuit output is doubled. However, note that the nonlinear term in (12.9) is largest when r = 1. Constant-Current Excitation By replacing the constant-voltage source (e.g., battery) with a constant-current source, the potentiometer circuit can be operated in constant-current mode. In this case, the ballast resistor is superfluous, since setting Rb = 0 does not change circuit performance. The voltage across the strain gage obeys Ohm’s law (12.2). Thus, going through the same derivation used with the elementary circuit arrives at the same expression for relative change in voltage drop across the gage

ΔRg ΔE = . E0 Rg

12.3.3 The Wheatstone Bridge Circuit As noted previously, even though the potentiometer circuit has several benefits, the drawback of having a large common-mode voltage on top of a small signal voltage makes its use impractical for static strain measurements, except in rare cases when used with semiconductor gages. Even then, separating ΔE from E 0 + ΔE is troublesome. By far the most common electrical circuit used with strain gages, and forming the basis of strain-gage instrumentation, is the Wheatstone bridge. Samuel Hunter Christie invented what was called the diamond method in 1833 and used the technique to compare resistances of wires with different thicknesses. The method went mostly unrecognized until 1843, when Charles Wheatstone proposed it in a paper to the Royal Society, for measuring resistance in electrical circuits. Although presented clearly as Christie’s invention, it was Wheatstone’s name that became associated with the bridge. As he used it, the bridge consisted of four resistors, a battery, and a galvanometer; he called it a differential resistance measurer; this same configuration is quite adequate for a simple straingage circuit. Wheatstone is also responsible for the curious habit of drawing the bridge in its familiar diamond shape. In developing the resistance equations particular to the bridge, it is helpful to begin with a battery power supply and galvanometer readout, and then discuss the influences of using more sophisticated supplies and electronic meters. This simple circuit is depicted in Fig. 12.4, where R1 , R2 , R3 , and R4 are the four arms of the bridge, Rm is the meter resistance, Rb represents the resistance of the battery circuit, plus any appreciaRb

B

1

2

Rm (12.10)

The meter resistance, effectively Rm in Fig. 12.3, can desensitize changes in gage resistance by shunting the gage. Two important benefits from using constantcurrent excitation with a high-impedance recorder instrument are, (1) reducing this shunting effect, and, (2) providing insensitivity of the circuit output to leadwire resistance.

V

E A

C

E0

4

3

D

Fig. 12.4 Basic Wheatstone bridge circuit including battery resistance Rb

287

Part B 12.3

Constant-Voltage Excitation For constant-voltage operation, a simple dry-cell battery is an excellent power source, providing an inexpensive alternative to high-end power supplies. For the unloaded, open-circuit output voltage, obtained when the meter resistance Rm is infinite,

12.3 Strain-Gage Circuits

288

Part B

Contact Methods

Part B 12.3

ble leadwire resistance in series with the battery, and i b and i m are the battery and meter currents, respectively. It can be shown that for small unbalances of the bridge, the value of Rb does not appreciably influence the effective bridge resistance as presented to the meter. Therefore, an expression for the current sensed by the meter can be written as Em , (12.11) im = R + Rm where R is the effective bridge-circuit resistance as presented to the meter and is found by writing the total resistance for a pair of parallel circuits connected in series, as shown in Fig. 12.5. Thus R=

R2 R3 R1 R4 + . R1 + R4 R2 + R3

(12.12)

The voltage presented at the terminals of the meter is then   R1 R2 − (12.13) , Em = E R1 + R4 R2 + R3 or written as the relative change in voltage Em R1 R3 − R2 R4 = . E (R1 + R4 )(R2 + R3 )

(12.14)

From this equation, it should be clear that the bridge output is zero when R1 × R3 = R2 × R4 .

(12.15)

Under this condition, the bridge is said to be balanced, which is important because it means the small voltage output caused by slight changes in resistance to one of the resistor elements (e.g., a strain gage) can be measured from a zero, or near-zero, starting condition. This small signal is then easily amplified for subsequent processing or recording. The voltage applied to the bridge E depends on the supply-circuit resistance (i. e., the resistance in series R1

R2

R4

R3 R

Fig. 12.5 Wheatstone bridge total resistance as presented

to an output meter

with the bridge and the voltage supply, and any internal battery resistance if a battery is used as the supply) and the total bridge resistance as presented to the power supply. For small unbalances, the bridge resistance as presented to the power source is Rc =

(R1 + R4 )(R2 + R3 ) . R1 + R4 + R2 + R3

(12.16)

Equation (12.16) evinces the series/parallel circuit arrangement of the two branches in a Wheatstone bridge (BAD and BCD in Fig. 12.4), hence, series resistors (R1 + R4 ) in parallel with series resistors (R2 + R3 ). The flow of voltage through this arrangement is described in the next section. It follows that the voltage applied to the bridge is then E=V

Rc . Rc + Rb

(12.17)

For the typical supply voltage used with strain gages (≈ 5–10 V), the useable bridge-output resolution is in the μV range, which prescribes a high-sensitivity meter for suitable display. Because the meter deflection, as dictated by (12.11), is based upon determination of the meter current, the circuit is said to be current sensitive. If the output is indicated with a high-impedance device, such as an electronic voltmeter or oscilloscope, current flow in the output circuit is essentially zero, because Rm → ∞ and i m → 0. But there can still be a resistive unbalance condition at the meter terminals, represented by the voltage across corners A and C of the bridge. In this case, the arrangement is called a voltage-sensitive deflection-bridge circuit [12.6], and the voltage indication is calculated by (12.13). Of course, the battery can be replaced by a constantvoltage power supply to achieve the same result, with the additional benefit of providing a zero, or near-zero, source resistance. Furthermore, given an essentially zero source resistance, the voltage applied to the bridge is equal to the power-source voltage. In application of the Wheatstone bridge with strain gages, the change in bridge output voltage caused by the change in gage(s) resistance(s) is the desired indication of strain. The exact bridge-output equations are particular to the arrangement and number of gages in the bridge, and these equations are provided in Sect. 12.4.8. However, the general expressions for constant-voltage and constant-current operation do illuminate differences between these types of excitation, which will now be discussed.

Bonded Electrical Resistance Strain Gages

R2 R1 = =r . R4 R3

(12.18)

As the gages are strained (and change resistance), the relative incremental gage output from (12.14) is

then that, when ΔR1 = ΔR3 = −(ΔR2 = ΔR4 ), the nonlinearity term is zero. Constant-Current Excitation Employing a constant-current power supply in place of a battery or constant-voltage supply converts the bridge into a current-dividing circuit, thus

I = I1 + I2 and its operation is governed by Ohm’s law, such that E m = I 1 R1 − I 2 R4 = I

(12.23)

which has the same balance condition as given in (12.15). The change in bridge output due to resistance changes in the bridge arms may be expressed as ΔE

r ΔE = E (1 + r)2   ΔR1 ΔR2 ΔR3 ΔR4 × − + − (1 − η) , R1 R2 R3 R4

=I

where 1

⎛ 1+⎝

1+r ΔR1 R1

ΔR3 ΔR4 2 + r ΔR R2 + r R3 + R4

a)

⎞.

 

R1 R3 R1 +ΔR1 +R2 +ΔR2 +R3 +ΔR3 +R4 +ΔR4 ΔR1 R1

R1

⎠

(12.24)

R2 Balance control

(12.20)

For the special case of all four gages having the same resistance Rg then r = 1 and (12.19) and (12.20) reduce to   1 ΔR1 − ΔR2 + ΔR3 − ΔR4 ΔE (1 − η) , = E 4 Rg

R4

R3

Amplifier

b)

R1

E

Display

R2

(12.21)

1 η=  . 2Rg 1+ ΔR1 + ΔR2 + ΔR3 + ΔR4



ΔR3 ΔR1 ΔR3 ΔR4 2 − ΔR R2 + R3 − R4 + R1 R3  4 . − ΔRR22 ΔR R4

×

(12.19)

η=

R1 R3 − R2 R4 , R1 + R2 + R3 + R4

(12.22)

Equations (12.21) and (12.22) are the elementary bridge equations for constant-voltage excitation of the Wheatstone bridge circuit. The nonlinearity term η will be zero under the simultaneous conditions of ΔR1 = −ΔR4 and ΔR2 = ΔR3 = 0. Likewise, the term will be zero when ΔR2 = −ΔR3 and ΔR1 = ΔR4 = 0. It follows

E R4

R3

Amplifier

Display

Voltage injection circuit

Fig. 12.6 Two popular circuit arrangements for commer-

cial strain-gage instruments

289

Part B 12.3

Constant-Voltage Excitation Employing a constant-voltage power supply mimics the general analysis presented above, when a battery was the considered supply. Referring to Fig. 12.4, an input voltage is applied across the power corners (BD) and divided between the parallel circuits (BAD and BCD) through the series resistances (R1 + R4 ) and (R2 + R3 ), providing an output across the signal corners (AC) that is a function of the input voltage and the individual arm resistances. If the bridge is initially balanced, or nearly balanced, an output voltage ΔE is developed when any one of the four arms of the bridge changes resistance. Rearranging (12.15),

12.3 Strain-Gage Circuits

290

Part B

Contact Methods

Part B 12.3

Examination of (12.19) and (12.24) for small resistance changes reveals that the bridge outputs are similar with either excitation. Using constant-current power is preferred because the output linearity error is roughly half that experienced when using constant-voltage excitation, which for large strains (i. e., large unequal changes in arm resistance) is an obvious plus. Practical Considerations Its simple construction and high resolution make the Wheatstone bridge circuit a popular choice for many commercial strain-gage instruments (e.g., two arrangements are shown in Fig. 12.6). Constant-voltage excitation is most commonly used. The bridge circuit is operated in either a balanced or unbalanced mode. For balanced circuits, the bridge arms are resistively symmetric about an axis bisecting the output corners of the bridge, such that



R1 R4



 =1= nom

R2 R3

 .

(12.25)

nom

The linearity error exists when resistance conditions in the bridge arms cause unequal changes in the currents through the arms. Given these asymmetrical current changes, the voltage output of the bridge is not proportional to the resistance changes and thus the bridge output is nonlinear with strain and the instrument indication is in error. Fortunately, this error is ordinarily small and can usually be ignored when measuring elastic strains in metals. However, the percentage error increases with the magnitude of strain being measured and can become significant at large strains. For example, in a quarter-bridge circuit the error is about 0.1% at 1000 με, 1% at 10 000 με, and 10% at 100 000 με (or, as a convenient rule of thumb, the error in percent is approximately equal to the strain, in percent, for quarter-bridge measurements). Considering the 50-fold strain sensitivity increase with semiconductor gages, it should be evident that, when operated in the unbalanced condition, large errors in the strain data will be contributed by the bridge nonlinearity. Therefore, special consideration must be given when using the Wheatstone bridge with these unique sensors, which is discussed in Sect. 12.5.

Nonlinearity Errors. One benefit of the balanced, con-

stant-voltage Wheatstone bridge circuit is the equal current flow through both branches of the bridge, which introduces no linearity error to the bridge output. In what’s called a quarter-bridge strain-gage circuit, the strain gage forms one arm of the bridge and fixed or variable resistors make up the other three arms. It should be obvious that, even if the circuit is initially balanced, as the gage changes resistance with strain, the bridge becomes unbalanced. Rarely, a static strain indicator works on the principle of rebalancing the bridge by adjusting a variable resistance in the adjacent arm, whereby the amount of resistance necessary to bring the bridge output to zero is the indication of strain. Under this condition, there is no linearity error associated with the bridge output. Most often, the Wheatstone bridge circuit found in commercial strain indicators is operated in an unbalanced mode. Even when voltage injection circuits are used, where the bridge output voltage is nulled by an equal and opposite voltage injected into the measurement circuit (amplifier), the bridge is resistively unbalanced and the linearity error is present. Voltage injection is frequently used to zero a strain indicator prior to starting a test. Note from Fig. 12.6 that in both circuits the balance control used to obtain an initial zero reading is outside the bridge, which indicates that, even with a balanced output condition, the bridge can be resistively unbalanced.

Noise Control. Electrical noise is a frequently encoun-

tered nuisance during strain-gage measurements. Some common causes are thermoelectric voltages (thermocouple effect at electrical junctions), high-frequency (and energy-level) radio waves, alternating-current (AC) or direct-current (DC) motors, florescent lights, and welding machines. Less common are exotic sources such as high-intensity x-rays or nuclear radiation. For sound data, it is important to recognize and eliminate noise. When using the Wheatstone bridge, disconnecting the power source is a simple and effective way of finding self-generating noise. With no power applied, obviously the bridge has no input with which to operate normally. Therefore, an output signal present when no power is applied must be caused by something other than strain or temperature acting on the bridge resistances (strain gages). If the observed noise signal is in strain units, it can simply be algebraically subtracted from acquired strain data, but the best course of action is to eliminate the signal altogether by removing the cause and/or shielding the strain gage, leadwires, and instrumentation system. For convenience in applying this noise detection method, most modern commercial strain-gage instrumentation has the function of disconnecting the power from the bridge circuit.

Bonded Electrical Resistance Strain Gages

12.4 The Bonded Foil Strain Gage

As previously alleged, strain gages do not actually report strain, but work through electrical circuits to relate a measured resistance value to the coercing strain. Therefore, it is important to understand the accomplishment of this transformation, as well as the circuits used for interrogation. During this study, certain influences on the basic resistance parameter other than strain will be encountered, and these too must be understood, so they can either be eliminated or compensated, or perhaps even exploited to our benefit, leaving only our intended measurement goal: accurately attaining the mechanical strain present in the test specimen, underneath the strain gage.

12.4.1 Manufacturing Some early bonded metallic electrical-resistance strain gages were produced by unwrapping the fine wire found inside a wire-wound resistor and forming that wire into a sinuous pattern on a piece of cigarette paper, affixing the wire to the paper with Duco cement. This paper gage was then bonded to the surface of a test piece, again with Duco cement. So then, how hard can it be to manufacture strain gages? Actually, if the experimental conditions are favorable (laboratory environment, stable room temperature, reasonable strain level – not too high, nor too low – uniform uniaxial stress field, etc.) and especially if the desired accuracy from the measurement is modest, say no better than around 15% uncertainty, then it is not at all difficult to produce something which will work. However, to achieve the precision measurements available from the modern strain gage demands exacting production methods and calibration procedures. The production requirement of primary interest to a user of strain gages is consistency, because a purchased gage is not the one calibrated by the manufacturer. Therefore, it is only through excellent repeatability in the production process that sufficient consistency in the finished product allows the extraordinarily precise measurements available today. The simplistic appearance of modern foil strain gages is somewhat deceiving. Included in each gage is a wealth of engineering expertise. From the choice and formulation of the insulating carrier, to the adhesive which holds the highly processed foil in place, every element and production step can influence the ability of the finished product to accurately sense and respond to strain. Every manufacturer has highly detailed and pro-

prietary procedures for achieving the desired result. Foil is especially noteworthy. Modern strain-gage construction, particularly modern foil technology, provides the almost mindless ability to achieve better than 5% uncertainty strain data [12.7]. The two most common foils used in strain gages are a copper–nickel alloy, usually called constantan, and a nickel–chrome alloy, which is a modified version of the trademarked name Karma (here forward referred to as just karma). Of these two, constantan is by far the most used. Foils are chosen first for their processibility (for example, chemical machinability); second for the ability to satisfy a particular test specification (for example, stability and survivability of a temperature profile, or ability to withstand high elongation without failure or appreciable change in performance properties); third for the ability to respond to thermal and mechanical processing, which allows matching to a specific test specimen material for greatest stability and ease of use by removing the requirement for minute attention to details such as precise knowledge of test temperature; and finally the foil must also satisfy the resistivity requirements by allowing designs for desirable resistances in a suitable size, which typically requires the metal to be rolled to a gage as thin as 2.5 μm. At the forefront of all foil requirements is the demand for consistency and repeatability in performance properties, such as temperature response and fatigue. The proceeding is only a cursory list of foil requirements. Next, the foil must be attached to a carrier. Referring to Fig. 12.7, manufacturers start with a thin insulating layer. Any suitably thin, flexible material is a candidate, and over the years everything from paper to highly formulated, reinforced epoxyphenolic has been used. Today, use of polyimides Overlay Adhesive Foil Adhesive Backing Adhesive

Specimen Fig. 12.7 Schematic of strain-gage construction stack and

attachment to specimen

Part B 12.4

12.4 The Bonded Foil Strain Gage

291

292

Part B

Contact Methods

Part B 12.4

environment. In fact, to expect the best results from every step of the strain gage manufacturing and testing process, the facility must be hospital clean.

12.4.2 Gage Factor – A Practical Theory In 1856 William Thomson (Lord Kelvin) discovered the basis of operation for resistance strain gages: stretching a metal wire changes its electrical resistance. This relationship between change in wire length and change in resistance can be derived from the expression for the resistance of a metal conductor as a function of its dimensions and resistivity L (12.26) , A where R is the conductor resistance (Ω), ρ is the resistivity (Ω cm), L is the conductor length (cm), and A is the conductor area (cm2 ). Differentiating (12.26) with respect to resistance gives R = ρ×

Fig. 12.8 Section from a finished plate of foil strain gages

and epoxy-based materials is most common. Typically, this insulating layer (often called the backing) is 12.5–50 μm thick. The processed foil is attached to the backing with an adhesive chosen to provide the best solution for a given set of problems. For example, upper temperature limit, maximum elongation, maximum fatigue, cost, or flexibility. Once the foil is attached, the finished basic, as it is normally called, is put through a photolithography process to image the desired pattern onto the foil and then machine the image into the foil. What remains is called a plate of gages, an example of which is shown in Fig. 12.8. Chemical machining is the most common method of forming strain-gage patterns in the foil. However, some foils, like platinum–tungsten, require other techniques, such as ion milling. At this stage, the plate is ready for secondary processing, which includes resistance adjustment to a specified value with tight tolerance (usually better than 0.5%). Additional features can be added, such as solder options and protective encapsulation. At the end of the production process, the individual gages are cut from the plate and go through a stringent quality-control (QC) process, ensuring that all manufacturing specifications have been met. Included in the QC procedure is the necessary calibrations (gage factor, transverse sensitivity, temperature response, and so on) needed for strain measurement. Finally, the finished gages are packaged with the information needed for successful use in experimental mechanics. To ensure the highest production standards, all primary and secondary work is conducted in a clean-room

dρ dL dA dR = + − . R ρ L A Recognizing that, for elastic deformation,  2 dA dL 2 dL = −2ν +ν A L L

(12.27)

(12.28)

and, ignoring the higher-power term and substituting (12.28) into (12.27), dL dρ dR = (1 + 2ν) + . R L ρ

(12.29)

Equation (12.29) shows that the relative change in resistance for a metal conductor is controlled by two separate parameters. The first term is a function of the conductor geometry and the second term is a function of the conductor material. Recognizing that, in the limit, the relative change in length term ( dL/L) is equivalent to the axial strain εx (12.1), and dividing (12.29) by εx , yields dρ/ρ dR/R = (1 + 2ν) + . εx εx

(12.30)

Observing that accurate calculation of the second term in (12.30) is impossible, because the term is itself a function of strain, then this equation suggests that, in order to obtain strain readings from resistance measurements, calibration is required. This calibration is performed by measuring the relative change in resistance of a strain gage stretched by a reference strain

Bonded Electrical Resistance Strain Gages

dR/R =F, εx

(12.31)

where F is the calibrated strain sensitivity of the strain gage, called the gage factor. An interesting deduction can be attained by substituting the conductor volume, V = AL, into (12.26) and essentially going through the same derivation as before, arriving at the result dρ dL dV dR = +2 − . R ρ L V

(12.32)

From (12.32), several different assumptions and approaches can be taken towards further derivation of a sensitivity expression, all subject to the fact that, in plastic deformation, volume does not change ( dV/V = 0). Regardless of the chosen approach, the end result is the same: under plastic deformation, the strain sensitivity of a metal conductor is, theoretically, close to 2. The importance of this conclusion should be obvious. It demonstrates that, to achieve a high degree of linearity over the largest range of deformation, strain sensitivity in the elastic range should to be as close to 2 as possible; this is indeed the case for the two most popular strain gage alloys: constantan and karma. While the previous simplistic analysis arrives at some interesting and useful conclusions, accurate gage factor calibration is a difficult task and demands certain compromises and bounding conditions. If not properly considered during the analysis of strain data, some of the necessary restrictions for running this calibration can subtly influence strain measurement results, possibly causing erroneous test conclusions.

Fig. 12.9 Gage factor measuring system per ASTM E 251

guidelines

Manufacturers adjust gages to a desired resistance with low uncertainty. For proper operation, the actual starting value is unimportant, but strain gage instrumentation is frequently built to accept predetermined resistance inputs. Typical gage resistance values offered by manufacturers are 120, 350, and 1000 Ω, and straingage instrumentation commonly accepts these initial resistances. These package resistance values can be shifted slightly during the gage installation process, but most strain gage instrumentation have adequate balance range to re-zero normal offsets caused by installation. This balanced value is then used as the reference for subsequent changes in resistance caused by strain (R and ΔR, respectively from (12.31)), or by other effects. To achieve accurate strain measurements, it is necessary to know with sufficient accuracy the relationship between applied strain and the resulting changes in gage resistance. From (12.31), it should be obvious that a critical element in the gage factor test is the applied strain. Unlike many other engineering parameters, there has never been an accepted standard for generating and precisely measuring strain. However, several researchers have addressed this void and have developed techniques for generating a single strain value in tension and in compression of approximately 1000 με with sufficient precision to allow gage factor uncertainties of 0.5% or better [12.8, 9]. These techniques are then used to verify a manufacturer’s gage factor measurement system. Figure 12.9 shows a test system used for determination of the gage factor. This system supplies a uniform, biaxial strain to gages attached to the constant-stress tapered steel cantilever beam. This system has the benefit of providing convenient determination of gage factor in both tension and compression, which is important because the two values are usually slightly different. The difference is small, normally 0.2% or less, and manufacturers normally report a single-value average of the two. In the following discussion, other subtleties of gage factor and its measurement are presented. Gage factor values supplied with the gages (package value) are normally entered into strain-gage instrumentation to scale the instrument output in units of microstrain (μm/m). However, it is not necessary – or sometimes even possible – to use the package gage factor value. Often, the instrument gage factor is set to a convenient number, like 2.000. This would be necessary, for example, if gage types having different gage factors are connected to a single channel instrument through a switch unit. Regardless, the instrument read-

293

Part B 12.4

input. Thus,

12.4 The Bonded Foil Strain Gage

294

Part B

Contact Methods

Part B 12.4

ing can easily be corrected using the relationship Fi εa = εi × , (12.33) Fa where εa is the strain corrected for gage factor, εi the indicated strain from the instrument, Fi the gage factor set on the instrument, and Fa the package gage factor.

12.4.3 Strains from Every Direction Strain gages are designed and intended to allow accurate determination of the strain beneath the gage, in the direction of its primary measurement axis, which is the direction defined by the gage length (Fig. 12.1). Note that the derivation of (12.31) is not limited to a particular direction of the conductor; it is only based on the correct application of dimensional measurements in the direction of interest. Recognizing this, and noting that the initial starting resistance is not critical to strain measurement because the electrical measurement is relative to any starting value, imagine the sensing grid of a strain gage as a summing transducer, where each infinitesimal length of metal foil acts as its own gage length and initial resistance value, and consider the quasihomogeneity of the metal foil; it should be evident that deformation of the foil from any direction will cause changes in electrical resistance. Hence, the gage responds not only to strains in the intended direction, but also to strains from every direction. (Strain gages are also sensitive to strain normal to their plane, but for common stress analysis gages, this sensitivity is negligible when compared with the in-plane sensitivity.) Since strains acting at every angle between 0◦ and 90◦ to the primary sensing axis are simply cosine combinations of the two orthogonal values, to allow correction for off-axis strains it is only necessary to know the cross-direction contribution; specifically, it is only necessary to know the percentage cross-direction sensitivity, relative to the primary axis sensitivity. This percentage is termed the transverse sensitivity, and is generally written Ft × 100 , (12.34) Kt = Fa where K t is the transverse sensitivity in percent, Ft is the sensitivity in the direction 90◦ to the primary sensing axis, and Fa is the sensitivity in the direction of the primary sensing axis. The practical combinations of variables which influence the magnitude of transverse sensitivity are numerous and complicated. Several design and construction parameters/properties contribute, including

grid geometry, alloy, conductor geometry (wire versus foil), and the shear stress between the conductor and the bonding carrier (gage backing). For example, a single strand wire strain gage bonded to a specimen has essentially zero sensitivity in the cross-direction. A single-line metal foil gage bonded to a specimen has varying cross-sensitivity, dependent on the line width. For a given alloy, short gage length gages tend to have a more positive transverse sensitivity than longer gage length gages. Within a similar geometrical design, constantan foil gages will have a more positive transverse sensitivity than karma foil gages. While it is theoretically possible to design a metal foil strain gage with zero transverse sensitivity, the result would be impractical to produce and not necessarily the best strain sensor. Because of the aforementioned attributes, similar to gage factor, transverse sensitivity must be measured. Unlike gage factor, the calibration of transverse sensitivity requires the application of a uniaxial strain (not uniaxial stress). A device for producing the required strain is shown in Fig. 12.10, and a method for measuring and reporting transverse sensitivity, along with gage factor, is provided in [12.3]. While an explanation is beyond the scope of the present work, suffice to say that it is not possible to generate a uniaxial strain as precisely, consistently, and uniformly as a uniaxial stress, which is why gage factor calibration is performed using the latter, and also why transverse sensitivity calibration is typically an order of magnitude less certain than gage factor calibration. Note that the measurement beam in Fig. 12.10 is short and wide. Plates attached to each enlarged end of the beam supply a bending force. Sufficiently inward from the sides of the beam (≈ 25 mm), strain is purely unidirectional. A strain magnitude of +1000 μm/m is

Fig. 12.10 Typical test rig for measuring transverse sensi-

tivity per ASTM E 251 guidelines

Bonded Electrical Resistance Strain Gages

Kt =

(ΔR/R)t × 100 , (ΔR/R)a

(12.35)

where K t is the transverse sensitivity in percent, (ΔR/R)t is the relative change in gage resistance in the transverse direction, and (ΔR/R)a is the relative change in gage resistance in the axial direction. When principal strain directions are unknown and multiple gages are installed at various angles to one another (e.g., a rosette gage; Sect. 12.4.10), it is almost certain that at least one grid will have a significant portion of a principal strain acting transverse to its measuring axis. If the transverse sensitivity of the gage elements in these gages is other than zero, the individual strain readings will be in error. From the standpoint of principal stress and principal strain calculation, one of the most convenient rosette arrangements is known as the rectangular rosette, where the grids are at 45◦ to one another. Correction for the effects of transverse sensitivity can be made either on the individual strain readings or on the calculated principal strains or principal stresses. Numbering the gage elements consecutively, elements 1 and 3 correspond directly to the two-gage 90◦ rosette and the same equations can be used for transverse sensitivity correction. The center grid of the rosette requires a unique correction relationship since there is no direct measurement of the strain perpendicular to this grid. The correction equations for all three grids are 1 − ν0 K t (ε1 − K t ε3 ) , 1 − K t2 1 − ν0 K t [ε2 − K t (ε1 + ε3 − ε2 )] , εˆ 2 = 1 − K t2 1 − ν0 K t (ε3 − K t ε1 ) , εˆ 3 = 1 − K t2 εˆ 1 =

(12.36) (12.37) (12.38)

where ε1,2,3 are the indicated strains from the respective grid elements and εˆ 1,2,3 are the corrected strains along the respective grid axes. It should be noted that (12.36–12.38) assume transverse sensitivity is the same, or effectively so, in all grid elements, which is often the case in stacked rosettes. This may not be true for planar rosettes, since each grid is oriented at a different angle with respect to the rolling direction of the foil used in making the gage.

However, even for planar rosettes, transverse sensitivity of the individual grids is normally close (and usually equal for at least grids 1 and 3) and a suitable average value for all grids can be used with these equations to provide adequate correction accuracy in most cases. For the fully general case of unequal transverse sensitivity among the grids, the reader is directed to Vishay Micro-Measurements [12.10].

12.4.4 Gage Factor – The Manufacturer’s Value With transverse sensitivity defined, perhaps it becomes clear that the manufacturer’s normal method of measuring gage factor in a uniaxial stress field yields a result dependent upon the Poisson’s ratio of the test beam. Strictly speaking, the manufacturer’s value is only valid on materials with the same Poisson’s ratio. Fortunately, ignoring this restriction normally results in a small error which can be disregarded for most stress analysis applications. However, if the material under test and the beam used to measure the gage factor have a significantly different Poisson’s ratio, or if appreciable transverse strains relative to the axial strains exist in the test part (i. e., biaxial loading), or the highest strain data accuracy is required, corrections should be made for transverse sensitivity. In general, then, a strain gage has two gage factors, Fa and Ft , which refer to the gage sensitivities as measured in a uniaxial strain field (not uniaxial stress) with, respectively, the gage axes aligned parallel to and perpendicular to the principal strain direction. For any strain field, the output of the strain gage can be expressed as ΔR (12.39) = Fa εa + Ft εt , R where εa is the strain parallel to the gage axis, εt is the strain perpendicular to the gage axis, Fa is the axial gage factor, and Ft is the transverse gage factor, or ΔR = Fa (εa + K t εt ) , R

(12.40)

where K t = FFat is the transverse sensitivity coefficient. When the gage is calibrated for gage factor in a uniaxial stress field on a material with Poisson’s ratio ν0 , εt = −ν0 εa . Therefore, ΔR = Fa (εa − K t ν0 εa ) , R

295

Part B 12.4

normally used. Test gages of the same type are bonded with their principal measurement axis parallel to and perpendicular to this applied strain, and gage sensitivity is measured in both directions. Transverse sensitivity is calculated and supplied as a percentage

12.4 The Bonded Foil Strain Gage

296

Part B

Contact Methods

Part B 12.4

or ΔR = Fa εa (1 − ν0 K t ) . R

(12.41)

Strain-gage manufacturers commonly write this as ΔR = Fε , R

(12.42)

where F is the manufacturer’s gage factor. This is deceptively simple in appearance, since, in reality F = Fa (1 − ν0 K t ) .

Error – % of axial strain 60 εt/εa = 5

50 –5 = εt/εa

4

–4

3

30 –3 20

–1

0

0

10

1

– 20

2

–2

+ 10

1 0 –1 –2

2 –3 3

30 40

εa

4

–4 –5

5

 ϕε =

(12.43)

Furthermore, ε, as used by the manufacturer, is actually εa , the strain along the gage axis (and only one of two strains sensed by the gage during calibration) when the gage is aligned with the maximum principal stress axis in a uniaxial stress field, on a material with Poisson’s ratio equal to that of the beam used for gage factor calibration. Errors and confusion can occur due to failure to fully comprehend and always account for the real meanings of F and ε as they are used by manufacturers. To judge the effect on subsequent test data caused by the conditions present during gage factor calibration, it is important to recognize that, for a gage with

40

nonzero transverse sensitivity, there is always an error in the strain indication unless the gage is used under the exact same conditions present during gage factor calibration. In some instances, this error is small enough to ignore. In others, it is not. The error due to transverse sensitivity for a strain gage oriented at any angle, in any strain field, on any material, can be expressed as

εt

Kt

εt εa

+ ν0

1 − ν0 K t

 ,

(12.44)

where ϕε is the error (as a percentage of the actual strain along the gage axis), ν0 is the Poisson’s ratio of the material used by the manufacturer for gage factor calibration, and εa and εt , respectively, are the actual strains parallel and perpendicular to the primary sensing axis of the gage. For convenience in judging whether the error magnitude is significant for a particular test condition, (12.44) is plotted in Fig. 12.11.

12.4.5 Transverse Sensitivity Error – Numerical Examples The effects of transverse sensitivity should always be considered in measurement applications where the strain field and/or test material do not match that which was used during gage factor calibration. Either it should be demonstrated that the error associated with transverse sensitivity is negligible and can be ignored, or if not negligible the proper correction should be made. As a simple first example, consider a tension test on an unfilled plastic using a single linear gage. With the instrument gage factor set to match the package gage factor supplied by the manufacturer, the test parameters and strain indication are

• • • • •

plastic beam in tension ν = 0.48 K t = 2.9% ν0 = 0.285 εI = 1200 μm/m

50 60 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10

–

+ Transverse sensitivity Kt (%)

Fig. 12.11 Error in indicated axial strain from transverse sensitivity for various ratios of principal strains

Equation (12.41) provides the specific relationship for relative resistance change of a strain gage in a uniaxial stress field for the particular case of ν0 (Poisson’s ratio for the gage factor calibration beam). This relationship can be generalized by substituting a generic ν for the specific ν0 . Further, solving (12.43) for Fa and

Bonded Electrical Resistance Strain Gages

ΔR Fε(1 − νK t ) . = R 1 − ν0 K t

(12.45)

All Wheatstone-bridge-based strain-gage instrumentation work by the relationship ΔR (12.46) = Fεi , R where εi is the indicated strain as displayed or recorded by the instrumentation. Equating (12.45) and (12.46), εi = ε

1 − νK t . 1 − ν0 K t

(12.47)

Solving for the actual strain ε along the gage axis, 1 − ν0 K t (12.48) 1 − νK t and substituting the example numerical information, ε = εi

1 − 0.285(0.029) 1 − 0.48(0.029) = 0.0012 × 1.00574 = 1206.8 με

ε = 0.0012 ×

results in a possibly negligible error of less than 0.6% for this example. For a second example of correcting strain gage data for transverse sensitivity, assume a stacked rectangular rosette, with each grid in the rosette having a transverse sensitivity of +5%. After loading, the three strain-gage readings are εx = 856 μm/m , ε3 = 306 μm/m .

ε2 = 195 μm/m ,

All three indicated strains contain small errors caused by the transverse sensitivity of the sensing grids. Rather than develop the correction equations for each grid in the rosette and then correct each individual reading before substituting into the rosette data-reduction equations, a simpler technique is to calculate the principal strains from the uncorrected strain measurements, and then correct the principal strains. For this example, the principal strains calculated from the three strain measurements are εpr = 1053 μm/m,

εqr = 108 μm/m ,

where εpr and εqr are the raw and uncorrected principal strains, respectively. Since transverse sensitivity errors in the calculated principal strains reflect the same errors as the individual strain readings, and since the principal strains are

two perpendicular strains, they can be corrected using (12.36) and (12.37). Thus, ignoring the (1 − K t2 ) term in the denominator, since it is less than 0.25% away from unity, one obtains εp = (1 − ν0 K t )(εpr − K t εqr ) , εq = (1 − ν0 K t )(εqr − K t εpr ) .

(12.49) (12.50)

Using the transverse sensitivity value of +5% and assuming a Poisson’s ratio of 0.285 for the gage factor calibration beam, the corrected principal strains are calculated as εp = (1 − 0.285 × 0.05)(1053 − 0.05 × 108) = 1033 με , εq = (1 − 0.285 × 0.05)(108 − 0.05 × 1053) = 55 με .

12.4.6 The Influence of Temperature Changes Thus far, attention has been focused on the response of strain gages at temperatures close to normal laboratory environments (≈ 24 ◦ C). Of the possible stimuli responsible for causing resistance changes in metal conductors, and therefore in strain gages, temperature is by far the most impressive. A sufficient change in temperature can easily cause resistance changes much larger than those resulting from normal elastic deformations in metal specimens. Left unchecked, this bias in strain data renders test results, at the least meaningless, and often dangerous. Fortunately, the signal generated by temperature is additive to the signal generated by strain and can therefore effectively be corrected or compensated. To begin an examination of the effect temperature has on strain gage readings, consider that all three independent variables of conductor resistance in (12.26) are themselves functions of temperature. Metal conductors have a temperature coefficient of resistance (TCR), which causes their electrical resistance to change with temperature, and a temperature coefficient of expansion (TCE), which causes dimensional changes with temperature. Thus, it should be evident that, since (12.26) is used to derive (12.31), this relationship is also a function of temperature. Dependent on the alloy used to make a gage, strain sensitivity can either increase or decrease with a temperature rise. Manufacturers normally supply a room-temperature gage factor and a slope for percentage change in sensitivity per degree of temperature change, sometimes called the temperature coefficient

297

Part B 12.4

substituting into (12.41),

12.4 The Bonded Foil Strain Gage

298

Part B

Contact Methods

Part B 12.4

(TC) of gage factor, or just the TC of gage factor. When strain gages are used at temperatures other than that which the manufacturer used during gage factor calibration, a correction may be needed. Equation (12.33) serves this purpose. In using (12.33) to correct gage factor variation with temperature, Fa is the actual gage factor at the test temperature, not the package gage factor. If the test temperature is known in advance, the gage factor control on the strain indicator can initially be set at Fa and no further correction is necessary. When using this technique, it should be remembered that, if thermal output corrections (see later discussion) are to be made using the thermal output correction data supplied by the manufacturer, these correction data must be adjusted from the gage factor value used by the manufacturer to collect the data, if that value is different from Fa . As a brief numerical example, assume a roomtemperature gage factor of 2.05 and a TC of gage factor equal to − 1.0%/100 ◦ C. With the instrument set at the room-temperature gage factor value, the strain indication at +230 ◦ C is 1820 με. Then, the percentage change in gage factor is −0.01 × (230 − 24)/100 = −0.0206 , and the gage factor at the test temperature is Fa = 2.05(1 − 0.0206) = 2.01 . Substituting into (12.33) yields εa = 1820 ×

2.05 = 1856 με . 2.01

In addition to the temperature effect on sensitivity (TC of gage factor), there is also the more complicated combined affect from the TCR of the alloy used in the gage, and the TCE of the strain-gage system (metal foil, backing, adhesive, and specimen). Before analyzing these two effects on gage response, the question of why it is important to consider temperature changes in the first place should be answered, and one should also look at benefits the strain gage can provide when temperature does change during an experimental stress analysis. Subjected to a temperature change, all materials experience an associated dimensional change. If the material is unrestrained (free to expand without restraint), this dimensional change causes no corresponding stress in the material. Dependent on the material and temperature excursion, the deformation associated with a temperature change can be large when compared with

typical elastic strains in metals. If, for example, a traditional extensometer is used to measure strain during this temperature excursion, the reading will include not only the strain caused by any mechanical loading, but also the free expansion resulting from the temperatureinduced dimensional change. Typically, the only desired result from strain-gage measurements is the one relating solely to strain in the test specimen caused by mechanical loading (mechanical stress). However, like most sensors, strain gages are sensitive to many influences besides that which is intended, temperature being prominent. Unlike many other sensors (e.g., the extensometer just mentioned), strain gages can be tailored to minimize temperature response within a specific scope and, at the same time, compensate for the unimportant temperature-induced deformations in the test specimen. Once an installed gage is connected to a strain indicator and the instrument zeroed, a subsequent change in temperature of the gage installation can produce a resistance change in the gage and a corresponding reading on the instrument. This temperature-induced resistance change is independent of, and unrelated to, any mechanical (stress-induced) strain in the test object. This response, resulting purely from a temperature change, is often called the thermal output, or more expansively the apparent thermally induced strain of the strain gage. Thermal output is potentially the most serious error source in the practice of static strain measurement with strain gages. In fact, when measuring strains at temperatures far from room temperature (or from the initial balance temperature of the gage circuit), the error due to thermal output, if not controlled, can be much greater than the magnitude of the strain being measured. Therefore, it is usually necessary to either use some type of compensation for thermal output or make subsequent numerical corrections to the acquired strain data. Several techniques are available to reduce or eliminate errors caused by thermal output. The easiest technique is simply to zero the strain gage circuit at the test temperature, and then not change the temperature again during data collection. A key to success when using this technique is ensuring that the test part remains completely free of mechanically or thermally induced stresses before zeroing the circuit. Obviously, this is not always possible. Many factors affect the thermal output of strain gages: test specimen material and shape; foil alloy and batch; gage construction and pattern geometry; transverse sensitivity; installation procedures and environmental protection; heating rate; and so on. To

Bonded Electrical Resistance Strain Gages

Thermal output με (Fi = 2) 4000 3000 Isoelastic

Nichrome V

2000 Karma (full hard) 1000

several key elements of the test are not specifically addressed. One key issue is heating rate, which is simply instructed to be slow. Dependent on the test material, heating rate can noticeably affect thermal output results [12.11]. Another concern is the test material itself. While most manufacturers use standard reference specimens, it is not at all certain that the properties of these materials will closely match a user’s test specimen, especially with regard to size and shape, even if within the same class of material. Additionally, manufacturers mainly want to give a representative thermal output characteristic. Rarely will more than one gage type be used to acquire thermal output data for a given batch of foil. It is almost a certainty that thermal output data supplied by the manufacturer will not adequately match the user’s specific test parameters. If it is determined, however, that the thermal output data provided by the manufacturer is sufficient, either because the manufacturer’s test conditions closely match the user’s, or the user’s accuracy demands are low (≈ ±10%), then correcting acquired strain readings for thermal output is simple. Using the test temperature at which the strain reading was acquired, determine from the supplied graph or equations what the value of thermal output is at that temperature, and then algebraically subtract that value from the strain readings. As noted previously, remember that consideration must be given to the instrument gage factor setting, ensuring that the acquired strain data and the supplied correction data use gage factors which have been properly corrected for temperature. From the above discussion it should be evident that in order to achieve the most accurate correction for thermal output, it is generally necessary to obtain thermal

24 °C 0 Constantan (full hard)

75 °F –1000 –2000 –3000

Alloys bonded to steel specimen –4000 –100

0

100

200

300

400

500

Temperature (°F) –50

0

50

100

150

200

250

Temperature (°C)

Fig. 12.12 Thermal output of several strain gage alloys (in the as-rolled metallurgical condition) bonded to steel

Fig. 12.13 Representative thermal output test system per

ASTM E 251 guidelines

299

Part B 12.4

help users, strain-gage manufacturers supply representative data for thermal output, usually presented in the form of a graph with accompanying curve-fit equations, or just the equations themselves (or perhaps just the coefficients). Representative thermal output plots for the special case of several as-rolled (hardtemper) strain gage alloys bonded to steel are shown in Fig. 12.12. Note that the curves for both the karma and constantan alloy gages pertain to non-self-temperaturecompensated foil (self-temperature compensation is discussed later in this section, where the benefits will become obvious when compared with the curves in Fig. 12.12). This data is used to correct acquired strain readings by providing a method for determining the value of thermal output at the temperature point(s) of interest. To complete the correction, this value is algebraically subtracted from the acquired strain readings. Because of the myriad affects on thermal output, it is important to understand fully how the manufacturer collects this data in order to determine its relevance to a particular set of test conditions. ASTM standard test method E 251 [12.3] presents suggestions for standardized testing of thermal output (Fig. 12.13 shows a representative thermal output test system following ASTM E 251 guidelines). However,

12.4 The Bonded Foil Strain Gage

300

Part B

Contact Methods

Part B 12.4

output data from the actual test gage installed on the actual test part. For this purpose, a thermocouple or resistance temperature sensor is installed immediately adjacent to the strain gage. With the gage connected to the strain indicator and no loads applied to the test part, the instrument is balanced for a zero strain reading. Subsequently, the test part is subjected to the test temperature(s), again with no loads applied, and the temperature and indicated strain are recorded under equilibrium conditions. If throughout this process the part is completely free of mechanical and thermal stresses and the temperature measurement accurately reflects the strain-gage temperature, the resulting strain indication at any temperature is the thermal output at that temperature. If the instrument gage factor setting during subsequent strain measurement is the same as that used for thermal output calibration, the calibrated thermal output at any test temperature can be subtracted algebraically from the indicated strain to arrive at the corrected strain. Otherwise, the thermal output data should be adjusted for the difference in gage factor settings – as previously described – prior to subtraction. Referring to Fig. 12.12, one can conclude that both constantan and karma in the hard-temper state have a high TCR. Fortunately, and unlike the other two alloys

shown in this figure, the TCR of both constantan and karma responds to heat treatment. With appropriate heat treatment, the thermal output curves for these two alloys can be rotated to minimize the temperature response over a wide temperature range when the gages are used on materials for which the foil heat treatment was targeted. Strain gages employing these specially processed foils are referred to as self-temperature compensated (STC). Representative curves for self-temperaturecompensated constantan and karma alloy strain gages bonded to 1018 steel are shown in Fig. 12.14. Note the ordinate scale and the markedly reduced output when compared with Fig. 12.12. While self-temperature compensation does not completely eliminate thermal output, it does improve performance to the point of reducing the number of cases where correction is mandatory. Thermal Output Compensation Techniques Exploiting the fact that thermal output is dependent on the difference between specimen and gage thermal expansion provides for a sometimes useful technique of minimizing temperature response, particularly when using gages in extreme cold (e.g., cryogenic temperatures). By installing the gage on a material other than that for which it was intended, the thermal outThermal output (με) 200

Thermal output με (Fi = 2) 500 400

0

300 200

K-alloy

24 °C

100

–200

0 –100

75 °F

–200

– 400 A-alloy

–300 –400

– 600

–500

Aluminum, TCE = 23.2 με/ °C Brass, TCE = 20.0 με / °C

–600

– 800

–700 –800

Test specimen – 1018 steel

–900 – 1000 –200

–100

0

100

200

300

400

500

–1000 –400 –300 –200 –100

0

100 200 300 Temperature (°F)

Temperature (°F) –100

–50

0

50

100

150

200

250

Temperature (°C)

Fig. 12.14 Representative thermal output of self-temper-

ature-compensated constantan (A-alloy) and modified Karma (K-alloy) strain gages

–250 –200 –150 –100 –50

0

50 100 150 Temperature (°C)

Fig. 12.15 Example of STC mismatch (thermal output curve rotation caused by using a gage on brass that was processed for use on aluminum)

Bonded Electrical Resistance Strain Gages

εTO(B) = εTO(A) + (αS(B) − αS(A) )ΔT ,

(12.51)

where εTO(B) is the expected thermal output of a strain gage intended for material A but installed on material B, εTO(A) is the manufacturer’s calibrated thermal output, measured on standard material A, αS(B) is the TCE of material B, and αS(A) is the TCE of material A. If certain conditions are expressly met, one of the best techniques for compensating thermal output is using a compensating gage, often called a dummy gage. The technique exploits the arithmetic capabilities of the Wheatstone bridge by placing an exact duplicate of the active gage in an adjacent arm of the bridge (Fig. 12.16). This gage is subjected to precisely the same temperature profile experienced by the active gage, but is mounted on an unstrained piece of duplicate test material and connected to the instrumentation using the same type of leadwire system routed beside the active gage leads (to allow an equal temperature profile on all leadwires). Under these conditions, two adjacent arms of the bridge (R1 and R4 in Fig. 12.4) will experience identical resis-

Active gage

L1

L2 Compensating gage

e0

E

L3

L4

Fig. 12.16 Circuit diagram for the dummy-gage thermal output compensation technique

tance changes caused by thermal output. From (12.25), this condition does not unbalance the bridge and, hence, the temperature response self-cancels in the bridge and there is no change in reading on the strain indicator. The effect from specimen surface curvature is the final topic covered on temperature effects. Hines [12.13] demonstrated that, when a strain gage is installed on a sharply curved surface, thermal output is different when compared to the same gage installed on a flat surface. As will be shown in the next section, thermal output is a function of the difference between TCE of the gage and specimen. This difference causes a real strain in the gage. The curvature-induced change in thermal output is the result of the strain-sensitive grid being above the specimen surface by an amount equal to the gage backing layer plus the installation-adhesive layer thicknesses, effectively causing a neutral-axis shift when compared to the flat-surface case. The curvature-induced change in thermal output is a second-order effect that can ordinarily be ignored, but it can become significant when the radius of curvature is small. As a rule of thumb, when the radius of curvature is greater than 12 mm, the effect is of little consequence. Employing the same basic approach and approximations used by Hines, but generalizing to account for any combination of adhesive and backing properties, an expression for estimating the curvature-induced change in thermal output can be written as follows ΔεTO =

1 [(1 + 2νA-B )(h A αA + h B αB ) R − 2νA-B αS (h A + h B )]ΔT ,

(12.52)

where, in consistent units, ΔεTO is the curvatureinduced change in thermal output, R is the test surface radius of curvature at gage site, νA-B is the average Poisson’s ratio of adhesive and backing, h A and h B are the adhesive and backing thickness, respectively, αA and αB are the TCEs of the adhesive and the backing, respectively, αS is the TCE of the substrate (specimen material), and ΔT is the temperature change from the reference temperature. Approximate values for some common strain gage adhesive and backing parameters used in (12.52) are given in Table 12.1. The sign of the change in thermal output is obtained from (12.52) when the signs of ΔT and R are properly accounted for; that is, an increase in temperature from the reference temperature is taken as positive, and a decrease negative; and correspondingly, a convex curvature is positive, and a concave curvature is negative. The calculated result ΔεTO is added algebraically with the standard thermal output correction

301

Part B 12.4

put curve is rotated. This technique is termed, STC mismatch [12.12]. The curves in Fig. 12.14 show that both constantan and karma strain gages have a steep thermal output slope below room temperature, and that the slope progressively increases as the temperature decreases. Because it does have a lower slope at cold temperatures, karma alloy is normally used at or near cryogenic temperatures, but the thermal output is still high. When a gage processed for use on a material with a certain TCE is used instead on a material with a lower TCE, the original thermal output curve is effectively rotated clockwise (Fig. 12.15). Conversely, using the gage on a material with a higher TCE will rotate its thermal output curve counter-clockwise. The following generalized form approximately accomplishes this transformation analytically

12.4 The Bonded Foil Strain Gage

302

Part B

Contact Methods

Part B 12.4

Table 12.1 Typical strain gage component values (used with (12.52)) Adhesives

Backings

Type Cyanoacrylate

HA , mm (in) 0.015 (0.0006)

Two-component 100%-solids epoxy Two-component, solvent-thinned, epoxy-phenolic

0.025 (0.001)

0.005 (0.002)

ppm/◦ C (◦ F)

αA , 227 × 10−6 (126 × 10−6 ) 81 × 10−6 (45 × 10−6 ) 81 × 10−6 (45 × 10−6 )

Type Cast polyimide

HB , mm (in) 0.030 (0.0012)

Single-thickness, glass-fiber-reinforced, epoxy-phenolic Double-thickness, glass-fiber-reinforced, epoxy-phenolic

0.025 (0.001)

0.038 (0.0015)

αB , ppm/◦ C (◦ F) 90 × 10−6 (50 × 10−6 ) 18 × 10−6 (10 × 10−6 ) 18 × 10−6 (10 × 10−6 )

νA-B = 0.35 for all combinations

data, prior to correcting indicated strains, as previously described. Because the adhesive and backing parameters given in Table 12.1 are approximate and are affected by gage installation technique and other variables, the curvature correction defined by (12.52) has limited accuracy. When the surface curvature is severe enough to warrant correction, the actual thermal output should be measured following the prescribed method given earlier.

12.4.7 Control of Foil Strain Gage Thermal Output In this section, an explanation is given for how a manufacturer controls thermal output to achieve selftemperature compensation. As noted above, there are two effects controlling the response of a strain gage to changes in temperature. The first, the TCR of the alloy used in making the gage, is a material property and can be expressed as ΔR

= β × ΔT , R β

(12.53)

where β is the temperature coefficient of resistance (Ω/Ω /◦ C) and T is the temperature (◦ C). The second effect is caused by a mismatch between the specimen and gage TCE. When present, as is usually the case, a change of temperature causes a thermally induced strain in the gage. Additionally, temperature affects materials isotropically, so an account must be made for both axial and transverse strains. Thus, assuming equal coefficients of thermal expansion in both directions for the specimen and gage (not necessarily true, but a necessary simplification to contain the

derivation), ΔR/R|a ΔL

(12.54)

= (αS − αG )ΔT = α L a Fa and, ΔL

ΔR/R|t . (12.55)

= (αS − αG )ΔT = L αt Ft Solving (12.54) and (12.55) for ΔR/R and summing both with (12.53), the total relative resistance change of the gage due to thermal output is ΔR

= [β + Fa (αS − αG ) + Ft (αS − αG )]ΔT . R TO (12.56)

Converting Fa and Ft to the manufacturer’s gage factor F,    1 + Kt ΔR

(αS − αG ) ΔT .

= β+F R TO 1 − ν0 K t (12.57)

For some alloys used in the production of strain gages, most notably constantan and karma, β can be controlled via thermal and mechanical processing. And this is precisely how manufacturers’ produce STC strain gages. An obvious consequence of the temperature-respondent TCR is the establishment of upper temperature bounds. For constantan, when the temperature exceeds 65 ◦ C for several hours, the initial room-temperature resistance will begin to drift (zero drift) as β changes (from thermal reprocessing). Similarly, when karma is exposed to temperatures in excess of 200 ◦ C, its initial zero will drift. Additionally, both foils, when exposed to sufficiently high temperatures for a significant period, can be reprocessed into another STC, but long before this occurs the zero drift will have rendered any strain data meaningless.

Bonded Electrical Resistance Strain Gages

12.4 The Bonded Foil Strain Gage

Bridge/strain arrangementa 1

ε E0

2

E

ε E0

E

–νε

3

ε E0

E

–ε

4

ε E0

E ε

5

ε

–νε E0 ε

–νε

6

E

ε

–νε E0

7

E

–ε

νε

ε

–ε E0

–ε

E ε

Nonlinearity b,c , η, where E0 /E = K  × 10−3 (1 − η)

Bridge output b,c E0 /E (mV/V)

Description

K=

Single active gage in uniaxial tension or compression.

E0 E

=

F ×10−3 4+2F ×10−6

Two active gages in uniaxial stres field – one aligned with maximum principal strain, one Poisson gage.

E0 E

=

F (1+ν)×10−3 4+2F (1−ν)×10−6

Two active gages with equal and opposite strains – typical of bending-beam arrangement

E0 E

=

F

2

Two active gages with equal strains of same sign – used on opposite sides of column with low temperature gradient (bending cancellation, for instance).

E0 E

=

F ×10−3 2+F ×10−6

Four active gages in unaxial stress field – two aligned with maximum principal strain, two Poisson gages (column)

E0 E

=

F (1+ν)×10−3 2+F (1−ν)×10−6

Four active gages in unaxial atress field – two aligned with maximum principal strain, two Poisson gages (beam).

E0 E

=

F (1+ν)×10−3 2

Four active gages with pairs subjected to equal and opposite strains (beam in bending or shaft in torsion)

E0 E

= F × 10−3

× 10−3

η=

K= η=

K=

K= η=

K= η=

K=

Corrections c

F 4 F 10−6 2+F ×10−6

=

2 i 2−F i ×10−6

F(1+ν) 4 F (1−ν)×10−6 2+F (1−ν)×10−6

=

2 i 2(1+ν)−F i (1−ν)×10−6

η=0

=

i 2

F 2 F ×10−6 2+F ×10−6

=

2 i 4−F ×10−6

F(1+ν) 2 F (1−ν)×10−6 2+F (1−ν)×10−6

=

2 i 4(1+ν)−F i (1−ν)×10−6

=

i 2(1+ν)

=

i 4

F 2;

F(1+ν) ; 2

η=0

K = F; η = 0

1 2

a, 4 3 (R /R ) 1 4 nom = 1; (R2 /R3 )nom = 1 when b Constant-voltage power supply is assumed c

two or fewer active arms are used

and i are expressed in microstrain units (m/m × 106 )

12.4.8 Foil Strain Gages and the Wheatstone Bridge Recalling the relationship between input voltage, arm resistances, and bridge output given in (12.13), and using (12.42), if the bridge is populated with four gages in

what’s called a full bridge, which have impressed strains ε1 , ε2 , ε3 , and ε4 , then (12.13) for constant-voltage excitation can be written as 1 + Fε1 1 + Fε2 E0 = − . E 2 + F(ε1 + ε4 ) 2 + F(ε2 + ε3 )

(12.58)

Part B 12.4

Table 12.2 Bridge output relationships for several common Wheatstone bridge arrangements used with strain gages

303

304

Part B

Contact Methods

Part B 12.4

Note that for clarity the gage factor has been assumed equal in all four gages. While not necessarily true, this is a common condition (e.g., any installation using four gages of the same type and construction) and this assumption greatly simplifies the equation. Several common strain gage bridge arrangements are shown in Table 12.2. With appropriate substitution of active and fixed gages into (12.58), the bridge output can be calculated and plotted to show the nonlinearity associated with each case. For reference, these relationships are provided in the table. Note that, for all relationships presented in the table, strain is scaled in m/m × 106 (e.g., 1500 m/m × 106 = 1500 μm/m = 0.001500 m/m). For each of the bridge arrangements shown, the nondimensional circuit output, in mV/V, can be expressed as E0 = K ε × 10−3 (1 − η) , E

(12.59)

where K is a constant, determined by the gage factor of the strain gage(s) and the number of active arms in the bridge circuit; and η, when not zero, represents the nonlinearity caused by unequal changes in the currents through the bridge arms. The fourth column of the table gives the expressions for calculating the nonlinearity associated with each bridge arrangement. Multiply this result by 100 to Percent nonlinearity 10 9 8 7 6 5 4 3

Case #1, #4 2

Case #2, #5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.1 1000

10 000

100 000

Strain (με)

Fig. 12.17 Error in indicated tensile strain caused by Wheatstone bridge nonlinearity

obtain the percentage nonlinearity in the circuit output. Examination of this column shows that only cases 3, 6, and 7 are linear. This is true because only in these three cases are the resistance changes due to strain such that the currents through the bridge arms remain equal. It can also be noted from this column that, on a percentage basis, the nonlinearity magnitudes are identical for cases 1 and 4, and for cases 2 and 5, although the circuit outputs differ. For convenience in quickly judging nonlinearity magnitudes, the relationships in cases 1, 2, 4, and 5 are plotted in Fig. 12.17, assuming positive (tensile) strains, a gage factor of 2.0, and Poisson’s ratio (where involved) of 0.30. The nonlinearity for compressive strain is opposite in sign and somewhat different in magnitude, but can always be calculated from the provided relationships. The final (rightmost) column in Table 12.2 holds relationships for converting the indicated strain εi , as registered by a strain indicator or other instrumentation system, to the actual surface strain under a single active gage, ε. These expressions numerically correct for Wheatstone bridge nonlinearity (when present) and for the number of active gages in the circuit. Linearity Error Correction Examples Three numerical examples of correcting foil strain-gage data for Wheatstone bridge nonlinearity follow. For simplicity, each assumes a quarter-bridge circuit with a single active gage, and a gage factor of 2.0. However, the procedures used in the examples are general and applicable to all circuit arrangements in Table 12.2. For the first example, the quarter-bridge circuit is initially resistively balanced and no load is applied to the test piece. The test piece is then loaded until the strain indicator displays a tension reading of 10 000 με. From Fig. 12.17 for case 1 (single active gage), the nonlinearity is approximately 0.99% or about 99 με. Substituting F = 2.0 and εi = 10 000 με into the correction equation from Table 12.2 for case 1 gives the actual surface strain in the test piece as 10 101 με. If the indicator had registered −10 000 με for compression, the same procedure would yield a surface strain of −9901 με. These calculations demonstrate that Wheatstone bridge nonlinearity causes indicated tensile strains to be low, and indicated compressive strains to be more positive. The previous example assumed the Wheatstone bridge was initially resistively balanced. However, it is not unusual during the bonding of a strain gage for the as-manufactured resistance to be significantly shifted; or assembly or preload stresses can strain the gage into the plastic range prior to making measurements. In such

Bonded Electrical Resistance Strain Gages

Leadwire Attenuation An often-overlooked source of potential error in straingage measurements are the wires used to connect the

gage(s) to instrumentation. There is potentially both a static concern (attenuation caused by the leadwire resistance) and a dynamic concern (change in leadwire resistance caused by a temperature change in the wire). When the Wheatstone bridge is used to indicate the resistance change of a strain gage, the gage and leadwire system constitute one arm of the bridge circuit. Therefore, leadwire resistance appears to the bridge as part of the measurement circuit by adding inactive (from the specimen-strain perspective) electrical resistance to the initial gage resistance. As the gage changes resistance with applied strain, the bridge circuit interprets the change as a function of the total arm resistance. The inactive leadwire resistance effectively reduces, or desensitizes, the gage factor by presenting strain-insensitive resistance to the bridge circuit, which increases the effective R in (12.42), without commensurately increasing the ΔR. Clearly, long runs of small-diameter and/or high-resistance wire are most problematic. The desensitization factor D is computed as D=

RG , RG + RL

(12.60)

where RG is the gage resistance and RL is the total initial resistance of leadwires in series with the gage (both lengths of wire for a two-wire circuit; one length of wire for a three-wire circuit). When the leadwire resistance is known (e.g., using a wire table value), leadwire errors can be corrected by multiplying indicated strains by 1/D. Alternatively, leadwire desensitization can be compensated by using an adjusted instrument gage factor setting, FG , instead of the package value, where FG = DFG . The percentage error in strain reading, e, caused by leadwire resistance can be expressed as e=

RL /RG RG × 100 . 1 + RL

(12.61)

The temperature coefficient of resistivity of metal leadwire introduces another complication if the leadwire temperature changes after zeroing the instrument. For leadwire attenuation correction using (12.60) when there is a temperature change after zeroing the instrument, RL and RG must be computed as the resistances at the test temperature. A temperature change in the leadwire system during a test can result in a resistance change of the leadwire that gets added to the resistance change of the gage. Knowing the wire TCR, this effect can be compensated by measuring the temperature change, calculating

305

Part B 12.4

cases, the gage resistance does not match the bridge completion resistors, causing a state of resistive unbalance. If not known to be insignificant, the initial unbalance should be measured. When large enough to warrant consideration, this initial unbalance (expressed in units of strain) must be added algebraically with any subsequent observed strains so that the nonlinearity correction is based on the total unbalance of the Wheatstone bridge. For the next example, assume that when the installed strain gage is connected in a quarter-bridge circuit, the strain indicator displays an initial unbalance of −3726 με. This is an indicated unbalance and, therefore, includes a small nonlinearity error. Substituting as before into the relationship from Table 12.2 for a single active gage, the actual unbalance (in strain units) is ε = −3712 με. With the instrument remaining in this state of unbalance, a specified load is applied to the test object. The instrument strain indication after the applied load is −12 034 με. The calculated correction for this strain indication is −11 891 με. Algebraically adding the corrected initial reading to the corrected final reading, the actual applied strain in the test object is −1891 − (−3712) = −8179 με. As a final example, consider a case where the indicated initial unbalance of the quarter-bridge circuit after installing the strain gage is +2200 με. The test object is then installed into a structure and the indicated reading is −47 500 με. Subsequent loading of the structure produces an indicated reading of −44 500 με, or an apparent load induced strain of +3000 με. What corrections should be made to determine the actual strain caused by loading the structure? Prior to loading the structure, the Wheatstone bridge is unbalanced by an indicated total of −47 500 με. Substituting into the correction expression for a single active gage from Table 12.2, the actual resistive unbalance in strain units prior to loading is ε = −45 346 με. After loading, the indicated unbalance of the bridge is −44 500 με. The calculated correction for this indicated strain yields −42 604 με. The applied strain due to loading the structure is then −42 604 − (−45 346) = +2742 με. This example demonstrates that, even with relatively modest working strains, the nonlinearity error can be significant (about 10% in this case) if the Wheatstone bridge is operated far from the resistive balance condition.

12.4 The Bonded Foil Strain Gage

306

Part B

Contact Methods

Part B 12.4

L1 L3 L2

e0

E

Fig. 12.18 Three-wire system for strain gage connection to the Wheatstone bridge

the change in leadwire resistance at the test temperature, and then computing the equivalent strain value resulting from this resistance change and algebraically subtracting it from the indicated strain reading. The apparent strain εAPL associated with leadwire temperature change can be computed as εAPL =

ΔRL /(RG + RL ) (ΔT β RL )/(RG + RL ) = , Fi Fi (12.62)

where ΔT is the change in leadwire temperature, after zeroing the instrument, β is the leadwire temperature coefficient of resistivity, and Fi is the instrument gage factor setting. To correct for leadwire apparent strain, algebraically subtract the result from (12.62) from the measured strains. Effects from leadwire resistance can be rendered negligibly small by using a three-wire connection to the strain gage, which is represented in Fig. 12.18. This wiring method affords two benefits. First by reducing the offending resistance to only one wire length L 1 , and second by placing equal resistances of leadwire into adjacent arms of the Wheatstone bridge, thus maintaining electrical balance with respect to the leadwire system, even with temperature changes, when the adjacent arm leadwires see these changes equally.

tionally, each is a function of gage construction and geometry, whereby a favorable specification for one can degrade performance in another. Therefore, a good understanding of what can affect sensor performance in each category will allow the user to make an informed choice when picking the best gage for the task. Fatigue Determining structural fatigue life is a common stress analysis measurement. Typically, these measurements constitute cycling the structure sufficiently long and at a sufficiently high mechanical strain level until it breaks. The working life for the structure at that strain level is then set at some fraction of the measured failure life. Like all metals, the sensing grid of a strain gage can break when cyclically strained. However, long before visible damage (cracking or breaking), the gage will exhibit erroneous resistance changes, rendering it unusable as an accurate strain sensor [12.14]. It is the useful measuring life that determines the fatigue resistance of strain gages. The number of cycles sustainable by a particular gage before it begins yielding poor data is commonly referred to as the fatigue life of the gage. When subjected to cold work during cyclic loading, the metal sensing grid in a strain gage will shift from its initial unstrained resistance. This change is permanent and is termed zero shift. Continued cyclic loading will cause small cracks to form in the foil, indicated by spike changes in the resistance as the gage is flexed. This condition is called supersensitivity. Resistance changes caused by zero shift and supersensitivity are inseparable from resistance changes caused by strain. Long before the gage ultimately fails (electrically open) data

12.4.9 Performance Characteristics The preceding sections discussed corrections for typical errors encountered when using strain gages in stress analysis applications. This section discusses strain gage performance characteristics associated with three specific parameters: fatigue, elongation, and reinforcement. Unlike the correctable errors, these parameters are inherent characteristics of the sensors, and can determine applicability to a particular test environment. Addi-

Fig. 12.19 Cracks (circled) in strain-gage foil grid lines, which cause supersensitivity

Bonded Electrical Resistance Strain Gages

Elongation All bonded resistance strain gages are limited in elongation capability by characteristics of the grid alloy, the backing system, and the installation adhesive. Highstrain failure usually occurs through grid cracking,

Cyclic strain level (με) ±5000

Isoelastic ±4000

Modified nickel – chromium (S TC condition)

±3000

±2000 Iron– chromium– aluminum Constantan (S TC condition) ±1000 102

103

104

105

106

107

108

Number of cycles (N)

Fig. 12.20 Representative fatigue life curves for four common strain gage alloys

matrix cracking, adhesive cracking, or loss of bond. Because of close strain coupling, any crack in the specimen surface, adhesive, or matrix will immediately propagate through the thin foil grid. High-performance gages using a glass-fiber-reinforced backing system are usually limited to maximum strains of 1–2% (10 000–20 000 με). Regular STC gages using constantan foil and a compliant polyimide backing system are capable of 5–10% strains (50 000–100 000 με), especially in larger gage lengths (> 3 mm). Certain gages are specifically designed for the measurement of very high (single-cycle) strains. These are often called either high-elongation or post-yield strain gages, and employ fully annealed constantan alloy

Fig. 12.21 Typical strain-gage fatigue failure (cracked solder tab, tangent to solder joint)

307

Part B 12.4

no longer represents what is happening on the surface of the test part. Zero shift manifests itself as an offset in the strain reading. As such, it is normally not of concern for purely dynamic testing, because the peak-to-peak readings remain accurate. However, if the test combines static and dynamic components, zero shift is important because the static results are biased by the shift. Supersensitivity results from cracks forming in the grid area of the foil, often at the transition between the straight grid line and the curved end loop (Fig. 12.19). These cracks open during the tension phase of the loading cycle. Monitoring the strain-gage output on an oscilloscope can display supersensitivity; when present, the waveform will become distorted on the tension loop, usually represented by a spike at the top of the loop. Often, the zero-load gage reading is nearly normal, because the cracks close at no load. If unexpectedly large strain signals appear during a cyclic test, the waveform should be checked for supersensitivity before interpreting the data as a possible failure in the test part. In the light of the preceding considerations, straingage fatigue failure is usually defined as a positive zero shift of either 100 or 300 με, dependent on the application (quasistatic versus purely dynamic, respectively). As would be expected, different strain-gage alloys exhibit differences in fatigue life as a function of their mechanical strength. Representative fatigue life curves for four common strain gage alloys are presented in Fig. 12.20. Each alloy represented in this graph was tested using a high performance strain-gage construction (fully encapsulated, glass-fiber-reinforced epoxy-phenolic resin system), for best performance. It should be noted that fatigue life is affected not only by alloy choice, but also by the strain-gage backing system. Furthermore, no other test environment is more demanding on gage installation technique than fatigue. Thin, unfilled adhesives are the best choices (cyanoacrylate or solventthinned unfilled epoxy). Since the most common fatigue failure mode of a strain gage is a tangent crack in the solder tab, adjacent to the solder joint on the grid side (Fig. 12.21), leadwires and solder connections should be as small as practicable, with the leadwire exiting the tab 90◦ to the direction of principal strain.

12.4 The Bonded Foil Strain Gage

308

Part B

Contact Methods

Part B 12.4

grids (non-STC) on a low-modulus unfilled polyimide film. Using careful installation techniques and a highelongation adhesive, these gages are capable of strains in excess of 20% (200 000 με) before grid failure. This performance requires a smooth defect-free test surface and small, flexible leadwire attachment with correspondingly small solder joints, preferably using a soft solder (e.g., 63% tin, 37% lead). The usual adhesive choice for high-elongation work is a two-component 100%-solids epoxy system. This adhesive can be cured over an extended period (≈ 2 weeks) at room temperature (≈ 24 ◦ C), or in about 2 h at +75 ◦ C. Most of these adhesive systems have some filler to regulate glue-line thickness. Prior to use, large particles should be removed from the mix. Reinforcement Although strain gages appear small and delicate, they are actually fairly stiff. Dependent on backing type, foil type, stacking elements, and encapsulation options, stress analysis gages range in thickness from slightly less than 0.025 mm, to about 0.18 mm. The composite modulus of elasticity (combined grid, backing, and encapsulation if present) usually lies between 5 and 30 GPa for standard organic-backing strain gages. On most metallic specimens of at least a few millimeters thickness, the gage stiffness is negligible, but if the specimen has a low modulus when compared to the gage, and/or is only a few times thicker than the gage, the presence of the installed gage can significantly alter the specimen strain distribution. For example, Fig. 12.22 shows the strain field disruption on a 0.25 mm-thick specimen of fully annealed copper in pure tension, caused by the presence of a post-yield strain gage installed using a high-elongation epoxy adhesive. The copper strain level is approximately 3% at the gage site, but the bulk specimen strain level (outside

Fig. 12.22 Moiré fringe pattern displaying strain field disruption in thin, annealed copper at high elongation, caused by the presence of a strain gage (located on opposite side at center of photo)

the gage area) is approximately 10% higher (3.3% total strain). This reduction of strain level at the gage site is caused by local reinforcement from the strain gage. The deformed fringes, which show lines of constant strain in the moiré pattern, clearly outline the gage boundary, which is bonded to the opposite side of the specimen from where the moiré measurements are taken. Obviously, strain-gage measurements on plastics require consideration of reinforcement. A somewhat subtle concern when making measurements on plastics is the self-heating generated by passing an electrical current through a strain gage. If high enough, the heat generated by the strain gage can soften (reduce the Young’s modulus of) the plastic specimen, thus altering its strain distribution relative to the unheated bulk of the part. Therefore, it is important to use the lowest gage excitation voltage possible (keeping the power density on the gage below 0.03 kW/m2 ) when making measurements on plastics. A general guideline for setting instrument voltage when testing any material with poor heat conduction is to start at a very low power density (≈ 0.01 kW/m2 ) and gradually increase the voltage until the zero reading becomes unstable (drifts or fluctuates), then back off about 5% below the highest value that results in a stable zero reading. A qualitative measure of the impact from reinforcement on any material subjected to plane strain can be calculated as Pr =

tb E b + tf E f + ta E a × 100 , ts E s

(12.63)

where Pr is the percentage reinforcement, tb is the gage backing thickness, tf is the gage foil thickness, ta is the installation adhesive layer thickness, ts is the specimen thickness, E b is the gage backing modulus of elasticity, E f is the foil modulus of elasticity, E a is the adhesive modulus of elasticity, and E s is the specimen modulus of elasticity. Because the accuracy of this calculation is restricted by many assumptions, not the least of which includes the requirement for the specimen and gage to have equal width, the result cannot be used for quantitative assessment. However, if the percentage reinforcement calculated from (12.63) is greater than about 10%, steps should be taken to compensate the strain measurements for this local increase in stiffness. For any low-modulus material including plastics once self-heating is avoided, a common method of compensating room-temperature strain readings is to calibrate the gage (measure gage factor) on a specimen made from the same material which will be used during

Bonded Electrical Resistance Strain Gages

typically make the measurement in a biaxial strain field on a steel beam with a Poisson’s ratio close to 0.28. Although a complete discussion is beyond the scope of the present work, readers are directed to [12.15–17]

Table 12.3 Performance parameters for certain strain gage components (used to facilitate gage selection) Type of test or application

General static or static–dynamic stress analysisa

Highelongation (post-yield)

Cyclic loading

Operating temperature range (◦ C)

−45 to +65

−45 to +205 −270 to +230 < 315 < 700 −45 to +65

−20 to +260 −270 to +260 −75 to +65 −195 to +260

Test duration (h)

< 104 > 104 > 104 > 104 < 103 > 103 > 103 < 102 < 10 < 10 > 103 > 103 > 102 < 103 < 104 < 104 < 104 < 104

Accuracy requiredb

Moderate Moderate Very High High Moderate High Moderate Moderate Moderate Moderate Moderate Moderate Moderate Moderate Moderate Moderate Moderate Moderate

Cyclic endurance required

Typical selection

Maximum stain, με

Gage construction, foil/backing A/P A/P A/G K/G A/G K/G K/G K/G K/G A/P Y/P Y/P A, K/G K/G D/P D/G D/G D/G

±1300 ±1300 ±1600 ±2000 ±1600 ±2000 ±2000 ±1800 ±1500 ±50 000 ±100 000 ±200 000 ±15 000 ±10 000 ±20 000 ±24 000 ±20 000 ±23 000

Number of cycles < 106 < 106 > 106 > 106 < 106 < 106 < 106 < 106 < 106 1 1 1 1 1 107 107 107 < 105

Adhesive No.

Description

1 2 3 4 5

Cyanoacrylate Two-component, 100%-solids epoxy (room- or elevated-temperature cure) Two-component, 100%-solids epoxy (elevated-temperature cure) Two-component, solvent-thinned epoxy-phenolic (elevated-temperature cure) Two-component, 100%-solids epoxy, high elongation (room- or elevated-temperature cure)

Adhesive

1 or 2 2 or 3 3 or 4 3 or 4 4 4 4 4 4 2 3 5 4 4 1 or 2 2 or 3 4 4

Foil/backing A K Y D P G a

Constantan (Cu-Ni) Karma (Ni-Cr) Constantan (Cu-Ni), treated for high-elongation, post-yield performance Isoelastic (Ni-Fe) Cast polyimide Glass-fiber-reinforced epoxy-phenolic

Category includes most testing situations where some degree of stability under static test conditions is required. For absolute stability with constantan gages over long periods of time and temperatures above +65 ◦ C, it may be necessary to employ half- or full-bridge patterns. Protective coatings may also influence stability. b It is inappropriate to quantify accuracy as used in this table without consideration of various aspects of the actual test program and the instrumentation used. In general, “moderate” for stress analysis purposes is in the 2–5% range, “high” in the 1–3% range, and “very high” is 1% or better.

309

Part B 12.4

the actual test, and then use this value for subsequent testing of that material with the same gage types. This also has the added benefit of avoiding the inherent errors associated with gage factor, since manufacturers

12.4 The Bonded Foil Strain Gage

310

Part B

Contact Methods

Part B 12.4

for more information on using strain gages successfully with plastics and other low-modulus materials.

12.4.10 Gage Selection Procedure Proper strain-gage selection is important for achieving reliable and valid data. One shortcoming with strain gages is the ability to relatively easily obtain a reading, even if that reading is meaningless. The only way to avoid useless data is to plan the test meticulously and choose the best gage carefully for the test conditions. Recognizing that compromises are often necessary in gage choice, and utilizing the fundamental gage properties and characteristics thus presented, the goal for this section is to provide a logical checklist that results in the selection of a gage with the greatest chance of success under the established test conditions. When beginning the gage selection process, there are six general considerations:

• • • • • •

type of strain measurement (static, dynamic, high elongation, etc.) strain field operating temperature at gage installation test duration required accuracy required cyclic endurance

Meeting these test considerations requires choice of certain gage design and construction parameters

• • • • • • •

alloy backing material gage length/overall size geometry self-temperature compensation grid resistance options

Performance expectations from several of these parameters have been discussed earlier. The information in Table 12.3 summarizes these expectations and provides the basic means for preliminary selection of the gage for several conventional applications. Also included are recommendations for adhesive, since the adhesive in a strain gage installation becomes part of the gage system, and correspondingly affects the performance of the gage. Of the selectable gage construction parameters, alloy is primary in determining gage operating characteristics. Cyclic endurance, elongation, temperature range, long-term stability, and even cost are a few of the gage characteristics determined by alloy choice.

As previously stated, constantan is the most widely used modern strain-gage alloy. This is true because it has the best overall combination of properties needed for many strain-gage applications. For example, it has an adequately high strain sensitivity, or gage factor, which is relatively insensitive to strain level and temperature. Additionally, its resistivity is high enough to achieve suitable resistance values, even in small grids, and it can be processed for self-temperature compensation to match a wide variety of test materials. Constantan can also be processed (annealed) to survive very large strains (> 20%), when used in gage lengths of 6 mm and longer, and its cyclic endurance is adequate for many test conditions. However, constantan is not the best choice for high-cycle, purely dynamic applications. Another shortcoming of constantan is that it drifts in resistance at temperatures above +65 ◦ C, rendering it a bad choice for good zero stability over a period of hours or days above this temperature. The second most widely used strain gage alloy, karma, is characterized by good fatigue life and excellent stability; it is the preferred choice for accurate static strain measurements over long periods of time (months or years) at room temperature, or shorter periods at elevated temperature. It is recommended for extended static strain measurements over the temperature range from −270 to +260 ◦ C. For short periods (a few minutes), an encapsulated karma-alloy strain gage can operate reliably when exposed to temperatures as high as +400 ◦ C. An inert atmosphere can improve stability and prolong useful life at these high temperatures. Karma has a higher resistivity than constantan and can therefore achieve higher gage resistances in similarly sized patterns, or the same resistances in smaller patterns. Like constantan, karma alloy can be selftemperature compensated for many common structural materials, and has a flatter thermal output curve, permitting more accurate correction for thermal output errors at temperature extremes. One drawback to the alloy is solderability. Either a plating feature (nickel or copper are most common) or acid solder flux must be used to attach leadwires. This is because, after cleaning/abrading the solder tabs, the alloy almost instantaneously forms a protective surface corrosion, which inhibits solder wetting. When making purely dynamic strain measurements – that is, when it is not necessary to maintain a stable reference zero – a nickel–iron–chrome alloy called isoelastic offers advantages. Principal among these are superior fatigue life, when compared with constantan or karma, and a high gage factor (≈ 3.2), which improves

Bonded Electrical Resistance Strain Gages

sonable accuracy (better than ±5◦ ). Of course, multiple single-grid gages can be installed at desirable relative angles and spacings to solve many stress analysis problems, but most gage manufacturers supply multigrid gages on a common backing to simplify the installation process. A good example is the dual-grid pattern where the grids are ±45◦ to the axis of the gage. When attached to a specimen and connected as a summing half-bridge, the resultant reading from the gage is equal to the shear strain in the specimen along the direction of the gage axis. For the most general measurement case, when the specimen stress state and principal axes are unknown, a three-element rosette should be used to determine the principal stress magnitudes and directions. When using a rosette, consideration should be given to the difference in measurement characteristics between single-plane and stacked rosettes. For any given gage length, the single-plane rosette is superior to the stacked rosette in terms of heat transfer to the test specimen, generally providing better stability and accuracy for static strain measurements. Furthermore, when there is a significant strain gradient perpendicular to the test surface (as in bending), the single-plane rosette will produce more accurate strain data because all grids are as close as possible to the specimen surface. Additionally, single-plane rosettes are more conformable than stacked rosettes. However, when there is a large strain gradient in the plane of the test surface, as is often the case, the single-plane rosette can produce errors in strain indication because the grids sample the strain field at different points. The stacked rosette is most advantageous in this case (but can result in a compromise with neutral-axis shifts for bending strains), and also when there is limited space for attaching the gage. Strain Peak strain

Indicated strain

x

Fig. 12.23 Reduction in indicated strain caused by strain averaging along the gage length

311

Part B 12.4

signal-to-noise ratio in dynamic testing. Isoelastic is not subject to self-temperature compensation and its thermal output is high (about 145 με/◦ C), typically making it unusable for static strain measurement. When possible, use of a half- or full-bridge configuration can reduce the temperature response. Isoelastic is magnetoresistive, which introduces another error source in high magnetic fields. Also, its gage factor is nonlinear, becoming significantly so at strains beyond ±5000 με. The strain-gage backing serves several important purposes. Foremost, it provides a way of stabilizing the delicate metal pattern and offers a means for handling the sensor during installation. The backing also affords a readily bondable surface for adhering the gage to a specimen and provides electrical insulation between the metal pattern and electrically conductive specimens. Good backing materials also allow excellent strain transmission into the grid of the gage, which supports good fidelity between specimen strain and gage reading. The two most popular strain gage backing materials are polyimide (cast or film) and glass-fiber-reinforced epoxy. Cast polyimide offers excellent flexibility and toughness and can easily be contoured to fit small radii (≈ 0.5 mm). With its typical operating temperature of −195 to +175 ◦ C, polyimide is an ideal choice for general-purpose static and dynamic stress analysis. It is also capable of high elongations, and can be used to measure plastic strains in excess of 20%. For higher-temperature operation and improved stability, cast polyimide can be reinforced with glass fibers, but flexibility and elongation are reduced. Higher-modulus film polyimide has slightly less flexibility and elongation, but has excellent long-term stability, partly as the result of lower moisture absorption, when compared with cast polyimide. Glass-fiber-reinforced epoxy offers outstanding performance over the widest range of temperatures. This backing can be used for static and dynamic strain measurement from −270 to +400 ◦ C. However, its flexibility is limited and the backing cannot be conformed over small radii (< 5 mm) or survive large elongation (> 2%). The pattern geometry, including gage length, is most often dictated by the strain field that the strain gage will sense. Many different patterns are commercially available to solve specific test problems with minimum effort and complication. For example, a single-grid gage would normally be used only when the stress state at the point of measurement is known to be uniaxial and the directions of the principal axes are known with rea-

12.4 The Bonded Foil Strain Gage

312

Part B

Contact Methods

Part B 12.4

Table 12.4 Checklist for strain gage selection

Considerations for parameter selection Selection step: 1 Parameter: Gage length

Selection step: 2 Parameter: Gage pattern

Selection step: 3 Parameter: Gage series

Selection step: 4 Parameter: Options

• • • • • • • • • • • • • • • •

• • • • • • • • •

Selection step: 5 Parameter: Gage resistance Selection step: 6 Parameter: S-T-C number

• • • • • • • •

Strain gradients Area of maximum strain Accuracy required Static strain stability Maximum elongation Cyclic endurance Heat dissipation Space for installation Ease of installation Strain gradients (in-plane and normal to surface) Biaxiality of stress Heat dissipation Space for installation Ease of installation Gage resistance availability Type of strain measurement application (static, dynamic, post-yield, etc.) Operating temperature Test duration Cyclic endurance Accuracy required Ease of installation Type of meassurement (static, dynamic, post-yield, etc.) Installation environment – laboratory or field Stability requirements Soldering sensitivity of substrate (plastic, bone, etc.) Space available for installation Installation time constraints Heat dissipation Leadwire desensitization Signal-to-noise ratio Test specimen material Operating temperature range Accuracy required

Since a strain gage reports the average strain reading from the area of its grid (Fig. 12.23), gage length and width must always be carefully considered relative to the expected strain gradients. For steep gradients, such as those found near stress concentrations, a grid that is large compared to the slope of the gradient will report significantly lower strain values than the peak strains in the specimen. As a rule of thumb, when practicable, the gage length should be no greater than 0.1 times the radius of a hole, fillet, or notch, or the corresponding dimension of any other stress raiser near which the gage will be installed. Obviously, for even a moderately small stress raiser, say, 12 mm, this rule will lead to a very small, if not impractical, gage size – another example where compromise is necessary. In the opposite case, for nonhomogeneous materials such as concrete, long gage lengths (> 100 mm) are used to span several periods of any repeating feature (aggregate, in the case of concrete), so that the bulk material strain can be measured, not a locally perturbed value biased by the repeating feature. Using the above considerations a checklist can be produced, which is shown in Table 12.4. While not allinclusive, this checklist is a good starting point for the gage selection process. Simply check all bullets that are relevant at each selection step, and then with consideration to available gages, determine which considerations must be compromised to choose the most likely gage that will be successful in achieving the desired test results.

12.4.11 Specific Applications The following sections present and discuss specific measurement applications and the best use of foil strain gages in solving these problems, resulting in the collection of the best experimental data. These applications will demonstrate many of the techniques, corrections, and characteristics presented in earlier sections. Plane Shear Measurement A conventional, linear, strain gage does not respond to pure shear strain as such, since shear strain simply changes the shape of the gage without altering the lengths of the grid filaments. However, shear and normal strains are related through the laws of elasticity; and it is thus possible, by proper orientation of the gages on the strained surface, and proper arrangement of the gages in a Wheatstone bridge circuit, to produce an instrument indication that is directly proportional to the shear strain in the surface. First, a theoretical basis for deriving shear strains from the normal strain

Bonded Electrical Resistance Strain Gages

12.4 The Bonded Foil Strain Gage

2

θ2 1

θ1 X

(12.68)

Fig. 12.24 Arbitrarily oriented strain gages in a biaxial

strain field

It is thus evident that, if the gage axes are oriented symmetrically with respect to, say, the X-axis (Fig. 12.25), θ1 = −θ2 = α

measurements is presented, and then several practical applications of the procedure are given.

and γ XY = −

Shear Strains From Normal Strains. Consider an array

of two strain gages oriented at arbitrarily different angles with respect to an X–Y coordinate system, which, in turn, is arbitrarily oriented with respect to the principal axes, as in Fig. 12.24. From elementary mechanics of materials, the strains along the gage axes can be written as ε X + εY ε X − εY γ XY + cos 2θ1 + sin 2θ1 , ε1 = 2 2 2

ε1 − ε2 ε1 − ε2 ε1 − ε2 = = . sin 2θ2 sin 2θ1 sin 2α

a) Y

X 2 1

1

(12.64)

ε2 =

(12.69)

ε X + εY ε X − εY γ XY + cos 2θ2 + sin 2θ2 . 2 2 2 (12.65)

Subtracting (12.65) from (12.64) and solving for γ XY gives γ XY =

2(ε1 − ε2 ) − (ε X − εY )(cos 2θ1 − cos 2θ2 ) . sin 2θ1 − sin 2θ2 (12.66)

2

b) 1 4

Y

1

2

4

3

2 3

2

α X

α 1

Fig. 12.25 Defining the X-axis as the bisector of the angle

between the gage axes

Fig. 12.26a,b Strain-gage shear bridges, with corresponding Wheatstone bridge connection: (a) 90◦ rosette for direct indication of shear strain, and (b) full shear bridge, typically for force measurement

Part B 12.4

It is now noticeable that if cos 2θ1 ≡ cos 2θ2 the term in ε X and εY vanishes, and 2(ε1 − ε2 ) . (12.67) γ XY = sin 2θ1 − sin 2θ2 Since the cosine function is symmetrical about the zero argument and about all integral multiples of π, cos 2θ1 ≡ cos 2θ2 when,       π π nπ θ1 + α = − , (0), , = θ2 − α . 2 2 2

Y

313

314

Part B

Contact Methods

Part B 12.4

The preceding results can be generalized as follows: The difference in normal strain sensed by any two arbitrarily oriented strain gages in a uniform strain field is proportional to the shear strain along an axis bisecting the strain gage axes, irrespective of the included angle between the gages. When the two gages are 90◦ apart, the denominator of (12.69) becomes unity and the shear strain along the bisector is numerically equal to the difference in normal strains. Thus, a 90◦ , two-gage rosette constitutes an ideal shear half-bridge because the required subtraction ε1 − ε2 is performed automatically for two gages in adjacent arms of the Wheatstone bridge circuit. When the gage axes of a two-gage 90◦ rosette are aligned with the principal axes, (Fig. 12.26a), the output of the half-bridge is numerically equal to the maximum shear strain. A full shear-bridge (with twice the output signal from the circuit) is then composed of four gages as shown in Fig. 12.26b. The gages may have any of several configurations, including the cruciform arrangement and the compact geometry illustrated in the figure. Principal Strains. It should be kept in mind that, with

the shear bridges described above, the indicated shear strain exists along the bisector of any adjacent pair of gage axes, and it is not possible to determine the maximum shear strain or the complete state of strain from any combination of gage outputs unless the orientation of the gage axes with respect to the principal axes is known. In general, when the directions of the principal axes are unknown, a three-gage 45◦ rectangular rosette can be used. Referring to Mohr’s circle for strain (Fig. 12.27), it is apparent that the two shaded triangles are always

–

γ 2

3 B 2

45°

A

22.5° 1

Fig. 12.28 45◦ rosette used to determine γA and γB , the shear strains in the A and B directions, respectively

identical for a 45◦ rosette, and therefore the maximum shear strain is equal to the vector sum of the shear strains along any two axes, which are 45◦ apart on the strained surface. Looking at the 45◦ rosette as shown in Fig. 12.28, it can be seen that the shear strains along the bisectors of the gage pairs 1–2 and 2–3 are in fact 45◦ apart, and thus the maximum shear strain is

γmax = γA2 + γB2 and, considering (12.69),      ε2 − ε3 2 ε1 − ε2 2 γmax = + sin 45◦ sin 45◦ or, γmax =

√  2 (ε1 − ε2 )2 + (ε2 − ε3 )2

(12.70)

ε1 ε2 2 1

2θ

2θ ε

0 ε3

3

ε1 +ε3 2

ε1 – ε3 2

ε1 +ε3 – ε2 2

Fig. 12.27 Mohr’s circle for strain used to determine maxi-

mum shear strains

Fig. 12.29 Setup for shear-buckling testing of a composite panel (courtesy of the National Research Council, Canada)

Bonded Electrical Resistance Strain Gages

d

0° fibers

w h

90° fibers R

d

L L h d R w

= 76.2 mm = 19.05 mm = 3.81 mm = 1.27 mm = 11.43 mm

Thickness = 2.54 – 5.08 mm (12.7 mm max)

Fig. 12.30 Schematic of v-notch (Iosipescu) specimen

used for shear modulus testing of composite materials

and, from Mohr’s circle again, the principal strains are obviously εp , εq =

ε1 + ε3  ± (ε1 − ε2 )2 + (ε2 − ε3 )2 . 2 (12.71)

Shear Measurement Applications. As described in the preceding material, shear strains are not measured directly, but inferred from the measurement of two normal strains at different angular orientations. An increasingly prominent area of shear strain determination occurs in research on shear buckling of thin panels of composite materials employed in satellites and other aerospace equipment generally. In such cases, it is always necessary to minimize the weight of the panel, while avoiding the danger of catastrophic failure by shear buckling. Figure 12.29 shows a small portion of a composite panel in a picture-frame loading fixture preparatory to undergoing a shear-buckling test. Shear strain magnitudes are obtained from the two centrally located strain gages. Since composite materials are not yet as standardized as the common structural metals, there is a frequent need for measuring basic material properties such as shear strength and shear modulus. The ASTM has developed a number of tests for evaluating these properties in composites (ASTM Annual Book of Standards,

a)

P A

a)

y Average A 0° fibers Conventional tee-rosette

b)

T C T

γxy

b)

y A–A

Average

C Bending stress

Shear stress

90° fibers

c)

T

C Front Rear

γxy

Fig. 12.31a,b Shear strain distributions for (a) 0◦ and (b) 90◦ unidirectionally reinforced graphite/epoxy speci-

mens (after Ifju [12.18])

C

T

Fig. 12.32a–c Schematics showing (a) the principal of the shear-web load cell, with (b) stress distribution and (c) circuit arrangement

315

Part B 12.4

90°

12.4 The Bonded Foil Strain Gage

316

Part B

Contact Methods

Part B 12.4

Volume 15.03). The goal has always been to achieve a small, inexpensive, easily produced specimen, which is characterized by a state of pure shear in its test zone. The V-notch (Iosipescu) specimen, shown in Fig. 12.30, initially appeared promising for this purpose [12.18]. However, it was soon found that the Iosipescu specimen did not produce reliable, material-independent results [12.19]. The reason for this outcome is seen in Fig. 12.31, which illustrates that the shear strain distribution varies across the test zone dependent on the fiber direction in unidirectionally reinforced composite materials. The strain gage will average the part of this distribution that it covers [12.20]. Therefore, different average shear strain readings are obtained dependent on the gage length. The solution to this problem was to design a shear gage specific to the specimen geometry such that both grids cover the full test zone height (from edge to edge). Since the average shear strain over the full test zone must be equal with either fiber direction, the new gages produced the same average shear strains regardless of fiber orientation. Probably the widest use of shear measurement is one in which the user does not care what the shear strain is at all, because he is measuring force or load that is proportional to shear strain. The reference here is to what are commonly called shear-web transducers, a schematic of which is shown in Fig. 12.32a, along with the corresponding stress distribution in the transducer (Fig. 12.32b) and the circuit arrangement (Fig. 12.32c). Such transducers are often made in the form of cantilever beams with recesses milled into both sides of the beam to leave a thin web, which is instrumented with two strain gages on either side, oriented to produce an output proportional to the shear strain. Shear-web transducers offer several advantages to the transducer designer. They are characteristically compact and robust, and capable of sensing high loads in minimal space. Furthermore, the transducer response is independent of the point of load application, because the vertical shear force is necessarily the same at every section of the beam to the left of the recess. Primarily because of the limited beam deflection, shear-web transducers also tend to exhibit very good linearity. Measurement of Thermal Expansion Coefficient The thermal expansion coefficient is a basic physical property of a material, which can be of considerable importance in mechanical and structural design applications. Although there are many published tabulations of expansion coefficients for the common metals and stan-

dard alloys, quite frequently the need arises to measure this property for a specific material over a particular temperature range. In some cases (e.g., composite materials, new or special alloys, etc.) there is apt to be no published data whatsoever on expansion coefficients. In others, data may exist (and eventually be found), but may encompass the wrong temperature range, apply to a somewhat different material, or be otherwise unsuited to the application. Dilatometry is the classical method for measuring thermal expansion, in which a dilatometer is used to determine the relative expansion between a test specimen and reference material. In this type of instrument, the difference in expansion between a rod made from the test material and a length of quartz or vitreous silica is compared. Differential expansion between the two materials is measured with a sensitive dial indicator or with an electrical displacement transducer, or some applicable optical method. When necessary, the expansion properties of the quartz or silica can be calibrated against the accurately known expansion of pure platinum or copper. The instrument is typically inserted in a special tubular furnace or liquid bath to obtain the required uniform temperatures. Making expansion measurements with the dilatometer is a delicate, demanding task, and is better suited to the materials science laboratory, rather than the typical experimental stress analysis facility. Given below is an alternate method for easily and quite accurately measuring the expansion coefficient of a test material with respect to that of a reference material having known thermal expansion characteristics. The technique described here uses two wellmatched strain gages, with one bonded to a specimen of the reference material, and a second to a specimen of the test material. The specimens can be of any size or shape compatible with the available equipment for heating and refrigeration (but specimens of uniform cross-section will minimize potential problems with temperature gradients). Under stress-free conditions, the differential output between the gages on the two specimens, at any common temperature, is equal to the differential unit expansion (in/in or m/m). Aside from the basic simplicity and relative ease of making thermal expansion measurements by this method, it has the distinct advantage of requiring no specialized instruments beyond those normally found in a stress analysis laboratory. This technique can also be applied to the otherwise difficult task of determining directional expansion coefficients of composite materials with anisotropic thermal properties.

Bonded Electrical Resistance Strain Gages

ΔL /L (ppm) 8

Thermal expansion: –80°C to +150°C 6 4

Principle of the Measurement Method. When a resis-

tance strain gage is installed on a stress-free specimen of any material, and the temperature of the specimen is changed, the electrical resistance of the strain gage changes correspondingly. As noted previously, this effect is referred to as the thermal output of the strain gage. The magnitude of the thermal output, for a particular strain gage type, is dependent on only two variables: the magnitude of the temperature change, and the difference between two thermal expansion coefficients (that of the strain gage, and that of the material on which the gage is bonded). Expressed mathematically  βG + (αS − αG ) ΔT , (12.72) εT(G/S) = FG where εT(G/S) is the thermal output of the gage, βG /FG is the temperature coefficient of resistance of the strain gage alloy, divided by gage factor, (αS − αG ) is the difference in expansion coefficient between the specimen material and the gage, and ΔT is the temperature change. Similarly, when the same type of strain gage is installed on a reference material of known thermal expansion coefficient,  βG + (αR − αG ) ΔT . (12.73) εT(G/R) = FG Subtracting (12.73) from (12.72) and rearranging, αS − αR =

(εT(G/S) − εT(G/R) ) . ΔT

(12.74)

Thus, the difference in expansion coefficients between the reference material and the test material, referred to a particular temperature range, is equal to the change in thermal output for the same change in temperature. Obvious from the derivation of (12.74) is that the output of the reference gage not only serves as a relative point from which thermal expansion can be measured, but the gage also serves as a compensating dummy to the specimen gage, thus precisely eliminating the TCR contribution from the gage.

2 0 –2 –80 –60 –40 –20

0

–20 –40 –60 –80 –100 –120 –140

Temperature (°C)

Fig. 12.33 Thermal expansion characteristics of titanium

silicate reference material (courtesy Corning Inc.) Measurement Procedures. Selection of the material

used as a reference standard is an important factor in the accuracy of this method, as it is for any form of differential dilatometry. In principle, the reference material could be almost any substance for which the expansion properties are accurately known over the temperature range of interest. In practice, however, it is often advantageous to select a material with expansion properties as close to zero as possible. Doing so will provide an output signal that closely corresponds to the absolute expansion coefficient of the test material, and permits a more straightforward test procedure. The thermal expansion of the reference material should also be highly repeatable and stable with time at any constant temperature. In addition, the elastic modulus of the material should be high enough that mechanical reinforcement by the strain gage is negligible. An excellent reference material with these and other desirable properties is ultra low expansion (ULE) titanium silicate, code 7972, available from Corning Incorporated, Corning, NY. As illustrated in Fig. 12.33, this special glass has a low thermal expansion coefficient, particularly over the temperature range from about −45 to +75 ◦ C. It should be noted, however, that the material also has a low coefficient of thermal conductivity, making it slow to reach thermal equilibrium. For optimum results, a dwell time of at least 45 min should be used at each new temperature point before taking data. Another potential disadvantage of titanium silicate as a reference material is its brittleness, since it will fracture readily if dropped on a hard surface. Because of the foregoing, a low expanding metal such as Invar may offer a preferred alternative.

317

Part B 12.4

Because typical expansion coefficients are measured in terms of a few parts per million per degree of temperature change, close attention to procedural detail is required with any measurement method to obtain accurate results; and the strain gage method is no exception to this rule.

12.4 The Bonded Foil Strain Gage

318

Part B

Contact Methods

Part B 12.4

a) Test material 1 2 3

E e0

1 2 3

Reference material

b) Test material 1

E 2

e0

3

Reference material

Fig. 12.34a,b Strain-gage circuits for measuring thermal expansion coefficients; (a) separate quarter-bridge circuits; (b) half-bridge cir-

cuit

In general, 350 Ω gages are preferable for this test to minimize self-heating by the excitation current. The gages should be of the same type, and should be from the same manufacturing lot for closely related thermal output characteristics. When expansion measurements must be made over an extended temperature range, or at high or low temperature extremes, the STC number of the gages should be carefully selected to obtain the best measurement accuracy. With excessive mismatch between the STC number of the gage and the expected thermal expansion of the specimen, the slope of the thermal output curve can become very steep at one or both temperature extremes. Judicious selection of the STC mismatch can be used to simultaneously keep the shapes of the thermal output curves for both the test and the reference materials under reasonably good control over the expected temperature range. The circuit arrangements shown in Fig. 12.34 offer the greatest convenience in reading the difference in thermal outputs required by (12.74). The two gages are connected in adjacent arms of a Wheatstone bridge circuit to form a half bridge, and the circuit output is a direct reading of the difference in thermal outputs. It is important that the leadwires be as short as possible, and the same length and wire size. It is also necessary that

Strain-Gage Selection and Measurement Circuits.

In principle, any standard strain gage of 3 mm length or longer, and with a backing material suitable for the expected temperature range, should suffice for this purpose. It should be remembered, however, that the procedure described here requires a pair of wellmatched strain gages: one for the test material, and one for the reference material.

–

γ ,τ 2

ε(φ– β) εφ ε(φ+ β)

+2β –2β

ε(φ – β)

2φ

εφ ε(φ+β) +β

Y

ε,σ

0 –β φ

Strain circle

X

Stress circle

εq(–) 1 Rε = – ν

σp

εp(+)

εq

εp

Fig. 12.35 Polar plot of strain distribution at a point in uniaxial

Fig. 12.36 Mohr’s circles of stress and strain for uniaxial

stress

stress

Bonded Electrical Resistance Strain Gages

Errors Due to Misalignment – Single Gage in a Uniform Biaxial Strain Field When a strain gage is installed on a test surface at a small angular error with respect to the intended axis of strain measurement, the indicated strain will be in error due to this gage misalignment. In general, for a single gage in a uniform biaxial strain field, the magnitude of the misalignment error depends upon three factors (ignoring transverse sensitivity):

1. the ratio of the algebraic maximum to the algebraic minimum principal strains εp /εq 2. the angle φ between the maximum principal strain axis and the intended axis of strain measurement 3. the angular gage-mounting error β between the gage axis after bonding and the intended axis of strain measurement These quantities are defined in Figs. 12.35 and 12.36 for the particular but common case of the uniaxial stress state. Figure 12.35 is a polar diagram of the strain at the point in question, and Fig. 12.36 illustrates the concentric Mohr’s circles for stress and strain at the same point. In Fig. 12.35, the distance to the boundary of the diagram along any radial line is proportional to the normal strain magnitude along the same line. The small lobes along the Y -axis in the diagram represent the negative Poisson strain for this case. It can be seen qualitatively from Fig. 12.35 that, when φ is zero or 90◦ , a small angular misalignment of the gage will produce a very small error in the strain indication, since the polar strain diagram is relatively flat and passing through zero slope at these points. However, for angles between 0 and 90◦ , Fig. 12.35 shows that the error in indicated strain due to a small angular misalignment can be surprisingly large because the slope of the polar diagram is quite steep in these regions. More specifically, it can be noticed from Fig. 12.35 that, when φ = 45◦ or 2φ = 90◦ , the same small angular misalignment will produce the maximum error in indicated strain, since ε is changing most rapidly with angle at this point. Writing the analytical expression for the polar strain diagram, and setting the second derivative equal to zero to solve for the angle at which the maximum slope occurs would obtain the same result. In fact, the general statement can be made that in any uniform biaxial strain field the error due to gage misalignment is always great-

est when measuring strain at 45◦ to a principal axis, and is always least when measuring the principal strains. The error in strain indication due to angular misalignment of the gage in a uniform biaxial strain field can be expressed as follows n = εφ±β − εφ ,

(12.75)

where n is the error in με, εφ±β is the strain on gage axis with angular mounting error ±β in με, and eφ is the strain on axis of intended measurement at angle φ from principal axis in με, or εp − εq (12.76) [cos 2(φ ± β) − cos 2φ] , n= 2 where εp and εq are the maximum and minimum principal strains, respectively. The error can also be expressed as a percentage of the intended strain measurement εφ εφ±β − εφ n = , (12.77) εφ Error in strain indication (με) β = –10°

200 –8°

– 6°

100

– 4°

– 2° – 1°

0 1° 2° 4°

–100 6°

8°

Uniaxial stress

–200

εp = 1000 με εq = –285 με

10°

0

10

20

30

40

50

60 70 80 90 Gage angle φ (deg)

Fig. 12.37 Error in indicated strain due to gage misalignment for the special case of a uniaxial stress field in steel

319

Part B 12.4

leadwire 2 be connected at the midpoint of the jumper between the gages so that half of the resistance of that wire is in each bridge arm.

12.4 The Bonded Foil Strain Gage

320

Part B

Contact Methods

Part B 12.4

which can be rewritten as n =

cos 2(φ ± β) − cos 2φ Rε +1 Rε −1

β

× 100 ,

Misaligned rosette axes (12.78)

ε

where Rε = εpq . However, from (12.77) it can be seen that n  becomes unmeaningfully large for small values of εφ , and infinite when εφ vanishes. In order to better illustrate the order of magnitude of the error due to gage misalignment, (12.76) is evaluated for a more-or-less typical case Given: Assume: Then, and

εq = −νp , ν = 0.285 . εp = 1000 με . εq = −285 με n = 642.5[cos 2(φ ± β) − cos 2φ] . (12.79)

Equation (12.79) is plotted in Fig. 12.37 over a range of φ from 0 to 90◦ , and over a range of mounting errors from 1 to 10◦ . In order to correct for a known misalignment by reading the value of n from Fig. 12.37, it is only necessary to solve (12.75) for εφ and substitute the value of n, including the sign, as given by Fig. 12.37. This figure is provided only as an example, and applies only to the case in which εq = −0.285 εp (uniaxial stress in steel). Equation (12.76) can be used to develop similar error curves for any biaxial strain state. Errors due to Misalignment – Two-Gage Rectangular Rosette While the above analysis of the errors due to misalignment of a single gage may help in understanding the nature of such errors, the 90◦ , two-gage (tee) rosette is of considerably greater practical interest. A tee rosette is ordinarily used by stress analysts for the purpose of determining the principal stresses when the directions of the principal axes are known from other sources, such as failure histories. In this case, the rosette should be bonded in place with the gage axes coincident with the principal axes. Whether there is an error in orientation of the rosette with respect to the principal axes, or in the directions of the principal axes themselves, there will be a corresponding error in the principal stresses as calculated from the strain readings. In Fig. 12.38, a general biaxial strain field is shown, with the axes of a tee rosette, misaligned by the angle β, superimposed. The percentage error in the principal stresses and maximum shear stress due to the misalign-

β

Principal axes

Fig. 12.38 Biaxial strain field with rosette axes misaligned

by the angle β from the principal axes

ment are σˆ − σp × 100 , σp (1 − Rε )(1 − ν)(1 − cos 2β) np = × 100 , 2(Rε − ν) σˆ − σq nq = × 100 , σq (Rε − 1)(1 − ν)(1 − cos 2β) nq = × 100 , 2(1 + ν Rε ) τˆmax − τmax nτ = × 100 , τmax n τ = −(1 − cos 2β) × 100 , n=

(12.80) (12.81) (12.82) (12.83) (12.84) (12.85)

where σˆ p , σˆ q , and τˆmax are the principal stresses and maximum shear stress inferred from the indicated strains when the rosette is misaligned by the angle β, and Rε = εp /εq is the ratio of the algebraic maximum to the algebraic minimum principal strain, as before. When the principal strain ratio is replaced by the principal stress ratio, where σp Rε + ν = (12.86) Rσ = 1 + ν Rε σq or, εp Rσ − ν Rε = = , (12.87) 1 − Rσ εq then 1 − Rσ (1 − cos 2β) × 100 , (12.88) 2Rσ Rσ−1 nq = (12.89) (1 − cos 2β) × 100 . 2 Equations (12.85), (12.88), and (12.89) will next be applied to an example in order to demonstrate the magnitudes of the errors encountered. np =

Bonded Electrical Resistance Strain Gages

σp /σq = 2

6 5 4 3

rσq

2 1 0 –1

rσp

–2 –3 –4

n τmax

–5 –6 –7 –8

0

1

2

3

4

5 6 7 8 9 10 Rosette mounting error β (deg)

Fig. 12.39 Error in calculated principal stresses and maximum shear stress for a biaxial stress field with σp /σq = 2.0 caused by angular misalignment of a tee rosette

Consider first a thin-walled cylindrical pressure vessel. In this case, the hoop stress is twice the longitudinal stress, and of the same sign. Thus σp = Rσ = 2 , (12.90) σq

where σˆ p is the maximum principal stress as calculated from the gage readings, σˆ q is the minimum principal stress as calculated from gage readings, and τˆmax is the maximum shear stress as calculated from τˆmax = (σˆ p + σˆ q )/2. While the errors in the above case were very small, this is not true for stress fields involving extremes of Rσ . In general, n p becomes very large for |Rσ |  1.0, as does n q for |Rσ | 1.0. The error in the shear stress is independent of the stress state. The preceding generalities can be demonstrated by extending the previous case of the pressurized cylinder. Consider an internally pressurized cylinder with an axial compressive load applied externally to the ends. If, for example, the load were 0.8πr 2 p, where r is the inside radius of the cylinder and p is the internal pressure, the principal stress ratio would become Rσ = 10. Then, from (12.88) and (12.89), n p = −0.45(1 − cos 2β) × 100 , n q = 4.5(1 − cos 2β) × 100 .

(12.88b)

In this case, a 5◦

error in mounting the rosette produces a 0.68% error in σp and a 6.75% error in σq .

and (12.85), (12.88), and (12.89) become n τ = −(1 − cos 2β) × 100 , n p = −1/4(1 − cos 2β) × 100 , n q = 1/2(1 − cos 2β) × 100 .

(12.87b)

Rectangular (12.84a)

(12.88a)

3

2

2

1 1

Equations (12.84a), (12.87a), and (12.88a) are plotted in Fig. 12.39. From the figure, it can be seen that the errors introduced by rosette misalignment in this instance are quite small. For example, with a 5◦ mounting error, τmax , σp , and σq are in error by only − 1.5%, − 0.38%, and + 0.75%, respectively. In order to correct for a known misalignment by reading the value of n from Fig. 12.39, or any similar graph derived from the basic error equations (12.81), (12.83), (12.85), (12.88), and (12.89), it is only necessary to solve (12.80), (12.82), and (12.84) for σp , σq ,

3

3

(12.87a)

1

2

Delta 3

3 3

2

1 1

2

2

3

2 1

1

Fig. 12.40 Geometrically different, but functionally equiv-

alent, configurations of rectangular and delta rosettes

321

Part B 12.4

and τmax , respectively, and substitute the value of n from Fig. 12.39, including the sign. That is, σˆ p σp = (12.91) np , 1 + 100 σˆ q σq = (12.92) nq , 1 + 100 τˆmax τmax = (12.93) nτ , 1 + 100

Stress error (%) 8 7

12.4 The Bonded Foil Strain Gage

322

Part B

Contact Methods

Part B 12.4

The errors defined and evaluated in the foregoing occur, in each case, due to misalignment of a single gage or of an entire tee rosette. The effect of misalignment among the individual gages within a rosette is not treated in this discussion. Strain-Gage Rosettes A strain-gage rosette is, by definition, an arrangement of two or more closely positioned gage grids, separately oriented so as to measure the strains along different directions in the underlying surface of a test part. Rosettes are designed to serve a very practical purpose in experimental stress analysis. It can be shown that for the not uncommon case of the general biaxial stress state, with the principal directions unknown, three independent strain measurements (in different directions) are required to determine the principal strains and stresses. Even when the principal directions are known in advance, two independent strain measurements are needed to obtain principal strains and stresses. For the purpose of meeting these requirements, strain-gage manufacturers typically offer three basic types of strain gage rosettes, each in a variety of forms:

• • •

Tee – two perpendicular grids Rectangular – three grids, with the second and third grids angularly displaced from the first grid by 45 and 90◦ , respectively Delta – three grids, with the second and third grids 60 and 120◦ away, respectively, from the first grid

As illustrated in Fig. 12.40, rectangular and delta rosettes may appear in any of several geometrically different, but functionally equivalent, forms. It must be appreciated that, while the use of a strain gage rosette is in many cases a necessary condition for obtaining the principal strains, it is not a sufficient condition for doing so accurately. Knowledge in the selection and application of rosettes is crucial to their successful use in stress analysis; the following information is intended to be helpful in this process. Rosette Selection Considerations. In addition to the

usual parameters that must be considered in the selection of any strain gage (i. e., the strain-sensitive foil, backing material, self-temperature-compensation number, gage length, etc.) two other parameters are important in rosette selection (1) the rosette type (tee, rectangular, or delta) and (2) the rosette construction (planar or stacked). The tee rosette can be used only when the principal directions are known in advance from other considera-

tions. Cylindrical pressure vessels and shafts in torsion are two classical examples. However, care must be exercised in all such cases that extraneous stresses (bending, axial stress, etc.) are not present, since these will affect the directions of the principal axes. Attention must also be given to nearby geometric irregularities (holes, ribs, shoulders, etc.) that can locally alter the principal directions. When necessary (and, using the proper datareduction relationship), the tee rosette can be installed with its axes at any known angle from the principal axes, but greatest accuracy will be achieved when alignment is along the principal directions. In the latter case, ignoring the error due to transverse sensitivity, the two gage elements in the rosette indicate the corresponding principal strains directly. The principal strains can also be expressed in the following form, which parallels those given later for rectangular and delta rosettes εp,q =

ε1 + ε2 1  (ε1 − ε2 )2 . ± 2 2

(12.94)

When the directions of the principal strains are unknown, a three-element rectangular or delta rosette is required, and the rosette can be installed without regard to orientation. Functionally, there is little difference between the rectangular and delta rosettes. Because the gage axes in the delta rosette have the maximum possible uniform angular separation (effectively 120◦ ), this rosette is assumed to yield the optimum sampling of the underlying strain distribution. Rectangular rosettes have historically been the more popular of the two, primarily because the data-reduction relationships are somewhat simpler. Currently, however, with widespread access to computers and programmable calculators, the computational advantage of the rectangular rosette is of little consequence. As a result of the foregoing, the choice between rectangular and delta rosettes is apt to be based on practical considerations such as availability from stock, convenience of solder tab arrangement, and other pragmatic considerations. Data-reduction relationships for the two types of rosettes are as follows Rectangular: εp,q =

ε1 + ε3 1  ± √ (ε1 − ε2 )2 + (ε2 − ε3 )2 , 2 2 

φp,q =

1 (ε2 − ε3 ) − (ε1 − ε2 ) tan−1 2 ε1 − ε3

(12.95)

 ,

(12.96)

Bonded Electrical Resistance Strain Gages

ε1 + ε2 + ε3 εp,q = √ 3 2 (ε1 − ε2 )2 + (ε2 − ε3 )2 + (ε1 − ε3 )2 , ± 3 (12.97)

√   3(ε2 − ε3 ) 1 −1 φp,q = tan (12.98) . 2 (ε1 − ε2 ) + (ε1 − ε3 ) For (12.94), (12.95), and (12.97), the first term on the right-hand side represents the distance to the center of Mohr’s circle for strain, and the second term represents the radius of the circle. All three types of rosettes (tee, rectangular, and delta) are manufactured in both planar and stacked versions. As indicated (for the rectangular rosette) in Fig. 12.41, the planar rosette is etched from the strainsensitive foil as an entity, with all gage elements lying in a single plane. The stacked rosette is manufactured by assembling and laminating two or three properly oriented single-element gages. When strain gradients in a)

Planar

1

b)

2

3

Stacked

the plane of the test part surface are not too severe, the normal selection is the planar rosette. This form of rosette offers a number of advantages in such cases:

• • • • • • •

thin and flexible minimal reinforcing effect superior heat dissipation to the substrate minimal sensitivity to strain gradients perpendicular to the surface (e.g., in bending) accepts all standard gage construction options optimal stability provides maximum freedom in leadwire routing and attachment

The principal disadvantages of the planar rosette arise from the larger surface area covered by the sensitive portion of the gage assembly. When the space available for gage installation is small, a stacked rosette may fit, although a planar rosette will not. More importantly, where a steep strain gradient exists in the surface plane of the test part, the individual gage elements in a planar rosette may sense different strain fields and magnitudes. For a given active gage length, the stacked rosette covers the least possible area, and therefore most nearly approaches measuring the strains at a point. It should be realized, however, that the stacked rosette is noticeably stiffer and less conformable than its planar counterpart. Also, because the heat conduction paths for the upper grids in a stacked rosette are much longer, the heat dissipation problem is more critical. This may necessitate reduction of the bridge voltage in order to achieve stable gage operation. Taking into account their poorer heat dissipation and their greater reinforcement effects, stacked rosettes are not commonly recommended for use on plastics and other nonmetallic materials. A stacked rosette can also give erroneous strain indications when applied to a thin specimen in bending, since the grid plane of the uppermost gage in a three-gage stack may be as much as 0.12 mm above the specimen surface. In short, the stacked rosette should ordinarily be reserved for applications in which the requirement for minimum surface area dictates its selection. General Errors and Corrections. Strain-gage rosettes

Fig. 12.41a,b Rectangular rosettes (of the same gage length) in (a) planar and (b) stacked construction

are subject, of course, to the same errors as singleelement gages (thermal output, leadwire resistance errors, etc.) and require the same controlling and/or corrective measures to obtain accurate data. There are, however, a few areas in the application of rosettes that require special attention. One, for example, is the sim-

323

Part B 12.4

Delta:

12.4 The Bonded Foil Strain Gage

324

Part B

Contact Methods

Part B 12.4

a)

b) 2

1

2

3

1

c)

3

d)

3

1

1

3 2

2

Fig. 12.42 A counterclockwise numbering scheme for

grids in strain-gage rosettes

ple matter of how the gage elements on the rosette are numbered. The foregoing equations for calculating the principal strains and the principal angle from rectangular and delta rosette strain measurements implicitly assume that the gage elements are numbered in a particular manner. A useful guide for numbering and analyzing strain gage rosette data is provided in [12.21]. Improper numbering of the gage elements will lead to ambiguity in the interpretation of φp,q ; and, in the case of the rectangular rosette, can also cause errors in the calculated principal strains. Treating the latter situation first, it is always necessary in a rectangular rosette to assign gage numbers 1 and 3 to the two mutually perpendicular gages. Any other numbering arrangement will produce incorrect principal strains. Ambiguities in the interpretation of φp,q for both rectangular and delta rosettes can be eliminated by numbering the gage elements as follows. In a rectangular rosette, gage 2 must be 45◦ away from gage 1; and gage 3 must be 90◦ away, in the same direction. Similarly, in a delta rosette, gages 2 and 3 must be 60 and 120◦ away, respectively, in the same direction from gage 1. By definition φp,q is the acute angle from the axis of gage 1 to the nearest principal axis. When φp,q is positive, the direction is the same as that of the gage numbering; when negative, the opposite applies. If εi is algebraically greater than the center coordinate of

Mohr’s strain circle, the axis of maximum principal strain is indicated; if less, the axis of minimum principal strain is indicated. Numbering the gage elements clockwise or counterclockwise is incidental to the interpretation of φp,q , but the direction must be established with some care. Examples of the grid numbering for several representative rosette types are illustrated in Fig. 12.42. In (a), the gages in this rectangular rosette are numbered counterclockwise; while in (b), after conceptually transposing gage 2 across the diameter to satisfy the 45 and 90◦ requirements, the gages are numbered clockwise. Similarly, in (c), the gages in this delta rosette appear to be numbered clockwise; but when gage 2 is transposed diametrically so as to be 60◦ from gage 1, the numbering is counterclockwise. In contrast, the gages in the stacked delta rosette (d) require no readjustment because gages 2 and 3 are already 60 and 120◦ , respectively, away from gage 1 in the same direction, and the gages are numbered counterclockwise. Because at least one of the gages in any rosette will in every case be subjected to a transverse strain that is equal to or greater than the strain along the gage axis, consideration should always be given to the transverse-sensitivity error when performing rosette data reduction. The magnitude of the error in any particular case depends on the transverse-sensitivity coefficient (K t ) of the gage grid, and on the ratio of the principal strains, εp /εq . In general, then, when K t ≤ 0.01, the transverse sensitivity error is small enough to be ignored. However, for larger values of K t , depending on the required measurement accuracy, correction for transverse sensitivity may be necessary. Procedures and relationships for making such corrections are given in Sect. 12.4.3. A potentially serious source of error can be created when the strain-gage user decides to make up a rosette on the specimen from single-element strain gages. The error is caused by misalignment of the individual gages within the rosette. If, for example, the second and third gages in a rectangular rosette configuration are not accurately oriented at 45◦ and 90◦ , respectively, from the first gage, the calculated principal strains will be in error. The magnitude of the error depends, of course, on the magnitude (and direction) of the misalignment, but it also depends on the principal strain ratio εp /εq and on the overall orientation of the rosette with respect to the principal axes. For certain combinations of principal strain ratio and rosette orientation, 5◦ misalignment errors in gages 2 and 3 with respect to gage 1 can pro-

Bonded Electrical Resistance Strain Gages

treme care should be exercised to obtain accurate gage alignment. When the principal strain directions can be known or determined, even approximately, alignment of gages 1 and 3 in a rectangular rosette, or alignment of any gage in a delta rosette, with a principal axis will minimize errors caused by intergage misalignment.

12.5 Semiconductor Strain Gages If etched-foil strain gages are the common workhorse of the experimental mechanics community, then semiconductor gages live in the ivory tower. Fascinating in theory and construction, with extraordinary sensitivity and range, their use can be equally frustrating in practice, with difficult data analysis and unique handling requirements, often leading to disastrous results unless great attention is given to the details. Developed in the 1950s as an outgrowth of work performed at Bell Laboratories on semiconductor devices for the electronics industry, semiconductor strain gages were first described and popularized by Smith [12.22] and Mason et al. [12.23]. Like foil strain gages, semiconductor gages work on the principle of piezoresistance. However, the similarities end there and trying to acquire and analyze data using the same techniques will inadvertently lead to gross errors in strain measurement. The primary performance differences between foil gages and semiconductor gages are (1) the extraordinarily high strain sensitivity found in semiconductor gages (> 50 × that of foil gages), which leads to (2) extraordinarily high nonlinearity in resistance change with applied strain and in temperature response. The very high changes in resistance also cause difficulties when using the Wheatstone bridge (the normal circuit found in strain-gage instrumentation), because of the bridge’s own inherent nonlinearity (although, as shall be discussed, with certain gage/bridge combinations, one nonlinearity opposes the other to provide some nonlinearity cancellation). High strain sensitivity, rock-solid repeatability, and impressive reliability are the hallmark characteristics of semiconductor gages; these very features make them the only choice for certain measurement problems, particularly for applications involving low strains, or high cycles, or very long (kilometers) leadwires. Additionally, since the gages are single-crystal elements, they have no measurable hysteresis or creep, which makes

them attractive for long-term installations, or for use in unique transducer applications, especially where long fatigue life is also a requirement. The fundamental understanding of semiconductor strain gages with respect to strain and temperature response was developed by Dorsey in the early 1960s and published by BLH Electronics of Waltham, MA in a series of eight papers under the common title, Semiconductor Strain Gage Handbook. The treatment of the operational characteristics of these unique sensors presented here follows that work.

12.5.1 Manufacturing Semiconductor strain gages are produced from either single-crystal silicon or germanium, with silicon being most widely used. Pure silicon is usually doped with another element to improve performance characteristics of the sensors. Since the single-crystal materials have a very high resistivity, most gages are simple singlestrand designs, unlike the sinuous patterns required with foil. Single-crystal silicon is grown in a rod shape called a boule, which is frequently referred to as an ingot because of its similar function as that found in the production of metal alloys. The boule is normally about 50 mm in diameter. Slices are precisely cut from the boule on what are labeled Miller indices, typically the (111) and (100) axis. The respective axis is chosen dependent upon the impurity to be added later. Pure silicon has several characteristics which can be improved for strain gage use by adding impurities from either the third or fifth group of the Periodic Table. This not only reduces the intrinsically high resistivity, but also reduces the nonlinear electrical resistance behavior inherent with the pure material. Two common additives are phosphorus and boron. When phosphorus is used, the electrical conduction mechanism of the resulting alloy is by negative elec-

325

Part B 12.5

duce an error of 20% or more in one of the principal strains. Since it is very difficult for most people to install a small strain gage precisely aligned, the user is well advised to employ commercial rosettes. For those cases in which it is necessary, for whatever reason, to assemble a rosette from single-element gages, ex-

12.5 Semiconductor Strain Gages

326

Part B

Contact Methods

Part B 12.5 Fig. 12.43 Three examples of finished semiconductor strain gages (the bottom example is unbacked)

trons, which is labelled n-conduction. Hence, gages manufactured from this material are commonly called n-type gages, and they happen to have negative strain sensitivity (decrease in electrical resistance with tensile strain). For n-type gages, the boule is sliced on the (100) axis. The use of boron as an additive results in conduction by positive holes, or vacancies, and is termed p-conduction. Therefore, strain gages made using phosphorus-doped silicon are typically called p-type gages, and have a positive gage factor. P-type gages are made from material sliced on the (111) axis. ΔR/R0

1

0.8

0.6

0.4

0.2 0.1 –5000

– 4000

–3000

Compression strain ( μm/m)

–2000

–1000

1000 – 0.1 – 0.2

2000

3000

Tension strain ( μm/m)

Fig. 12.44 Strain sensitivity of high resistance (undoped) p-type

silicon

Diffusion of the additions (doping), or impurities, is administered in a furnace at temperatures above 1000 ◦ C. Slices are placed in the furnace and an inert gas containing the desired dopant is passed through the chamber. In general, higher levels of the impurity further reduce resistivity and nonlinearity, but there is a practical limit, restricted by the ability to consistently and uniformly diffuse atoms into the silicon slices. Dopant levels are dictated by the desired roomtemperature resistivity of the finished slice. For p-type material, a value of 0.02 Ω cm is normal, and the diffusion time is chosen to achieve this value. As will be discussed later, n-type gages have a unique behavior (negative strain sensitivity), which allows processing for self-temperature compensation on select materials. Therefore, room-temperature resistivity varies with compensation number, ranging from a low of 0.01 to a high of 0.04 Ω cm. The finished slices are then processed by photolithographic masking and etching techniques, similar to those used in the manufacturing of etched-foil gages, to form a pattern of gages on each slice. Because of the high resistivity, these sensors are normally straight lengths of silicon, or infrequently U-shaped, less than 0.5 mm wide by about 0.05 mm thick, with some higher resistance sensors having a thickness of only 0.01 mm. Further masking and processing deposits a thin film of metal, usually aluminum or gold, to form connection pads. Pure gold leads are attached to these pads, usually through a welding operation. These leads serve as jumpers to the final instrument lead, which is welded to the gold jumpers. The individual gages are then separated from the slice by etching and lapping/polishing. It is during this stage that the flexibility and fatigue resistance of the sensor is developed. Great care is exercised to achieve a very smooth and defect-free surface. The bare silicon strip can be used as is, or to enhance handling and robustness it can be attached to an insulating carrier, much like foil strain gages. Some examples are shown in Fig. 12.43. At this point, some unique characteristics of semiconductor gages surface. For example, even when using very controlled and precise production methods, there is no guarantee the individual gages will be close in strain and temperature sensitivity. This is easily understood, considering the bulk nature of the impurity diffusion process. Also, there is not normally a simple method of adjusting the initial resistance of the individual gages. To overcome these limitations, manufacturers will provide matched sets, typically in lots of four. These matched sets can have a nominal initial resistance

Bonded Electrical Resistance Strain Gages

ΔR/R0 0.6 0.5

12.5.2 Strain Sensitivity

0.4

Analogous to foil strain gages, change in semiconductor gage resistance is dependent upon several parameters, including: absolute temperature (T ), strain level (ε), unstressed resistance (R0 ), material resistivity (ρ), and geometric design, all of which have some temperaturesensitive component. Since strain sensitivity is the primary concern here, this may be expressed qualitatively as   ε ε2 ε3 ΔR = f (12.99) , 2, 3 ... , R0 T T T which can be written quantitatively as ΔR = C 1 ε + C 2 ε2 + · · · , R0

0.2

– 4000

–3000

–2000

0.1 –1000

1000

2000

3000

4000

Tension strain ( μm/m)

– 0.1

Bonded gage condition

– 0.2 – 0.3 – 0.4 – 0.5

Fig. 12.45 Strain sensitivity of 0.02 Ω cm silicon (12.100)

where C1 and C2 are experimentally determined constants, themselves a function of temperature. Note that powers higher than strain squared in (12.99) are ignored, since the product of strain cubed and a fitted calibration constant C3 (≈ 35 000) is negligibly small. Equation (12.100) describes a parabola and is plotted in Fig. 12.44 for T = 25 ◦ C using C1 = 175 and C2 = 72 625, which are values given by Mason [12.23] for high-resistivity (low-doping) p-type silicon. Note that the inflection point for (12.100) is described by ε=

0.3

Compression strain ( μm/m)

−C1 ΔR −C12 , = . 2C2 R0 4C2

The most popular p-type silicon material has a roomtemperature resistivity of approximately 0.02 Ω cm. The experimental determination of the change in resistance versus strain for this material is plotted in Fig. 12.45. Note that the act of bonding semiconductor gages tends to move the initial resistance along the curve (denoted by a cross in the figure), which alters the starting sensitivity. Using a best-fit parabola for this curve yields C1 = 119.5 and C2 = 4000, which is different from the values found by Mason for low-doped silicon. Apparently, the many-valley theory that works well in describing low-doped p-type silicon does not hold for high doping levels. Since C1 is the slope of the parabola at the origin, it could be considered the basic strain sensitivity, or gage factor, for this material. However, a quick study

ΔR/R0 0.6 0.5 0.4 0.3 0.2 0.1 – 4000

–3000

–2000

Compression strain ( μm/m)

–1000

1000 – 0.1 – 0.2

327

Part B 12.5

tolerance of better than ±2% and a nominal matched temperature sensitivity of ±0.2% over a 110 ◦ C temperature range.

12.5 Semiconductor Strain Gages

2000

3000

4000

Tension strain ( μm/m)

– 0.3

Fig. 12.46 Strain sensitivity of 0.031 Ω cm n-type silicon

of Fig. 12.45 reveals an inherent nonlinear change in resistance with applied strain. N-type silicon gages behave similarly to the p-type just discussed, the most significant difference being a reduction in resistance when stretched. That is, n-type gages have a negative gage factor. This behavior is plotted in Fig. 12.46 for a highly doped, low-resistivity (0.031 Ω cm) n-type gage. In this example, the strain relationship is given by (12.100), where C1 = −110 and C2 = 10 000. An interesting observation from Fig. 12.46 is that, given the normal compressive strain locked into a bonded gage during the adhesive cure cycle, which

328

Part B

Contact Methods

Part B 12.5

is caused by the differential expansion of the gage and specimen at the cure temperature, the resistance will be shifted up the curve, toward the more linear portion. When n-type gages are pre-installed on a thin shim for subsequent spot-welding to a specimen, the manufacturer can prestress the shim and force the installed resistance into the more linear region of the curve, thus improving linearity response to service strains. This is beneficial, because as can also be seen from the figure, n-type gages are inherently more nonlinear than p-type.

Temperature changes affect the resistance (TCR) and the sensitivity (TC of gage factor) of semiconductor strain gages. Because gage sensitivity varies with applied strain, the process of determining this effect, suitable for use in strain data correction, is different from the method used with foil gages. To better understand the issues involved with temperature sensitivity, it is necessary to examine (12.100) more closely, which, by ignoring terms higher than squared, can be expanded as:     298 298 2  2 ΔR = C2 ε . (12.101) F0 ε + R0 T T In this equation, 298 refers to an arbitrarily selected reference at room temperature in Kelvin, F0 is the reference-temperature gage factor, and C2 is the reference temperature calibration constant; ε is, of course, the applied strain. Noting that F0 will vary inversely as a function of T , and that C2 will change with the ΔR /R0 0.35 0.3 0.25 0.05 Ω cm 0.2 0.02 Ω cm

0.1 0.01 Ω cm

0.05 0 23.8

50

75

100

125

150

Temperature (°C)

Fig. 12.47 Thermal output of p-type silicon, various resis-

tivities

0.3 0.25 0.04 Ω cm 0.2 0.031 Ω cm 0.15 0.25 Ω cm 0.1

0.018 Ω cm

0.05 0 –0.025 23.8

12.5.3 Temperature Sensitivity

0.15

ΔR /R0

0.01 Ω cm 50

75

100

125

150

Temperature (°C)

Fig. 12.48 Thermal output of n-type silicon, various resis-

tivities

square of the inverse of T , suggests that higher temperature changes result in more linear gage output. For p-type sensors, output becomes more linear with increased temperature for two distinct reasons. First, C2 decreases rapidly in comparison with gage factor and, second, most materials expand at a rate higher than p-type silicon, which has a TCE of 1.5–3.0 ppm/◦ C. Consequentially, a tensile strain is exerted on the gage as temperature rises, and as can be seen from Fig. 12.45, the higher the tensile strain, the more linear the gage response. The usual method of bonding a strain gage to a reference material and subjecting the installation to a temperature excursion of interest will yield false thermal output data from semiconductor gages, because the gage sensitivity changes so dramatically with temperature. Additionally, unless assurance can be given of a zero state of stress in the installed gage (very difficult), the initial resistance at room temperature R0 will not be that of the free filament. Therefore, the starting point for generating a thermal output curve is shifted (Figs. 12.45 and 12.46) from the origin, and the resistance–temperature relationship is not valid for general use. If the exact conditions present during such a calibration are also those encountered during a subsequent test (e.g., using the installed gage as a temperature sensor), then this procedure is adequate. Otherwise, and most typically, this is not the case, which leads to the inconvenient task of either measuring the resistance changes with applied temperature of an unbonded sensor, or employing a fixture suitable for correcting the temperature-induced strain in the gage, present because of different TCE values between the gage and specimen.

Bonded Electrical Resistance Strain Gages

Temperature Compensation Given the above discussion on the strain–temperature characteristics of semiconductor gages, it should be no surprise that both the determination of and implementation of temperature compensation for stress analysis applications is both different and somewhat more involved than that which is employed with foil gages. There is no heat-treatment option available for silicon-based sensors to finely alter the intrinsic TCR. However, the negative gage factor of n-type silicon can be exploited to provide a rudimentary compensation on certain materials. This is best explained by examining the bonded TCR as given by

βb = β + (αs − αg )F ,

Gage factor 0

100

125

150

175

200 1

1.05

3034 Aluminum

1.1 0.9 300 Series stainless steel 1.15 0.85

1018 Steel 1.2

Data correction factor

Fig. 12.49 TC of gage factor for 0.02 Ω cm p-type strain gages bonded to 1018 steel (11.7 ppm/◦ C), 300series stainless steel (17.1 ppm/◦ C), and 2024 aluminum (23.4 ppm/◦ C) with constant-current bridge excitation

range of −100 to −135, it is clear from this example how a select group of specimen materials can be self-temperature compensated. Although limited in application, self-temperature-compensated n-type sensors are the most economical way to achieve temperature compensation with semiconductor gages. Taking advantage of the opposite-sign strain sensitivities for p- and n-type gages, and the additive nature Gage factor (% of room-temp. value) 100

where βb is the bonded TCR, β is the unbonded TCR, αs is the specimen TCE, αg is the gage TCE, and F is the gage factor. The bonded TCR, βb , will be zero, or negative, when the differential expansion term in (12.102) is equal to or exceeds the unbonded gage TCR, β. Thus the condition for self-temperature compensation is approximated when STC ≈ αs , where

90

For example, a backed n-type gage having F = −110, αg = 2.5 ppm/◦ C, and β = 1080 ppm/◦ C is essentially self-compensated when bonded to a mild steel specimen (αs = 11.7 ppm/◦ C and STC = 12.3 ppm/◦ C). With highly doped n-type gages having a gage factor

75

0.95

110

(12.103)

Temperature (°C) 50

1

(12.102)

β STC = αg + . |F|

25

0

25

50

75

Temperature (°C) 100

125

150

175

200 1

1018 Steel

80

1.2

1.4

70 300 Series stainless steel

1.6

60 2024 Aluminum 50

1.8

2

Data correction factor

Fig. 12.50 TC of gage factor for STC n-type strain gages on 1018 steel (11.7 ppm/◦ C), 300-series stainless steel (17.1 ppm/◦ C), and 2024 aluminum (23.4 ppm/◦ C) with constant-current bridge excitation

329

Part B 12.5

In the former, care must be taken to avoid excessive self-heating of the free filament; use of a liquid environment is beneficial. In the latter, a fixture must be used to relieve the temperature-induced tensile strain in the gage, by applying a corresponding compressive strain in the specimen. Typical thermal output curves from various resistivity p-type and n-type unstressed gages are shown in Figs. 12.47 and 12.48, respectively. Essential to the interpretation of (12.101) and these figures is the understanding that R0 is the unstrained resistance at temperature T , and that it varies with T [12.24]. There will be a different R0 at each temperature. Interestingly, even though n-type semiconductor strain gages have a negative strain sensitivity, like ptype sensors, they too have a positive TCR. As is discussed in the next section, it is these unique characteristics of n-type gages which provide a means of rudimentary self-temperature compensation on select materials.

12.5 Semiconductor Strain Gages

330

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Part B 12.5

of adjacent arms in a Wheatstone bridge, provides another method to achieve self-temperature compensation. This technique is commonly referred to as circuit compensation with push–pull gages, and is similar to the dummy-gage method, but relieves the necessity of ensuring zero stress on the compensating gage (dummy gage). In the push–pull method, a p-type and an n-type gage are bonded side-by-side on the test piece and wired as adjacent arms in the Wheatstone bridge. Assuming equal gage TCE values, the difference in bonded gage TCR is βp − βn = (Fn − Fp )(αs − αg ) .

An obvious benefit of the push–pull method is the ability to combine more gage characteristics and, therefore, compensate for more specimen materials. While not usually a concern with semiconductor gages, the push– pull method also increases bridge output. TC of Gage Factor Compensation The temperature coefficient of gage factor is negative for both p- and n-type gages. Much of the drop in sensitivity with increased temperature is the result of the large, positive TCR of silicon, causing the gage resistance (R) to increase significantly, thus decreasing the

0 100

25

50

75

Temperature (°C) 100

125

110

25

50

100

150

175

200

75

Temperature (°C) 100

125

150

175

200 1

90 1.2

80

1.4

70

1.6 60 1.8 50

2

Data correction factor

Fig. 12.52 TC of gage factor for any STC n-type strain

gage bonded to the matching material with constantvoltage bridge excitation

apparent sensitivity (ΔR/R). An obvious way to eliminate this effect is by using constant-current excitation. This obvious conclusion illustrates a not-so-obvious issue when using semiconductor gages: the choice of excitation can dramatically influence correction of acquired strain data. Figures 12.49 and 12.50 show TC of gage factor curves for p- and n-type sensors, respectively, on several different materials and with constant-current excitation. Figures 12.51 and 12.52 provide the same information with constant-voltage excitation. From these figures the benefit of constant-current excitation is clear. However, most commercial strain gage instrumentation utilizes constant-voltage bridge excitation.

1

90 80

0

(12.104)

Temperature compensation is approximated when STCc ≈ αs , where, βp − βg STCc = αg + . (12.105) Fn − Fp

Gage factor (% of room-temp. value)

Gage factor (% of room-temp. value)

RS /2 1.25

70

V 1.5

60 1.75

RS /2

Data correction factor

Fig. 12.51 TC of gage factor for p-type silicon with constant-voltage bridge excitation

Fig. 12.53 Full-bridge semiconductor circuit showing the inclusion of compensation resistors for TC of gage factor

Bonded Electrical Resistance Strain Gages

|FTb | Rs = , RB mβb − |FT |

(12.106)

where Rs is the value of compensation resistor, RB is the total bridge resistance across the power corners, FTb is the bonded gage TC of gage factor, m = 1 for full-bridge or 0.5 for half-bridge installations, βb is the bonded gage TCR (12.102), and FT is the unbonded gage TC of gage factor. It is usual practice to insert one-half of Rs into each of the two power leads, although this is not a requirement.

12.5.4 Special Circuit Considerations Given their high sensitivity to strain, semiconductor gages do lend themselves to use with elementary or potentiometer circuits. However, the common-mode voltage is still problematic and must be subtracted away from the measurement. Still, there are recorded instances of accurate strain measurement, and especially troubleshooting, of semiconductor installations by use of a simple ohm-meter [12.25]. Apparent from previous discussions is the relatively large resistance shifts encountered with semiconductor gages, caused by any stimulant. In some ways these large changes make for simpler interrogation of the sensors, but it can also make the job more difficult, as almost all modern strain-gage instrumentation operates with a constant-voltage Wheatstone bridge. Nonlinearity Errors Because of the large resistance changes encountered with semiconductor gages, steps must be taken to address the inherent nonlinearity associated with this condition in the constant-voltage Wheatstone bridge, especially if the typical stress analysis quarter-bridge

circuit is employed. In fact, use of this circuit is rare with semiconductor gages, half- and full-bridge installations being the norm. One method to overcome the linearity concern in a constant-voltage, quarter-bridge circuit is to buck the inherent bridge nonlinearity with that of the gage. As shown in Fig. 12.45 for p-type silicon, the nonlinearity is concave up. The nonlinear bridge output for a quarter-bridge circuit is concave down. With the gage designated by Rg , and occupying the R1 (Fig. 12.4) position in the bridge, this can be written as ⎛ ⎞ ⎜ E0 = E ⎜ ⎝

⎟ ΔRg R3 ⎟. × ΔRg (Rg + R4 )(R2 + R3 ) ⎠ 1+ Rg + R4 (12.107)

To counteract the expected high resistance changes of Rg relative to the other bridge resistances, R3 and R4 can be made larger than Rg and R2 , by a multiplier. With the multiplier equal to 10, for example, the nonlinearity is reduced to about 10% of its normal value [12.26]. Although unattractive because of a loss in sensitivity, another method of linearization is simply to desensitize the output by shunting the active gage, which has the combined effect of lowering circuit output and reducing nonlinearity. However, for a shunt ratio of 1.5 (shunt resistance to installed gage resistance), which produces a quite linear output curve, the reduction in output sensitivity to applied strain is 44%. Like the multigage methods used for temperature compensation, use of push–pull gages and p/n-type pairs can also reduce circuit nonlinearity. Push–pull gages do not completely eliminate the nonlinear output, but through careful selection of gage pairs, placed in adjacent arms of the bridge, with one gage sensing compressive strain and the other sensing tensile strain, linearity is improved and output is increased. The same result is achieved when a full bridge is employed, as is the case in most semiconductor-based transducers. P- and n-type gages can be chosen and mounted side by side (sensing the same strain) to achieve partial circuit linearization, but the method is imperfect, because n-type gages are more nonlinear than p-type. Leadwire Attenuation Leadwire attenuation is less serious with semiconductor gages, because the gages have such large resistance changes when compared to any realistic leadwire contribution, and the sensitivity is so high in comparison to

331

Part B 12.5

Since all semiconductor gages have a positive TCR and a negative TC of gage factor, gage factor change with temperature can be compensated by increasing the bridge supply voltage in proportion to the loss in sensitivity. This is accomplished by inserting low-TCR, fixed resistors in series with bridge power leads, as shown in Fig. 12.53. In this circuit, the bridge is considered a single, variable resistor with positive TCR, and the circuit acts like a voltage divider. By choosing the ratio of series resistance to bridge resistance equal to the ratio of bridge TCR to TC of gage factor, the increase in bridge voltage resulting from the positive bridge TCR can be made to offset the drop in bridge sensitivity caused by the negative TC of gage factor, thus

12.5 Semiconductor Strain Gages

332

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Part B 12

any leadwire desensitization. When using the Wheatstone bridge circuit, however, it is still necessary to give consideration to leadwire affects, as discussed in Sect. 12.4.8 on foil gages.

12.5.5 Installation Techniques Installation techniques for backed semiconductor gages are no different from those used with backed foil gages [12.27]. However, care must be exercised to avoid undue resistance shifts during installation, or at least the installed resistance must be accurately measured to determine where on the strain-sensitivity curve the installed gage falls. Also, methods are available for prestraining the installation, to provide a specific initial condition. This is most frequently done to match the performance of one gage against another to aid linearity and/or temperature compensation. These methods are beyond the scope of this overview treatment, but can be studied in the literature [12.2, 28, 29]. Free-filament semiconductor gages are rarely used in experimental stress analysis work. They are, however,

popular in certain transducer applications. Installation of the free filament is straightforward. The specimen surface is prepared in accordance with normal straingage procedures, and then a thin precoat of adhesive is cured on the specimen surface to act as an insulator. This precoat is subsequently abraded to remove the glossy finish, and the gage is laid in a pool of adhesive on top of the prepared precoat layer. No pressure is necessary, as capillary action will form a smooth, thin glue line, even in the overhead position. Most often instrument leads are welded to the preattached wires supplied by the manufacturer. This is because the normal wire material used by the manufacturer is a delicate gold, which does not solder well. However, gages can also be supplied with heavy-gage lead ribbons already welded to the gold jumpers. These heavier ribbons accept the same soldering techniques as any strain-gage installation. Obviously, care must be taken to ensure stress relief in the instrument leads, so that subsequent loading of the part does not tear off the leadwire.

References 12.1

12.2 12.3

12.4

12.5

12.6 12.7

12.8

12.9

P.K. Stein: 1936 – a banner year for strain gages and experimental stress analysis – an historical perspective, Exp. Tech. 30(1), 23–41 (2006) A.L. Window, G.S. Holister: Strain Gage Technology (Applied Science, London 1982) ASTM: Standard Test Methods for Performance Characteristics of Metallic Bonded Resistance Strain Gages, Vol. E 251 (ASTM Int., West Conshohocken 2003) BIML: Performance Characteristics of Metallic Resistance Strain Gauges, Vol. 62 (Bureau International de Metrologie Legale, Paris 1985) VDE/VDI: Bonded Electric Resistance Strain Gauges with Metallic Measurement Grids – Characteristics and Testing Conditions, VDE/VDI – Richtlinien, Vol. 2635 (Verein Deutscher Ingenieure, Düsseldorf 1974) J.P. Holman: Experimental Methods for Engineers (McGraw-Hill, New York 1989) G.F. Chalmers: The dangers of the 5% mentality, Proc. Int. Conf., BSSM & The Institution of Production Engineers (UK 1980) R.B. Watson, D. Post: Precision strain standard by moire interferometry for strain gage calibration, Exp. Mech. 22(7), 256–261 (1982) R.B. Watson: Calibration techniques for extensometry: Possible standards of strain measurement, J. Test. Eval. 21(6), 515–521 (1993)

12.10

12.11

12.12

12.13

12.14

12.15

12.16 12.17

Vishay: Errors Due to Transverse Sensitivity in Strain Gages, Technical Note TN-509 (Document Number: 11051), (Vishay Micro-Measurements, Raleigh 2007) R.B. Watson, C.C. Perry, S.K. Harris: Effects of material properties and heating/cooling rate on strain gage thermal output observations, ICEM12 – 12th Int. Conf. Experimental Mechanics (Politecnico di Bari, Italy 2004) R.B. Watson, C.C. Perry: The intelligent use of strain gage thermal output, Proceedings of the 2002 Conf. on Experimental Mechanics (The Society for Experimental Mechanics, Bethel 2002) F.F. Hines: Effect of mounting surface curvature on the temperature coefficient of bonded resistance strain gages, Proceedings, Western Regional Strain Gage Committee (The Society for Experimental Mechanics, Bethel 1960), pp 39–44 ASTM: Standard Test Method for Ambient Temperature Fatigue Life of Metallic bonded Resistance Strain Gages, Vol. E 1949 (ASTM Int., West Conshohocken 2003) C. C. Perry: Strain gage reinforcement effects on low-modulus materials, Exp. Tech. 9(5), 25–27 (1985) E.G. Little, D. Tocher, P. O’Donnell: Strain gauge reinforcement of plastics, Strain 26(3), 91–97 (1990) A. Ajovalasit, B. Zuccarello: Local reinforcement effect of a strain gauge installation on low modulus materials, J. Strain Anal. 40(7), 643–653 (2005)

Bonded Electrical Resistance Strain Gages

12.19

12.20

12.21

12.22 12.23

P.G. Ifju, D. Post: A Compact Double Notched Specimen for In-Plane Shear Testing, Proc. 1989 Conf. on Experimental Mechanics (The Society for Experimental Mechanics, Bethel 1989) M.-J. Pindera, P.G. Ifju, D. Post: Iosipescu shear characterization of polymeric and metal matrix composites, Exp. Mech. 30(1), 101–108 (1990) R.B. Watson, C.C. Perry: Factors influencing the measurement of shear properties when using vnotch specimens with strain gages, Proc. 2nd European Conf. on Composites (ECCM-CTS2) (Hamburg 1994) ASTM: Standard Practice for Analysis of Strain Gage Rosette Data, Vol. E 1561 (ASTM Int., West Conshohocken 2003) C.S. Smith: Piezoresistive effect in silicon and germanium, Phys. Rev. 94(42) (1954) W.P. Mason, J.J. Forst, L.M. Tornillo: Recent Developments in Semiconductor Strain Transducers,

12.24

12.25

12.26 12.27

12.28

12.29

Vol. 15-NY-60 (Instrument Society of America, Research Triangle Park 1960), Preprint J.Dorsey: BLH Semiconductor Strain Gage Handbook (Theory), Section I, (Vishay Micro-Measurements, Raleigh 1964) J. Dorsey: BLH Semiconductor Strain Gage Handbook (Strain Measurement), Section VII, (Vishay Micro-Measurements, Raleigh 1965) H.K.P. Neubert: Strain Gauges (Macmillan, London 1967) ASTM: Standard Guide for Installing Bonded Resistance Strain Gages, Vol. E 1561 (ASTM, West Conshohocken 2003) J. Dorsey: BLH Semiconductor Strain Gage Handbook (Gage Application), Section IV, (Vishay MicroMeasurements, Raleigh 1964) Micron Instruments: Installation of Semiconductor Strain Gages, Technical Note, Rev. 1.01, (Micron Instruments, Simi Valley 2005)

333

Part B 12

12.18

References

335

Extensomete 13. Extensometers

The important characteristics and construction of the common types of extensometers are reviewed. Extensometers are the most convenient and widely used instruments for the measurement of strain in a material test specimen. Traditionally extensometers have been based around contacting techniques, however noncontacting optical techniques are becoming more important and the two approaches are compared in this chapter. Whichever technique is employed it is important to be aware of the possible influence of the extensometer on the test specimen and the need to match the extensometer to the specimen, environment, and test being carried out.

13.2.5 Linear Incremental Encoders ......... 338 13.2.6 Electronics and Signal Conditioning 338 13.3 Ambient-Temperature Contacting Extensometers ..................... 13.3.1 Clip-On Axial Extensometers .......... 13.3.2 Other Types of Clip-On Extensometers ............................. 13.3.3 Long-Travel (Elastomeric) Extensometers ............................. 13.3.4 Automatic Extensometers ..............

338 338 340 340 341

13.4 High-Temperature Contacting Extensometers ..................... 341 13.4.1 Longitudinal-Type High-Temperature Extensometers .. 341 13.4.2 Side-Loading High-Temperature Extensometers ............................. 341

13.1 General Characteristics of Extensometers 13.1.1 Accuracy...................................... 13.1.2 Resolution................................... 13.1.3 Stability ...................................... 13.1.4 Temperature Coefficient ................ 13.1.5 Operating Force ........................... 13.1.6 Contact Force ............................... 13.1.7 Weight ........................................ 13.1.8 Response Time/Bandwidth ............ 13.1.9 Kinematics and Constraints ...........

336 336 336 336 336 337 337 337 337 337

13.5 Noncontact Extensometers .................... 13.5.1 Optical Targets ............................. 13.5.2 Servo Follower Type Optical Extensometers ............................. 13.5.3 Scanning Laser Extensometers ....... 13.5.4 Video Extensometers .................... 13.5.5 Other Noncontact Optical Extensometers ............................. 13.5.6 Noncontact Extensometers for High-Temperature Testing ........

13.2 Transducer Types and Signal Conditioning 13.2.1 Strain-Gaged Flexures .................. 13.2.2 LVDTs .......................................... 13.2.3 Potentiometers ............................ 13.2.4 Capacitance Transducers ...............

337 337 337 338 338

13.6 Contacting versus Noncontacting Extensometers ..................................... 345

Extensometers provide the most convenient way of accurately measuring the average strain in the gage section of a material test specimen. Alternative methods such as attempting to determine strain from the movement of the test machine or using bonded strain gages are either much less accurate or are more complex and involve a significant amount of specimen preparation.

343 343 344 344 344 344 345

13.7 Conclusions .......................................... 346 References .................................................. 346

There are a wide range of different types of extensometer covering many materials, applications, and environments. Applications range from multi-axis cyclic fatigue testing to high-throughput, industrial quality control, tensile testing. Materials range from tissues and other biological materials to concrete and other construction materials. Environments range from

Part B 13

Ian McEnteggart

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Contact Methods

liquid-helium cryostats through saline baths at body temperature to ultrahigh-temperature furnaces. Extensometers have the potential to influence the behavior of the test specimen, and the strain measurements obtained are often influenced by the extensome-

ter’s mounting and environment. In order to obtain representative and repeatable strain results from an extensometer it is very important that the user of these devices understands their operating principles and characteristics.

Part B 13.1

13.1 General Characteristics of Extensometers 13.1.1 Accuracy The ability of a measurement system using an extensometer to measure accurately the strain in a test specimen depends on both the gage length and extension accuracy. The gage length accuracy of an extensometer can be verified by directly measuring the distance between the extensometer knife edges using, for example, a calibrated traveling microscope. Alternatively the marks made by the knife edges in a soft material, e.g., copper can be measured using the same type of equipment. Verification of extensometer extension measurement performance at ambient temperature can be achieved using calibrated displacement reference devices (e.g., precision micrometers or laser interferometers). Practical considerations dictate that most hightemperature extensometers are verified at ambient temperatures. It is, however, desirable to check the accuracy of these extensometers at temperature, in the test configuration in which they are to be used. Such a verification can be performed by providing extended rods that allow the reference device to be located outside the high-temperature environment (i. e., a temperature chamber or furnace) and the motion to be transmitted to the extensometer which is located inside the hightemperature environment [13.1]. Methods of verifying the accuracy of contacting extensometers are defined in various standards [13.1, 2]. Regular verification checks on extensometer accuracy are recommended, particularly in the case of extensometers which are used for testing materials which release large amounts of energy at failure and subject the extensometer to repeated shock loads.

13.1.2 Resolution The resolution of a measurement device or system is usually defined as the smallest discernible change in the measured value. For extensometers based on analogue transducers the resolution can be taken as being equal

to one-half of the noise level. For extensometers based on digital transducers (e.g., optical encoders) the resolution is usually equal to the resolution of the digital transducer. Signal processing techniques, such as oversampling, filtering, and resampling, can increase the effective resolution of a transducer system. It should be clear that a good resolution specification for an extensometer does not necessarily imply a corresponding high accuracy, and both specifications should be checked when selecting an extensometer for an application.

13.1.3 Stability The stability of an extensometer is the drift in the measured value as a function of time. Zero, or balance, stability can be expressed as a fraction of the full-scale change in output the over a given time period. Span stability is similarly expressed as a change in full-scale output sensitivity over time. Stability is important for extensometers used in long-term creep or fatigue testing.

13.1.4 Temperature Coefficient The temperature coefficient of an extensometer is a measure of the change in output as a function of temperature. The temperature coefficient is expressed as a fraction of the full-scale change in output for a defined change in temperature. As for stability, the temperature coefficient is generally unimportant in short-duration tension or compression tests but it is important in long-term testing. It should be noted that extensometer temperature coefficients are almost always quoted for isothermal conditions, i. e., when there are no temperature gradients across the extensometer. In some situations, e.g., high-temperature extensometers used with tube-type, furnaces there can be large temperature gradients across the extensometer; in these cases the temperature coefficient should be checked in the test configuration.

Extensometers

13.1.5 Operating Force

13.1.6 Contact Force The contact force is the force applied to the knife edges to keep them attached to the specimen and to prevent them from slipping. The required contact force is determined by both the weight and operating force of the extensometer.

13.1.7 Weight The weight of an extensometer can be important when testing delicate materials, as the weight of the extensometer has to be supported by the specimen via the knife edges. In some cases supporting springs or a counterbalance system can reduce the effective weight of the extensometer.

13.1.8 Response Time/Bandwidth The extensometer system (i. e., the combination of the extensometer and the associated signal condition-

ing electronics) must have a sufficiently fast response, to follow the behavior of the material in a test. Most modern systems have a sufficiently rapid response for monitoring strain in routine tensile testing. Extensometers used as strain feedback devices for machine control or for use in high-frequency cyclic testing (> 10 Hz) need to have a fast-response, wide bandwidth and be free of resonances within the control bandwidth. Reference [13.1] outlines a method that can be used to investigate the suitability of an extensometer for dynamic use.

13.1.9 Kinematics and Constraints The strain measured by an ideal extensometer will be unaffected by the presence of mechanical mounting constraints and will only be sensitive to specimen strain. In practice extensometers, apart from clip-on types, require some form of independent mechanical mounting. The extensometer mounting is a constraint which can influence the strain measurement, e.g., errors in the measured strain can be introduced by lateral movement of the specimen at the start of a tensile test in a machine with a self-aligning load string. The principles and details of kinematic design are explained in Chap. 3 of [13.3].

13.2 Transducer Types and Signal Conditioning Modern extensometers make use of a number of different transducers to convert mechanical deflections into electrical output signals that can be easily processed and recorded. Transducers in common use include 1. 2. 3. 4. 5.

strain-gaged flexures linear variable differential transformers (LVDTs) potentiometers capacitance transducers linear incremental encoders

13.2.1 Strain-Gaged Flexures In this type of transducer a number of foil-type strain gages (usually four) are bonded to a flexural beam and electrically connected in the form of a Wheatstone bridge. Bending of the beam causes a change in resistance of the gages [13.4]. The change in resistance is

small and the full-scale electrical output signal from the bridge is typically 2 mV/V, however modern signal conditioning electronics can measure the signals from such bridges with great stability and accuracy. Provided that the flexure is deformed elastically it has excellent linearity and repeatability [13.5]. Strain-gaged flexures have a significant operatingforce; however they have the advantages of being of small size and having low mass. The operating temperature range of foil-type strain gages, suitable for transducer use, ranges from cryogenic temperatures to about 200 ◦ C.

13.2.2 LVDTs A LVDT is basically a transformer with variable coupling between two or more inductance coils [13.4]. LVDTs are usually of a cylindrical form with a central

337

Part B 13.2

The operating force of an extensometer is the force required to displace the extensometer knife edges by a known amount. The operating force is usually expressed as a force divided by the full-scale displacement of the extensometer.

13.2 Transducer Types and Signal Conditioning

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Part B 13.3

plunger – the output signal is proportional to the position of the plunger. In principle LVDTs have a zero operating force; however most commercially available units incorporate a spring and have an appreciable operating force. Compared to strain-gaged flexures LVDTs are larger and more massive, and can operate over similar temperature ranges.

13.2.3 Potentiometers Potentiometers consist of a sliding contact running along a resistive track [13.4]. Rotary or linear potentiometers have limited resolution and linearity and are, generally, limited to use at ambient temperature. Multi-turn rotary types and types using conductive plastic tracks provide the highest available resolution, making them most suitable for use in extensometers. Potentiometers produce a large direct-current (DC) output signal, making them easy to interface to dataacquisition systems. The main extensometer application of potentiometers is in long-travel extensometers; they are unsuitable for dynamic use.

13.2.4 Capacitance Transducers The capacitance between two adjacent electrodes is a function of the overlapping area, the gap, and the dielectric occupying the space between the electrodes. The majority of capacitive transducers are based on the variation in capacitance as a function of the gap [13.4]. In general the electrical output from a capacitive transducer is a nonlinear function of displacement and the conditioning electronics must incorporate some form of linearization in order to produce acceptable system measurement accuracy. Capacitive transducers have a zero operating force and are available in small sizes with low mass. The

maximum operating temperature range of a capacitive transducer can be several hundred degrees Celsius.

13.2.5 Linear Incremental Encoders Linear incremental encoders are essentially digital devices which produce output pulses corresponding to defined changes in displacement. These encoders are often based on optical gratings along with optical sensors and interpolation electronics. Incremental transducers are usually very accurate and can have resolutions of less than 0.1 μm. They are, however, bulky and massive. Furthermore they are generally unsuitable for dynamic applications because they cannot follow rapid displacement changes. Incremental transducers are generally limited to operation at room temperature. It should be noted that linear encoders can make excellent reference transducers for extensometer calibrators.

13.2.6 Electronics and Signal Conditioning Almost all of today’s materials testing machines are based around digital electronics; however the majority of extensometers are based on analogue transducers, so some form of analogue-to-digital (AD) conversion is required. Once the signal is converted into a stream of sampled data it can be processed and recorded. Common processing operations include linearization to improve accuracy and filtering/resampling to reduce noise and improve resolution. As well as providing real-time signal conditioning functions modern testing machine and data-acquisition electronics will often provide automatic transducer identification and calibration. Automatic transducer identification eliminates the possibility of operator error when changing transducers.

13.3 Ambient-Temperature Contacting Extensometers The majority of extensometers in use today are contacting types. This section outlines the designs and characteristics of some of the main types of contacting extensometers.

13.3.1 Clip-On Axial Extensometers Clip-on extensometers are very widely used in the testing of rigid materials such as metals, plastics, and

composites, over a range of temperatures from cryogenic to several hundred degrees Celsius. They are designed to attach directly to, and be supported by, the test specimen, as illustrated in (Fig. 13.1). Because of their weight and relatively large operating and contact forces they are not generally suited to testing flexible or delicate materials such as foils/films, rubbers, fabrics, and many biological materials. The design of clip-on extensometers limits the maximum measurable

Extensometers

Fig. 13.2 Biaxial extensometer (courtesy of Epsilon Cor-

poration)

Fig. 13.3 Long-travel extensometer (courtesy of Instron)

strain to about 100% and hence they are not generally suitable for testing high-extension materials. Clip-on extensometers are available with single and multiple fixed gage lengths from below 10 mm to over 200 mm. Full-scale strains are in the range 5–100%. Clip-on extensometers incorporate a means of setting the reference gage length and a transducer to measure the extension of the test specimen. The extensometer illustrated in Fig. 13.1 has a manually operated spring-loaded gage length setting mechanism. Another common means of setting the gage length is a closefitting pin, which locks the extensometer to the required gage length. Clip-on extensometers use a pair of knife edges located on the specimen to define the gage length. These knife edges are usually quite sharp and are held in place using some form of spring or clip in order to reduce the chances of slip during the test. The unwanted stresses due to the presence of knife edges on the surface of a delicate specimen can influence the test itself. In addition to the stresses due to the knife-edge contacts, the weight and operating (or activation) force of the extensometer can exert an influence when testing delicate materials. With any type of clip-on extensometer the effect of energetic specimen breaks on the long-term reliability of the extensometer should be considered. In some situations, e.g., tensile testing of metals, it is possible to pause the test and remove the extensometer after the yield stress has been determined and determine the specimen strain for the rest of the test from the machine extension. The most common transducer mechanism employed in a clip-on extensometer is a strain-gaged flexure; however LVDTs, capacitive and incremental digital transducers are also used. Strain-gaged flexures are compact and lightweight, making them easy to integrate into a clip-on extensometer design. They have a very high inherent resolution; the limiting factor is the noise level or digital resolution of the conditioning electronics. The accuracy of clip-on extensometers using strain-gaged flexures will usually be sufficient to meet ASTM E83 [13.1] grade B1/B2 or ISO 9513 [13.2] grade 0.5 (i. e., accuracy is better than 0.5% of the reading). LVDT transducers are larger and more massive than equivalent strain-gaged flexures. They have a high resolution; however, in general, the linearity of LVDT transducers is not as high as a strain-gaged flexure. They are generally more robust than strain-gaged flexures and because of this they are often used in extensometers intended for testing materials, which release significant

339

Part B 13.3

Fig. 13.1 Clip-on extensometer (courtesy of Instron)

13.3 Ambient-Temperature Contacting Extensometers

340

Part B

Contact Methods

Part B 13.3

energy at breakage, e.g., steel plate and reinforcing bars. Extensometers for this type of heavy-duty testing are often designed to separate into two halves when the specimen breaks. Tension/compression testing of materials in the range −150 ◦ C to 500–600 ◦ C is usually carried out using a temperature chamber. The use of clip-on extensometers to test at temperatures above and below ambient using a temperature chamber is limited by the ability of the transducer to withstand the test temperatures. Standard extensometers using strain-gaged flexures can be used down to cryogenic temperatures; however they will require calibration at the test temperature because of the effect of temperature on the sensitivity of the strain gages and on the reference gage length. Using standard strain gages and wiring limits the maximum temperatures to about 200 ◦ C. Operation at temperatures above about 200 ◦ C requires either a strain-gaged extensometer with some form of cooling (e.g., water or air) or the use of a transducer capable of operating at the test temperature, e.g., a high-temperature encapsulated strain gage or capacitive type. Clip-on extensometers can be used when testing in environments other than air, e.g., in high-humidity air, fluids or gases. In these situations care should be taken to ensure that all the materials used in the construction of the extensometer are compatible with the particular environment in use.

13.3.2 Other Types of Clip-On Extensometers In addition to axial clip-on extensometers there are a number of other clip-on extensometers, some of these are outlined here. The averaging axial clip-on extensometer uses a pair of axial extensometers positioned across the width or diameter of the test specimen. The measurement of average axial strain eliminates strain measurement errors arising from the presence of bending in the test specimen caused by alignment errors in the load string [13.6]. Measurement of some materials properties, e.g., Poisson’s ratio or plastic strain ratio (r-value) requires the simultaneous measurement of axial and transverse strain using either strain gages bonded to the specimen or a combination of axial and transverse extensometers. Transverse clip-on extensometers are similar in design to axial types but they measure the change in diameter or width of the test specimen. A gage length setting mechanism is usually not required; the gage length (in

Fig. 13.4 Automatic extensometer (courtesy of Instron)

this case a gage width) is determined by the dimensions of the specimen. Biaxial extensometers (Fig. 13.2) which integrate both axial and transverse extensometers into a single unit are available and are more convenient to attach than two separate extensometers. Crack-opening clip gages, whilst not strictly extensometers, have much in common with clip-on extensometers. There is no requirement for a gage length setting mechanism and the operating forces are very high.

13.3.3 Long-Travel (Elastomeric) Extensometers Long-travel extensometers (Fig. 13.3) are contacting types with travels typically in excess of 250 mm. They are intended for strain measurement on materials such as rubber with high failure strains (e.g., 1000–2000%). Typically this type of long-travel extensometer is based on a pair of arms (which may be counterbalanced) with spring-loaded clamps which locate on the test specimen. The arms are free to move on a vertical track following the movement of the specimen gage length. The arms then operate either a potentiometer or an incremental digital transducer, which converts the arm separation into an electrical output. Gage length setting is provided by means of mechanical stops.

Extensometers

Long-travel extensometers are optimized for testing high-elongation materials – they can measure high elongations accurately, however, they have much lower resolution than clip-on extensometers.

13.3.4 Automatic Extensometers

can be automatically set to the required proportional gage length when testing specimens of differing crosssectional area. There are two main types of contacting automatic extensometers: the direct acting type and the servo follower type. Direct acting automatic extensometers have a set of arms which contact the specimen and link directly to a measurement transducer located inside the body of the extensometer. Servo follower type automatic extensometers have deflection sensors (usually strain-gaged flexures) incorporated into the arms; these sensors then drive servo systems which drive the arms to maintain zero deflection (Fig. 13.4). In this way the operating force of the extensometer can be reduced to a very low level. Once again arms are linked to a measurement transducer. A common feature found on an automatic extensometer is a transverse strain measurement transducer, which provides the ability to measure the plastic strain ratio (r-value) on sheet metals.

13.4 High-Temperature Contacting Extensometers The term high-temperature extensometer is commonly applied to extensometers designed to work at temperatures beyond 500–600 ◦ C. Below test temperatures of 500–600 ◦ C most testing takes place inside a temperature chamber and it is possible to use a suitable clip-on extensometer inside the chamber. Above these temperatures it is usual to locate the transducer outside the heated zone and to transfer the specimen motion to the transducer by means of refractory rods. Using this approach it is possible to produce extensometers capable of measuring strain on specimens being tested at very high temperatures. There are two main approaches to the design of contacting, high-temperature extensometer for use at high temperatures. Longitudinal high-temperature extensometers use rods oriented parallel to the test specimen to transmit the deflection to the displacement transducers. Side-entry high-temperature extensometers have rods oriented at right angles to the specimen.

13.4.1 Longitudinal-Type High-Temperature Extensometers

testing of metals. They are also employed in hot tensile testing. The design uses refractory metal rods, located on either end of the specimen gage length by machined ridges, or groves, or via knife edges clamped on to the specimen. The rods transmit the specimen strain to displacement transducers located outside (usually at the bottom) of the furnace. The transducers employed are usually LVDTs or capacitive displacement sensors, but strain-gaged clip-on extensometers are sometimes employed. This type of extensometer can measure strain on either side of the specimen, from which average strain can be obtained. The measurement of average strain in a materials test reduces errors in materials properties due to bending of the specimen caused by poor alignment of the load string. The metal rods and clamps used in the construction of the longitudinal high-temperature extensometers limit their use to temperatures below about 1100 ◦ C.

13.4.2 Side-Loading High-Temperature Extensometers

The other approach to high-temperature extensometry Longitudinal-type high-temperature extensometers [13.5], uses a side-loaded design. In this type of extensome(Fig. 13.5) are widely used for high-temperature creep ter the extensometer is provided with ceramic rods,

341

Part B 13.4

Automatic extensometers are contacting types with remote control of many or all of the extensometer functions. These extensometers are mainly used for the tensile testing of rigid materials and they find wide application in high-volume tensile testing of metals. Automatic extensometers are an essential part of any fully automated (robotic) testing system. As a minimum an automatic extensometer will be provided with an ability to engage and disengage from a test specimen. Fully automatic extensometers will have other remotely controlled functions, e.g., gage length setting and gripping force setting. The ability to control the gage length remotely means that it

13.4 High-Temperature Contacting Extensometers

342

Part B

Contact Methods

Part B 13.4

which pass through a cut-out in the side of the furnace or through a gap in an induction heating coil. Side-entry extensometers (Fig. 13.6) can be designed to attach directly to the specimen using high-temperature cords or they can be designed to be suspended from a support mechanism located outside the furnace or heating coils and pushed on to the specimen. Externally supported high-temperature extensometers may be provided with a slide mechanism, which allows the extensometer to be loaded onto the test specimen after the specimen has been heated to the test temperature. This so-called hot loading can be very useful as hightemperature load strings can be subject to a significant amount of movement due to thermal expansion during the heat-up phase. A slide loading mechanism also allows the extensometer to be withdrawn from the specimen prior to failure – this eliminates the possibility of the energy release at failure damaging the extensometer rods. Gage length setting on side-loaded high-temperature extensometers can be provide by a pin to lock the rods at the required gage lengths, the pin being removed after

loading the rods on to the specimen. Alternatively the extensometers rods are located on a gage length reference bar (i. e., a bar with a pair of grooves separated by the required gage length) and the output is set to zero. The extensometer is then transferred to the specimen and the rods adjusted until the output is, once again, zero. At this point the contact points are at the correct gage length. The transducers in common use with side-loading extensometers are strain-gaged flexures, capacitive sensors, and LVDTs. Capacitive and LVDT sensors both offer the advantage of zero operating force. In addition the capacitive sensor is small, light, and offers no resistance to out-of-axis movements. LVDT sensors are relatively heavy, which complicates the design of the extensometer, limiting the bandwidth and requiring high contact forces. The profile of the end of the rod in contact with the specimen is important if the possibility of knife edge slip is to be reduced and consistent results are to be obtained. The aim should be to have a kinematic contact (i. e., sufficiently constrained but not overconstrained contact). For flat specimens, conical point contacts are required. For round samples, two flat chisel ended rods or one rod with an inverted V chisel contact and one flat chisel rod. The ideal material for the rods of a high-temperature extensometer would have high modulus, low density, low thermal conductivity, good creep resistance, and be chemically compatible with the specimen and the test environment. For temperatures up to about 1000 ◦ C the material with the best thermal properties is quartz, which has a very low coefficient of thermal expansion and a low thermal conductivity, however it is

Fig. 13.5 Longitudinal-type high-temperature extensometer (courtesy of Instron)

Fig. 13.6 Side-entry high-temperature extensometer (courtesy of Epsilon Corporation)

Extensometers

13.5 Noncontact Extensometers

343

Table 13.1 Materials for high-temperature extensometer rods Material Quartz

E (MPa)

α (ppm/◦ C)

ρ (g/cm3 )

K (W/m◦ C)

1000

Very good thermal performance but very brittle Creeps above 1200 ◦ C Good for testing in air to 1600 ◦ C Vacuum/inert only Vacuum/inert only *Typical (anisotropic)

0.4

2.2

Alumina Silicon carbide

360 440

8.1 4.8

4.0 3.2

29 79

1200–1600 1500–1600

Graphite C–C (C3 )

207 71*

7.9 −0.4

2.3 1.7

5.0 21

2500+ 2500+

mon materials used in extensometer rods is shown in Table 13.1. In order to minimize drift the sensitive parts (e.g., pivots/flexures and sensors) of a high-temperature extensometer must be kept cool. Techniques to achieve this include the use of radiation shields, cooling fins, forced air or thermostatically controlled water-cooling systems. In the case of longitudinal extensometers the location of the transducers below the furnace assists in keeping them cool. For demanding long-term creep tests the use of a fully temperature-controlled environment is recommended.

13.5 Noncontact Extensometers There are a number of different approaches to optical, noncontact extensometry. Most noncontact optical extensometers, designed for use at ambient temperatures, operate in a reflection mode in which light is reflected off optical targets back into the detector. It is also possible to use a transmission mode in which the optical targets cast a shadow. Noncontacting extensometers are inherently free from spurious mechanical resonances and hysteresis, however there may be a time delay (lag) in the output signal which can affect stability when using the signal for closed-loop control.

13.5.1 Optical Targets The majority of noncontacting, optical techniques require that a contrasting mark or target be attached to the specimen to define the axial gage length. This is a critical area, as any slip or distortion of the target will generate a spurious strain signal. The use of contrasting ink or paint marks on the specimen surface is effective in many situations; how-

ever at large strains the size of the ink mark grows and the contrast may reduce. In situations where ink or paint marks are not suitable adhesive targets may be used. The size and shape of the target is important. Round targets are generally preferred over line targets because they eliminate errors that can arise from rotation of the mark. Line targets do, however, allow the possibility of measuring average strain. When testing delicate specimens the target size should be as small as possible in order to minimize the reinforcing effect of the target on the specimen. When applying targets to delicate specimens care should be taken to ensure that the adhesives and/or solvents used in the inks or paints are compatible with the specimen. Clearly any chemical incompatibility obviates one of the key benefits of noncontact optical technique. Another potential problem with optical targets is a mismatch between the mechanical gage length, as defined by the distance between points of contact on the specimen, and the gage length sensed by the optical measurement

Part B 13.5

Notes

70

a brittle material which is easily broken by mishandling or by the shocks from specimen failure. Alumina is a more durable material and can be used to above 1200 ◦ C. Silicon carbide is suitable for temperatures of up to 1600 ◦ C. Rods for use above 1600 ◦ C can be made from carbon, usually in the form of carbon– carbon composite. These materials oxidize rapidly in air so they must be used in a vacuum or inert gas environment. At very high temperatures chemical compatibility can often only be satisfied by using knife edges made from the same material as the specimen. A summary of the relevant properties of some com-

0.22

Tmax (◦ C)

344

Part B

Contact Methods

13.5.2 Servo Follower Type Optical Extensometers

Part B 13.5

The servo follower type optical extensometer is similar, in principle, to the servo follower type automatic extensometer described earlier. The extensometer uses a pair of optical sensors mounted on moving carriages that track contrasting targets on specimen. An analogue or incremental digital displacement transducer then measures the separation of the carriages.

13.5.3 Scanning Laser Extensometers The scanning laser extensometer is a reflection-mode system in which a beam of laser light is rapidly scanned across a pair of reflective optical targets on the specimen. An optical detector detects the reflected energy. The distance between the targets is determined from the time that elapses between the reflections. There are two alternative implementations of this principle. The first is intended to measure large extensions and uses a large-angle (45−90◦ ) fan beam of laser light reflected off a rotating mirror. This type of optical extensometer has a relatively low resolution (typically 10–100 μm) and is suitable for high-elongation testing of rubbers and elastomers. The second uses a more sophisticated optical system to produce a parallel scanned beam. This type of extensometer has a smaller extension range but much higher resolution.

13.5.4 Video Extensometers Video extensometers use real-time image processing of images from a video camera to measure the position of contrasting marks on a test specimen. In common with several other techniques the video extensometer measures strain by tracking two contrasting targets, which define the gage length. However unlike most noncontact optical systems, which measure along one axis, the video extensometer is an area scan device. Because of this it is possible to track the position of the target more precisely so that the measurement is less sensitive to target distortion, specimen lateral movement, or bending. Figure 13.7 shows a block diagram of a typical video extensometer system. The system is based on a MegaPixel charge-coupled device (CCD) digital camera sending 50 images per second to a high-performance personal computer (PC) via an IEEE 1394 (firewireTM ) interface. The CCD image sensor has approximately 1200 × 800 pixels (light-sensitive elements) and real-

time image-processing algorithms can determine the locations of the marks to subpixel accuracy, achieving a resolution of better than one-hundredth of a pixel. A key part of the system is the lighting system, which is designed to enhance the contrast of the optical targets whilst minimizing the sensitivity of the extensometer the surface finish of the test specimen and to changes in ambient lighting. The lighting system uses a large number of red light-emitting diodes (LEDs) pulsed synchronously with the camera’s electronic shutter. A red filter on the camera optimizes the sensitivity of the camera to the light from the LEDs, and a pair of crossed polarizing filters, one in front of the LEDs and a second in front of the camera, eliminate specular reflections from the specimen. This enhances the mark contrast, particularly when working with specimens with a bright, polished surface. The image analysis routines used in the video extensometer can track either dot or line targets. Dot targets are suitable for most tests but line targets can be used to measure an average strain across the width of a test specimen. When dot targets are used the video extensometer is capable of measuring both axial and transverse strain, making it suitable for determining rvalue (anisotropy ration) as part of a tensile test. The field of view (FoV) of the camera can be changed by simply changing the camera lens. Fields of view from 60 mm to 500 mm are available. A key element of the system is the method of calibration, which uses a precision two-dimensional (2-D) target to correct for any distortions in the optics. Calibration of the extensometer involves positioning the calibration target instead of the test specimen and running an automatic calibration routine.

13.5.5 Other Noncontact Optical Extensometers In addition to the techniques described other approaches have been used. One technique is a transmission technique based on optical diffraction. Optical flags are attached to the specimen defining the gauge length the flags are large and are arranged so there is a small gap between them. A laser beam is aimed to pass through the gap and on to a detector. The light is diffracted by the gap, forming a diffraction pattern at the detector. Measurement of the fringe spacing of the diffraction pattern can be used to determine the separation of the flags and hence the strain in the specimen [13.7]. Another approach that has been used successfully at very high temperatures (3000 ◦ C) has been to use opti-

Extensometers

Narrow band optical filter matched to illumination

13.6 Contacting versus Noncontacting Extensometers

345

Fig. 13.7 Video extensometer block diagram (courtesy of Instron)

Firewire digital camera

Forced air Firewire interface CDAT

Polarizing filter

Parallel port

Camera synch interface Controller

PC interface

Data synch

Pulsed LED lighting driver

cal flags, high-power laser back-lighting and a Zimmer electro-optical measurement system to track an image of the flags. A comparison of this approach with a contacting approach appears in [13.8, Chap. 14].

13.5.6 Noncontact Extensometers for High-Temperature Testing Noncontact optical extensometers are attractive instruments for measuring strain at nonambient temperatures; however they have limitations that need to be recognized. As mentioned earlier, tension/compression testing of materials in the range from −150 ◦ C to 500–600 ◦ C is usually carried out using a temperature chamber. It is possible, indeed it is common practice, to provide these chambers with viewing windows, usually consisting of a number of glass panels separated by gaps. In principle noncontact optical extensometers can see the test specimen through the window and can measure the required strain. In practice a number of factors need to be addressed including the quality of the glass used in the window and that the targets used to define the gage length must be able to operate at the required temperatures without degradation.

When testing at low temperatures, there is a tendency for atmospheric moisture to lead to ice build up on the window and specimen, obscuring the view through the window and decreasing the contrast of the optical targets. When used at temperatures above 1000 ◦ C optical extensometers require a powerful source of external illumination and some method of preventing the sensitive optical detectors from being overloaded by the intense black-body radiation from the furnace. Suitable sources of external illumination are laser light or a mercury vapor lamp. Optical filters matched to the wavelength(s) of the external light source are then used to protect the optical detectors from the radiation from the furnace. All optical systems are sensitive to noise caused by convection currents in the vicinity of hot objects. With all noncontact optical extensometers that are used to test at nonambient temperatures the resolution of the measurements reduces as the temperatures increase due to light refraction from the air currents inside the chamber. It is possible to isolate the light paths from the turbulent air flow around the chamber or furnace by enclosing them in evacuated tubes or using light guides, however these are expensive and complex to integrate with other equipment.

13.6 Contacting versus Noncontacting Extensometers Contacting extensometers are in direct mechanical contact with the specimen and do not require the attachment of optical targets. Furthermore the contact is, generally, unaffected by changes in the surface condition of the

specimen (as may be caused by oxidation in metals or microcracking in polymers). However, particularly with delicate materials, contact stresses at the knife-edge locations may lead to premature specimen failure and the

Part B 13.6

LED array

346

Part B

Contact Methods

weight of a contacting extensometer can generate some degree of bending in the specimen. Noncontacting measurement means that no unwanted loads are applied to the specimen. In practical

terms noncontact extensometers are generally more robust and less subject to damage and drift than contacting extensometers. They are also easier to use and less prone to operator error.

Part B 13

13.7 Conclusions There are a range of contacting and noncontacting techniques for measuring specimen strain. When selecting an extensometer for a specific test it is important to understand the characteristics of the properties of the material being tested and the characteristics of the extensometer. When correctly applied, both contact and noncontact extensometry are capable of producing reliable data.

We can expect to see continued advances in CCD cameras and real-time analysis of video images. These developments will generate exciting possibilities for noncontact extensometry leading to improvements in both the accuracy and flexibility of this technique. Nevertheless, there will continue to be a need for a range of high-performance contacting extensometry into the foreseeable future.

References 13.1

13.2

13.3 13.4

ASTM: ASTM E83-06 Standard Practice for Verification and Classification of Extensometer Systems (ASTM International, West Conshohocken 2006) ISO: ISO 9513:1999 Metallic materials – Calibration of extensometers used in uniaxial testing (ISO, Geneve 1999) S.T. Smith, D.G. Chetwynd: Foundations of Ultraprecision Mechanism Design (CRC, Boca Raton 1994) H.K.P. Neubert: Instrument transducers (Oxford Univ. Press, Oxford 1975)

13.5

13.6 13.7

13.8

B.F. Dyson, M.F. Loveday, M.G. Gee: Materials Metrology and Standards for Structural Performance (Chapman Hall, New York 1995) P. Han: Tensile Testing (ASM International, Metals Park 1992) B.F. Dyson, R.D. Lohr, R. Morell: Mechanical Testing of Engineering Ceramics at High Temperatures (Elsevier, Amsterdam 1989) R. Lohr, M. Steen: Ultra High Temperature Mechanical Testing (Woodhead, Cambridge 1995)

347

Optical Fiber 14. Optical Fiber Strain Gages

Chris S. Baldwin

14.1 Optical Fiber Basics ............................... 348 14.1.1 Guiding Principals for Optical Fiber 348 14.1.2 Types of Optical Fibers ................. 349

14.2.3 14.2.4 14.2.5 14.2.6

Advantages of Fiber Optic Sensors . Limitations of Fiber Optic Sensors .. Thermal Effects ........................... Introduction to Strain–Optic Effect

352 353 354 354

14.3 Interferometry ..................................... 14.3.1 Two-Beam Interference ............... 14.3.2 Strain–Optic Effect....................... 14.3.3 Optical Coherence ....................... 14.3.4 Mach–Zehnder ........................... 14.3.5 Michelson .................................. 14.3.6 Fabry–Pérot ............................... 14.3.7 Polarization................................ 14.3.8 Interrogation of Interferometers ...

354 355 355 356 357 357 357 358 358

14.4 Scattering ............................................ 359 14.4.1 Brillouin Scattering...................... 359 14.4.2 Strain Sensing Using Brillouin Scattering................................... 359 14.5 Fiber Bragg Grating Sensors .................. 14.5.1 Fabrication Techniques ................ 14.5.2 Fiber Bragg Grating Optical Response ................................... 14.5.3 Strain Sensing Using FBG Sensors .. 14.5.4 Serial Multiplexing ...................... 14.5.5 Interrogation of FBG Sensors, Wavelength Detection.................. 14.5.6 Other Grating Structures ...............

361 361 362 363 364

14.6 Applications of Fiber Optic Sensors ......... 14.6.1 Marine Applications..................... 14.6.2 Oil and Gas Applications .............. 14.6.3 Wind Power Applications ............. 14.6.4 Civil Structural Monitoring ............

367 367 367 368 368

366 367

14.2 General Fiber Optic Sensing Systems....... 351 14.2.1 Strain Sensing System Concept ...... 351 14.2.2 Basic Fiber Optic Sensing Definitions ................................. 351

References .................................................. 369

Many optical fiber sensors are based on classical bulk optic arrangements, as discussed in Chap. 18 – Basics of Optics. These arrangements are named after the in-

ventors who are credited with their invention. Other optical fiber sensors are based on physical phenomenon such as scattering effects that are inherent to the optical

14.7 Summary ............................................. 368

Part B 14

Optical fiber strain sensing is an evolving field in optical sciences in which multiple optical principles and techniques are employed to measure strain. This chapter seeks to provide a concise overview of the various types of optical fiber strain sensors currently available. The field of optical fiber strain sensing is nearly 30 years old and is still breaking new ground in terms of optical fiber technology, instrumentation, and applications. For each sensor discussed in the following sections, the basic optical layout is presented along with a description of the optical phenomena and the governing equations. Comprehensive coverage of all aspects of optical fiber strain sensing is beyond the scope of this chapter. For example, each sensor type can be interrogated by a number of means, sometimes based on differing technology. Furthermore, these sensors are finding applications in a wide variety of fields including aerospace, oil and gas, maritime, and civil infrastructures. The interested reader is referred to the works by Grattan and Meggit [14.1], Measures [14.2], and Udd [14.3] for more-detailed descriptions of interrogation techniques and applications of the various sensors.

348

Part B

Contact Methods

fiber material. In general, fiber optic sensors function by monitoring a change in an optical parameter as the optical fiber is exposed to the strain field. The three main types of optical fiber strain gages are

• • •

interferometry (changes in optical phase) scattering (changes in optical wavelength) fiber Bragg grating (changes in optical wavelength)

Other optical phenomena such as intensity variations are exploited for optical fiber sensors measuring other quantities such as chemical concentrations, applied loads/pressure, and temperature. The strain sensing mechanisms listed above are explored in the sections below, but first an introduction to optical fibers and a general overview of the basic layout, advantages, and disadvantages of optical fiber strain sensors is provided.

14.1 Optical Fiber Basics Part B 14.1

The first thought many people have when introduced to optical fiber is of a thin, fragile piece of glass. On the contrary, the ultrapure manufacturing that goes into producing the low-loss optical fiber of today creates a thin glass structure essentially free of defects with tensile strength values near 800 ksi (5.5 GPa) [14.2]. The following sections discuss the optical guiding properties of optical fiber and provide a brief description of the various types of optical fiber commercially available for sensing applications.

14.1.1 Guiding Principals for Optical Fiber Before any discussion of fiber optic strain sensors can be realized, an understanding of the light guiding principles of optical fiber should be presented. The basic material for optical fiber is fused silica [14.2]. Standard optical fiber consists of two concentric glass portions. The inner portion is called the core and the outer portion is called the cladding. The core region of the optical fiber typically contains dopants such as germanium (Ge) or boron (B) to increase the refractive index of the core (n core ) to a slightly higher value than that of the cladding (n clad ), which is pure fused silica. Optical fiber also uses protective layers of polymer coatings to protect the glass surface from damage. For most strain sensing applications, one of two coatings is applied to the optical fiber. Acrylate-coated optical fibers have a final outer diameter of 250 μm. Polyimide-coated optical fibers are available in a range of final outer diameters, typically around 180 μm. Standard telecommunications optical fiber has a germanium-doped core. The optical transmission spectrum for Ge–Si is shown in Fig. 14.1 [14.2]. The transmission spectra displays the characteristic decay of the Rayleigh curve (1/λ4 ); at higher wavelengths the transmission is limited by the absorption of energy by the silica structure. The multiple peaks in the attenua-

tion curve are due to OH− (hydroxyl) scattering in the optical transmission curve. The effect of the hydroxyl peaks is the creation of transmission windows for optical fiber. The telecommunications industry makes use of the wavelength regions around 1550 nm due to the lowloss properties and the availability of optical sources and detectors. This is commonly referred to as the Cband. The region around 1300 nm, termed the S-band, is also used for data transmission, but typically is reserved for shorter-length applications such as local area networks (LANs). A schematic diagram of standard single-mode optical fiber is shown in Fig. 14.2. The size of the core (radius = a) is dependent on the wavelength of light the optical fiber is designed to guide. For single-mode operation in the C-band, this value is around 4 μm. The cladding diameter is standardized at 125 μm for many optical fiber types. This allows for the use of common optical fiber tools such as cleavers, strippers, and fusion splicers to be used on different optical fiber types from different optical fiber vendors. In the simplest sense, light is guided via total internal reflection (TIR) between the core and cladding Attenuation (dB/km) S-band

0.6

0.8

1

1.2

C-band

1.4

1.6 1.8 Wavelength (μm)

Fig. 14.1 Optical transmission windows for optical fibers

Optical Fiber Strain Gages

14.1 Optical Fiber Basics

349

1 J0 (V ) 250 μm a

0.75

J1 (V )

0.5

125μm

V = 2.405

0.25 0

Fig. 14.2 Schematic of typical optical fibers –0.25

Therefore, the modal propagation of light in a stepindex optical fiber is dependent on the core size (a), the refractive index difference between the core and cladding, and the wavelength of light propagating in the core. The V number value to ensure single-mode operation is 2.405. If the V number is larger than 2.405 then

Acceptance cone

α

Cladding Core

Fig. 14.3 Total internal reflection between core and

cladding

–0.5 –0.75

–J1 (V ) 0

2 4 Singlemode region

6

8

10

Fig. 14.4 Bessel solution space for modal operation of op-

tical fiber

light in the optical fiber is not propagating as singlemode. Based on the formulation given in (14.1), a cutoff wavelength value can be determined for the optical fiber, which defines the lower limit of single-mode operation with respect to wavelength. The cutoff wavelength is given by:  2πa n 2core − n 2clad . (14.2) λcutoff = V Any wavelength of light lower than the cutoff wavelength will not propagate as single-mode in the optical fiber. Likewise, to determine the appropriate core size to ensure single-mode operation at a given wavelength, (14.1) can be written as   (14.3) a = 2.405λ/2πV n 2core − n 2clad . The vast majority of optical fiber strain sensors employ standard single-mode optical fiber or similar constructed optical fiber. The following section provides a brief description of optical fiber types that may be encountered in the field of strain sensing.

14.1.2 Types of Optical Fibers For the most part, fiber optic strain sensors make use of readily available, standard optical fiber and components. Optical fiber sensing has experienced a new thrust of interest and development in recent years. Much of the current research has been focused on the field of biosensing with the development of new optical fiber types and sensing techniques. Optical fiber strain sensing has been steadily finding and being integrated

Part B 14.1

of the optical fiber, as illustrated in Fig. 14.3. TIR was introduced in Chap. 18. Figure 14.3 shows a slab waveguide description for optical fiber. Based on the slab waveguide formulation, the standard Snell’s law can be employed to determine the guiding principles for the structure. This analysis leads to an acceptance cone where light entering the core within this angle will result in a guiding condition. In optical fiber terms, this is related to the numerical aperture (NA). Of course, the explanation of TIR for the guiding principles of optical fiber is very limiting. The derivation for the propagation of light through an optical fiber is based on a solution of the electromagnetic wave equation (18.1), as discussed in Chap. 18. This derivation is beyond the scope of this chapter. The important result from the derivation is that the propagation of light through the core is based on Bessel functions (Ji (x)). Single-mode propagation in optical fiber is defined by a parameter known as the V number, which is the argument of the Bessel function solution for guided core modes [14.2]. The solution plots of the first few Bessel functions are shown in Fig. 14.4. The V number for standard single-mode optical fiber is expressed as   2πa n 2core − n 2clad . (14.1) V = ka n 2core − n 2clad = λ

350

Part B

Contact Methods

a) Singlemode

b) Multimode

c) PM-elliptical core

d) PM-bowtie

Part B 14.1

Fig. 14.5a–d Cross-section views of various types of opti-

cal fiber

into more application areas including civil infrastructure monitoring, oil and gas application, embedded composite sensing, and maritime sensing. As discussed in Sect. 14.2.3, optical fiber sensing possesses some inherent advantages over traditional sensing techniques for these and other application areas. The development of low-loss optical fibers in the 1970s began a period of innovation in both telecommunications and sensing applications. The first optical fibers designed for telecommunication applications were manufactured by Corning in 1970 and possessed attenuation values just less than 20 dB/km [14.2]. Current attenuation values for telecommunication grade optical fiber are of the order of 0.2 dB/km [14.2]. Although the vast majority of fiber optic sensors use standard communications-grade single-mode optical fiber, it should be noted that multiple varieties of optical fiber exist and are being explored for various sensing applications. Figure 14.5 shows a cross-section view of several types of optical fiber. For example, a variety of polarization-maintaining optical fibers exist that possess an induced birefringence throughout the fiber length, thus maintaining two principal polarization axes for optical transmission. Various dopants and fiber structures have been explored throughout the past few decades to

decrease the influence of bend losses and improve signal quality within optical fiber. Using Fig. 14.2 as a reference, the optical fibers displayed in Fig. 14.5 can be compared to the standard single-mode optical fiber. Multimode optical fiber (illustrated in Fig. 14.5b) possesses a much larger core size compared to single-mode fiber. This aspect was discussed in Sect. 14.1.1 in terms of the V number dictating single-mode operation. Multimode optical fiber may either have a step index core (like the single-mode optical fiber) or a graded refractive index core. If the multimode fiber is step index, then the various propagation modes travel different path lengths due to the oscillatory nature of the propagation. Graded-index optical fibers allow the various modes propagating in the fiber to have equivalent path lengths. Figure 14.5 also displays two types of polarizationmaintaining fiber (PM fiber). Polarization of light is covered in both Chaps. 18 and 25 of this Handbook. The polarization-maintaining aspect is induced either through a geometric structure of the core (elliptical core fiber – Fig. 14.5c) or by inducing a permanent mechanical load within the fiber structure (such as bow-tie fiber – Fig. 14.5d). PM fiber develops two polarization states (fast and slow axis) whose responses to mechanical loads are slightly different. PM fiber has found applications in strain sensing, as discussed in Sect. 14.3.7. Most recently, the development of photonicbandgap (PBG) fiber and photonic-crystal fiber (PCF) has generated great interest in the sensing community, particularly in the field of chemical and biosensing. These optical fibers are commonly referred to as “holey fibers” because the core and cladding regions are developed by manufacturing a structured pattern of voids (holes) that traverse the entire length of the optical fiber. For more information on this type of optical fiber, the reader may visit www.crystal-fibre.com. In general, PCFs use a patterned microstructure of voids along the length of the optical fiber to create an effective lower-index cladding and guide light in a solid core by modified total internal reflection (M-TIR)[14.4]. PBG fibers guide light within a low-index region (longitudinal void) by creating a photonic bandgap based on the structure and pattern of the voids [14.4].

Optical Fiber Strain Gages

14.2 General Fiber Optic Sensing Systems

351

14.2 General Fiber Optic Sensing Systems 14.2.1 Strain Sensing System Concept

Interrorgation unit

Fiber optic sensor

Source Optics Feedback Data acquisition

Demodulator

Fig. 14.6 Basic configuration of an optical fiber sensor sys-

tem

a) Distributed

b) Discrete

Fig. 14.7a–c Fiber optic sensor classifications

14.2.2 Basic Fiber Optic Sensing Definitions Due to the wide variety of fiber optic sensors, some definitions of optical fiber strain sensing are required. Fiber optic sensors can be classified into distributed, discrete, or cumulative strain sensors, as illustrated in Fig. 14.7. A distributed strain sensor provides a measure of strain at potentially every point along the sensing optical fiber. Distributed sensors typically have a spatial resolution of approximately 1 m due to the resolution of the measurement systems. A discrete strain sensor provides a strain measurement at one location often based on a smallgage-length fiber optic sensor. By serially multiplexing these sensors, a distributed sensor array can be fabricated with the strain measurements being at discrete points instead of an average of the spatial resolution. A cumulative strain sensor provides a measure of strain that is an average strain value over the entire sensing length of the optical fiber. The selection of a fiber optic strain sensor for a particular application is driven by the sensing requirements for the application. For example, measurement of the hoop strain of a pressure vessel may use a distributed sensor to obtain strain values at multiple locations around the circumference. A discrete sensor can provide a single strain measurement at a single location on the circumference or multiple measurements if serially multiplexed. A cumulative strain sensor would provide a single strain measurement based on the overall strain induced in the fiber bonded to the circumference of the vessel. Cumulative sensors provide the highest sensitivity and are often employed when very small strain signals are of interest such as the measurement of acoustic fields. In many cases, the optical fiber serves as both the conduit for the optical signal and the sensing mechanism. In this situation, the sensor is referred to as an intrinsic sensor. An extrinsic sensor is the case where the optical fiber delivers the optical signal to the sensc) Cumulative

Part B 14.2

The basic building blocks of a fiber optic strain sensor system can be compared to the building blocks of a resistance strain gage (RSG) system discussed in Chap. 12. A resistance strain gage circuit requires a voltage supply, Wheatstone bridge, electrical wires connecting to the RSG, a voltmeter to monitor the circuit output, and a data-acquisition system to record the voltage changes. In terms of fiber optic sensors, an equivalent arrangement can be drawn where the voltage supply is equivalent to the light source (laser, light emitting diode, or other optical source). The Wheatstone bridge is equivalent to the optics that guide the light to and from the sensors or sensing region of the optical fiber. The wires that connect to the RSG are simply the lead optical fiber to the fiber optic sensors. In some cases, a single strand of optical fiber is used to monitor multiple (potentially thousands) fiber optic strain sensors (see multiplexing below). This is vastly different from RSG circuits where each uniaxial gage requires at least two conductors. The voltmeter for a fiber optic sensor system is the interrogation unit (also called a demodulator). The interrogation units typically have one or more photodetectors to transfer the optical signal to a voltage signal proportional to the optical intensity. The data-acquisition system is the one component that may be an exact duplicate between the RSG circuit and the fiber optic sensor system. This is typically a personal computer (PC) or laptop device with appro-

priate interface cards/boards to record the data from the interrogation unit.

352

Part B

Contact Methods

A

A/2

B/2

B

A/2 B/2

Fig. 14.8 Bidirectional optical fiber coupler

Part B 14.2

ing mechanism such as an air cavity. In both cases, the applied strain induces a physical change in the geometry of the sensor. For the case of an intrinsic sensor, a strain–optic effect is also induced and must be considered when determining the calibration coefficient (gage factor) of the sensor. Another differentiation between types of optical fiber sensors is whether the sensor works as a reflective or a transmission element. The majority of fiber optic strain sensors function in reflection mode, where light traveling in the optical fiber is reflected back towards the optical source from reflective elements or from inherent scattering of the optical fiber material. In order to direct the back-reflected light to the interrogation instrumentation, an optical fiber coupler device is employed. A bidirectional coupler is typically used in these applications, as shown in Fig. 14.8. The commercially available standard bidirectional coupler is termed a 3 dB coupler because it divides the optical signal equally into the two output arms as indicated in Fig. 14.8.

14.2.3 Advantages of Fiber Optic Sensors Optical fiber sensors offer many advantages over traditional electrical-based strain sensors. Many of these advantages stem from the pure optical nature of the sensor mechanisms. Traditional sensors depend on the measurement of variations of resistance or capacitance of the electrical sensors. Optical fiber sensors depend on changes to optical parameters of the optical fiber or the light traveling within the optical fiber. This difference leads to many advantages, including

• • • •

immunity to electromagnetic interference long transmission lead lines no combustion danger serial multiplexing

The following discussion expands on each of the advantages stated above.

Immunity to Electromagnetic Interference Optical fiber sensors function by measuring changes to the light that is traveling within the optical fiber. Unlike low-level voltage signals, resistance changes, or other electrical phenomena used for traditional sensors, electromagnetic influences do not cause measurable levels of noise. Therefore, when dealing with optical lead lines and cabling, no special shielding is required, thus greatly reducing the cabling weight compared to traditional sensors. Long Transmission Lines With the advances made in the telecommunications field for optical fiber transmission, current standards for optical signal transmission allow the light signal to be transmitted many kilometers without a detrimental level of signal degradation. Optical power loss in current communications grade optical fiber are on the order of 0.2 dB/km. Other factors that affect the transmission of optical signals in fibers such as chromatic and material dispersion have also experienced improvements. No Combustion Danger In some applications, there exists a risk of ignition or explosion of combustive materials from sparks or electrical potentials from sensors. The optical power guided within the core for fiber optic sensors is typically less than 1 mW, essentially eliminating any potential of spark or ignition danger. Without the requirement of a voltage potential at the sensor location and the ability to have long lead lines, optical fiber sensors are regarded as being immune to combustion danger. This is one reason why fiber optic sensors are being heavily explored for various oil and gas industry applications [14.5]. Multiplexing Some of the optical fiber sensors discussed in this chapter have the ability to be multiplexed (multiple sensors interrogated by a signal instrumentation system). The two main types of multiplexing are parallel and serial multiplexing. In parallel multiplexing (also called spatial multiplexing), the light source is separated into multiple optical fiber channels, with each channel containing an optical sensor. Typically, each sensor in a parallel multiplexing scheme will have its own detector and processing instrumentation. If the light source is guided to the multiple channels via a fiber optic switch component (as shown in Fig. 14.9), then the optical sensor signals are not monitored simultaneously but sequentially depending on the timing of the optical switch. In this case, a common detector and processing instru-

Optical Fiber Strain Gages

Optical switch Source

Coupler

Demodulation and data acquisition

Sensor arrays

Switch control signal

Fig. 14.9 Parallel multiplexing employing an optical

switch Coupler

#1 #2 #3

#n–1 #n

Sensor array

Instrumentation

Fig. 14.10 Serially multiplexed sensors

mentation may be employed for all the sensors. If the optical source is divided to all the sensors via a fiber optic coupler arrangement, then all the fiber optic sensors can be interrogated simultaneously through each sensor’s individual processing instrumentation. Serial multiplexing is a major advantage of some types of optical fiber sensors (in particular the fiber Bragg grating, Sect. 14.5). In serial multiplexing techniques, multiple sensors are positioned along a single optical fiber lead (in serial fashion), as illustrated in Fig. 14.10. The strain signals from each of these sensors are separated through optical means or via the processing instrumentation. The sensing system may also contain a secondary fiber sensor to provide reference and/or gating signals to decouple the serially multiplexed signals.

14.2.4 Limitations of Fiber Optic Sensors In most texts dealing with optical fiber sensors, the reader can find a similar list of advantages as displayed in Sect. 14.2.3. What is often omitted from these texts is a list of their limitations. For the reader (and potential user of these systems), it is imperative that these factors be discussed and explained so the strain sensing community achieves a common understanding of the present limitations of fiber optic strain sensors. The following lists the major limitations of fiber optic sensing technology. The degree to which each limitation affects fiber optic sensing technology is dependent on the particular style of fiber optic sensor

1

1.5

2 2.5 Bend radius (cm)

Fig. 14.11 Attenuation due to bend radius for SMF-28 optical fiber with one complete wrap 1550 nm (marks data, solid line theory)

• • • •

limited bend radius precise alignment of connections cost varying specifications

Limited Bend Radius In order to maintain optimum transmission of the optical power, the limited bend radius of the optical fiber must be adhered to. This is much more restricted compared to traditional electrical sensors where bending the electrical wires through a 90◦ turn does not damage the electrical wire. Thus placing optical fiber sensors near physical boundaries requires some consideration. A general rule of thumb is to keep the bend radius greater than 2 cm. When the optical fiber bend radius decreases below 2 cm, the level of attenuation sharply increases, as shown in Fig. 14.11. Manufactures have been developing new classes of optical fiber for various applications. Some of these are considered bend insensitive. Precise Alignment with Connections Unlike traditional electrical sensors, where twisting two copper conductors together is all that is needed to make a connection, optical fiber requires precise alignment of the two end-faces to ensure proper transmission of the optical power. For single-mode optical fiber, the alignment of the adjoining cores is critical. The end-faces of the fiber optic connectors must also be polished and cleaned to ensure low losses at the connection. Issues with misalignment can cause signal loss and also reduce signal-to-noise levels with the introduction of backscat-

353

Part B 14.2

Source

Power loss (%) 100 90 80 70 60 50 40 30 20 10 0 0 0.5

14.2 General Fiber Optic Sensing Systems

354

Part B

Contact Methods

tered light from the interface. Fortunately, a number of fiber optic connectors have become industry standards and these provide quality connections.

Part B 14.3

Cost In some cases, the cost of the optical fiber sensor is equivalent to the price of standard telecommunication fiber and components. With the high-volume production of optical fiber and certain optical components, these items are reasonably priced. However, the instrumentation systems required to interrogate these sensors are high-precision electro-optic systems that are relatively expensive, especially compared with RSG instrumentation. Varying Specifications As mentioned earlier, there are multiple choices for instrumentation systems for each fiber optic strain sensor type. This level of variability in the marketplace has led to a lack of industry standards, with companies continuing to market a particular technology to various application areas. This issue is of greatest importance for the fiber Bragg grating sensor discussed in this chapter where various types of the FBG sensors can be used to measure strain with different interrogation techniques.

respect to temperature. With RSG sensors, proper selection of materials and fabrication of the foil allows for the manufacture of sensors that are thermally compensated for a particular material. This same process of developing thermally self-compensating sensors cannot be implemented with fiber optic sensors. There is a definite limit on the material property selection for optical fibers, and the fabrication procedures for optical fibers do not allow for the development of material property variations that occur with metal coldworking processes. There are some athermal packages of fiber optic sensors available commercially, but these are not designed for strain measurement. The sensors are bonded into self-contained housings (not suited for bonding to other structures) that are typically composed of materials of differing thermal expansion coefficients, thus counteracting any expansion or contraction due to thermal variations. For the most part, thermal compensation is accomplished by using a secondary sensor to measure the temperature and subtract this effect from the measured strain reading of the fiber optic sensor. Often, the secondary sensor is another fiber optic sensor of similar design, but isolated from the strain field such that the sensor thermal response is well characterized for the application.

14.2.6 Introduction to Strain–Optic Effect 14.2.5 Thermal Effects Most fiber optic sensors also experience issues of thermal apparent strain readings. Extrinsic sensors, whose strain sensing is accomplished in an air gap or comparable structure, may have negligible thermal issues. Intrinsic sensors suffer from thermal influences due to the thermal expansion of the optical fiber and changes in the refractive index with respect to temperature. These quantities are comparable to the elongation and the resistance change of electrical resistance strain gages with

Central to many fiber optic strain sensors is the influence of the strain–optic effect. As a mechanical load is applied to the optical fiber, the length of the optical fiber changes and the refractive index also changes as a function of the strain field. The change in the refractive index is similar to the change in specific resistance of electrical strain gages with respect to the applied strain. Similar to the thermal effects, extrinsic sensors do not experience this influence due to the use of an air cavity as the sensing region.

14.3 Interferometry As low-loss optical fiber became more readily available, researchers in the late 1970s and 1980s began exploring the uses of optical fiber as a sensing medium. Some of the first sensors to be explored were optical fiber equivalents of classical interferometers such as those of Michelson and Mach–Zehnder (MZ) [14.1]. Another interferometer, the Sagnac fiber optic sensor, has enjoyed commercial success as the fiber optic gyroscope [14.6].

Many of the basic concepts employed in fiber optic interferometry are discussed in Chap. 18 in this Handbook. Each interferometer discussed in this section makes use of fiber optic components to create two optical paths that interfere to provide a measurement of strain. Similar to full-field strain measurement discussed in other chapters of this Handbook, fiber optic interferometers must extract the strain data from the resulting fringe information. In the case of fiber op-

Optical Fiber Strain Gages

tic interferometers, the resulting interference signal is detected via a photodetector and transformed to a voltage signal, which is processed via an instrumentation system.

14.3.1 Two-Beam Interference

where nd is the optical path length for the wave, ω is the frequency of the light wave, and φ is the phase of the optical wave. Interference is generated by the combination of two coherent light waves with different optical phases. In general, the phase difference is created by having the two optical waves travel different paths forming a path length difference (PLD) as discussed in Chap. 18. The derivation of the intensity response of a general interferometer begins with two coherent light waves that differ by the optical phase term, as given in (14.6). E1 = A1 cos(φ1 − ωt) , E2 = A2 cos(φ2 − ωt) .

(14.6)

The intensity that results from the interference of these two electric field vectors is given by I = |E1 + E2 |2 . I= 2A1 A2 cos(φ1 − ωt) cos(φ2 − ωt) . A21 cos2 (φ1 − ωt) +

identity: 2 cos(α) cos(β) = cos(α + β) + cos(α − β). Applying these two considerations to (14.8), the intensity function becomes I = A¯ + A1 A2 cos(φ1 + φ2 − 2ωt) + . . . A1 A2 cos(φ1 − φ2 ) .

I = A + B cos(Δφ) ,

(14.8)

The first two terms in (14.8) are the intensity of the two individual electric field vectors and only contribute an effective constant to the measurement signal as described in Chap. 18. The final term in the above expression can be replaced by the product of cosines

(14.10)

where the phase difference (Δφ) contains the strain information of interest in the terms of a PLD. The differences between the types of fiber optic interferometers are based on how the two coherent beams are derived from the system and where the two optical paths are recombined. In classical interferometers, the output of a light source is split via a beam-splitter and the two paths are recombined on a screen. In fiber optic interferometers, the beam splitting and recombining are accomplished within the optical fiber components so that the optical signals are always guided. The interference signal is then coupled to one (or more) photodetectors for signal acquisition and data processing.

14.3.2 Strain–Optic Effect The following derivation examines how the change in optical phase is related to the strain on the optical fiber. Where the stress–optic law discussed in Chap. 25 examines the optical effect on a two-dimensional structure (i. e., plate) under a state of plane stress, the strain–optic effect discussed here examines the influence of a strain state on an optical fiber. Light traveling in an optical fiber can be classified by an optical modal parameter called the propagation constant β given by [14.7] β = nkw =

A22 cos2 (φ2 − ωt) + . . .

(14.9)

The second term in (14.8) is at twice the optical frequency and only adds to the constant term in the equation. The final term in (14.8) provides a means for measuring the difference in optical phase between the two coherent light waves with intensity. Equation (14.8) can now be written in the more familiar form for interferometers

(14.7)

Substituting (14.6) into (14.7) yields:

355

2πn core , λ

(14.11)

where n core is the refractive index of the core of the optical fiber and kw is the wavenumber associated with the wavelength of light propagating in the optical fiber. The optical phase (φ) of light traveling through a section of optical fiber is given by φ = βL =

2πn core L. λ

(14.12)

Part B 14.3

As discussed in Chap. 18, the basic equation for the electric vector of a light wave is given by (18.1) and is rewritten here for convenience in (14.4):   2π (14.4) (z − vt) , E1 = A cos λ where A is a vector giving the amplitude and plane of the wave, λ is the wavelength, and v is the wave velocity. Since the wave is traveling in the optical fiber, the refractive index of the core is included in the path length and wave velocity as    2π c  nd − t E1 = A cos λ n

2π 2π nd − ct = A cos λ nλ (14.5) = A cos(φ − ωt) ,

14.3 Interferometry

356

Part B

Contact Methods

When a mechanical load is applied to the optical fiber, a phase change is experienced due to changes in the length of the optical fiber (strain) and due to changes to the propagation constant (strain–optic). These influences are the basis for interferometric strain sensors. The change in optical phase (Δφ) can be written as [14.8] Δφ = βΔL + LΔβ .

(14.13)

Part B 14.3

The influence of the βΔL term is directly related to the strain on the optical fiber (ε), as a change in length and can be written as βεL. The change in the propagation constant is attributed to two sources. One is a waveguide dispersion effect due to the change in the optical fiber diameter. This effect is considered negligible compared to the other influences on the optical phase change [14.8]. The other source is the change in the refractive index, written as dβ (14.14) Δn . LΔβ = L dn The dβ/ dn term is reduced to the wavenumber, kw , but the Δn term is related to the optical indicatrix, Δ[1/n 2 ]2,3 , as [14.8, 9] 

n3 1 (14.15) Δn = − , Δ 2 2 n 2,3 where light is propagating in the axial direction of the optical fiber as indicated in Fig. 14.12. For the case of small strains, the optical indicatrix is related to the strain on the optical fiber as:   1 = pij ε j , (14.16) Δ 2 n eff i where pij is the strain–optic tensor of the optical fiber and ε j is the contracted strain tensor. In the general case, the strain–optic tensor will have nine distinct elements, termed photoelastic constants related to the normal strains (shear strain is neglected). Fortunately, for a homogenous, isotropic material (such as optical fiber), the strain–optic tensor can be 2

Fig. 14.12 Indicial directions for optical fiber

For the case of a surface-mounted sensor, it has been shown that the contracted strain tensor with zero shear strain may be written as [14.2, 9] ⎛ ⎞ 1 ⎜ ⎟ (14.18) ε j = ⎝−ν ⎠ εz , −ν where ν is Poisson’s ratio for the optical fiber. If transverse loads are of interest, they may be incorporated through the strain tensor. Due to the symmetry of the optical fiber, the optical indicatrix is equivalent for i = 2 or 3. Incorporating the contracted strain tensor into (14.16) leads to the following expression for a surfacemounted optical fiber sensor   1 = εz [ p12 − ν( p11 + p12 )] . (14.19) Δ 2 n eff 2,3 After substituting (14.19) into (14.13), the result is Δφ = βεz L −

2π L n 3 εz [ p12 − ν( p11 + p12 )] . λ 2 (14.20)

Reducing the above equation leads to:   n2 Δφ = βL 1 − [ p12 − ν( p11 + p12 )] εz . 2 (14.21)

Normalizing (14.21) by (14.12) yields the standard form of the phase change with respect to the original phase:   n2 Δφ = 1 − [ p12 − ν( p11 + p12 )] εz . (14.22) φ 2

14.3.3 Optical Coherence

1 3

represented by two photoelastic constants p11 and p12 as [14.9] ⎞ ⎛ p11 p12 p12 ⎟ ⎜ pij = ⎝ p12 p11 p12 ⎠ . (14.17) p12 p12 p11

Another important concept for fiber optic interferometers is optical coherence. Interference of optical waves can only occur if the two waves are coherent, meaning that the phase information between the two waves is related. When light is emitted from a source, it is composed of coherent packets of energy. Coherence of

Optical Fiber Strain Gages

Fig. 14.13 Optical source spectrum

λ 20 Lc ≈ Δλ

357

Sensing path

Δλ λ0

14.3 Interferometry

Reference path

Fig. 14.14 Basic Mach–Zehnder fiber optic sensor arrangement

λ

14.3.5 Michelson

Lc ≈

λ20 Δλ

.

(14.23)

14.3.4 Mach–Zehnder As optical fiber and optical fiber components became available for sensing purposes, the recreation of classic interferometers was the first choice for early researchers. The first reported fiber optic strain gage by Butter and Hocker was an optical-fiber-based Mach– Zehnder [14.8]. In this early version, the output from two optical fibers was recombined on a screen to monitor the resulting interference pattern. As displayed in Fig. 14.14, light from a coherent light source is split into two optical paths via a 3 dB coupler. A second 3 dB coupler is used to recombine the optical paths and create the interference signal. The interference signal passes through both outputs of the second 3 dB coupler to the photodetectors. The use of both outputs is beneficial to some interrogation techniques. The path length difference for the Mach–Zehnder interferometer is dependent on the strain applied to the sensing path.

The history of the Michelson interferometer was discussed in the basic optics chapter of this handbook (Chap. 18). Like the Mach–Zehnder interferometer, the Michelson interferometer is constructed in a similar fashion to its classical namesake. The Michelson interferometer, illustrated in Fig. 14.15, uses a 3 dB coupler to divide the optical source into two optical paths. Each optical path has a mirror that reflects the optical signal back to the 3 dB coupler where the two paths interfere. The intensity of the interference signal is then monitored via the photodetector. The reference path is usually isolated from strain, so changes to the sensing path due to the applied load create an optical phase change with respect to the reference path.

14.3.6 Fabry–Pérot The basis of a Fabry–Pérot interferometer is the interference of a light wave reflected from a cavity. French physicists Fabry and Pérot analyzed this type of optical structure in the 19th century. As displayed in Fig. 14.16, the basic fiber optic Fabry–Pérot sensor functions from the interference of a reflection from a first surface, fiber to air, and a second surface, air to fiber. The air–glass interface leads to a reflection of approximately 4% of the input intensity. With such a low reflectivity percentage, the Fabry–Pérot interferometer can be assumed to be a two-beam interferometer. If higher-reflectivity coatings are applied, then a multipass interferometer is created. The particular Fabry–Pérot sensor displayed in Fig. 14.16 is an extrinsic Fabry–Pérot interferometer

Sensing path

Reference path

Fig. 14.15 Basic Michelson fiber optic sensor

Part B 14.3

an optical source is a measure of the length of the optical wave packets. Each wave packet can be thought to have a different starting optical phase; thus, light from two different wave packets have different starting phases and will not produce an interference pattern. For very narrow-band sources such as lasers, the coherence length is very long. For broadband optical sources such as light bulbs, the coherence length is very short. In order for an interferometer to function and produce an interference signal, the coherence length of the optical source must be greater than the resulting PLD of the interferometer. A general rule of thumb for determining the coherence length (L c ) of an optical source is to divide the square of the center wavelength (λ0 ) by the bandwidth of the source (Δλ), as given by (14.23). A depiction of the optical source spectrum is shown in Fig. 14.13.

358

Part B

Contact Methods

signal is generated when the output of the PM fiber is launched back into standard single-mode fiber. It should be noted that PM fiber and PM optical components are also used to make Michelson, Mach– Zehnder, and Fabry–Pérot interferometers with the light being restricted to a single polarization axis. This type of construction alleviates issues due to polarization fading that are common among the optical fiber interferometers. Fig. 14.16 Schematic of a fiber optic Fabry–Pérot sensor

14.3.8 Interrogation of Interferometers Part B 14.3

(EFPI). The sensor is extrinsic because the sensing region is the air cavity formed by the two optical fiber end faces. There are intrinsic Fabry–Pérot sensors where the sensing cavity is formed between partial mirrors fused within an optical fiber. These sensors are much more difficult to fabricate and have not experienced the same commercial availability as the EFPI sensors. The low-finesse interferometer allows for a twobeam interferometer to be assumed for mathematical analysis. With the sensing region taking place in an air cavity for the EFPI, the influence of the strain–optic effect can be neglected and the refractive index is assumed to be equal to 1. Thus, the optical phase change with respect to applied strain can be written as Δφ = βΔL =

4π L FP ε. λ

(14.24)

For the case of intrinsic Fabry–Pérot sensors, the strain– optic effect will be included in the phase change formulation.

14.3.7 Polarization As discussed in Sect. 14.1.2, there exist specialty optical fibers that are designed to preserve the polarization of light as it is guided along the optical fiber. These polarization-maintaining (PM) fibers induce a birefringence state in the optical fiber where the refractive index along the two principal axes of polarization are different. Thus, light traveling in the optical fiber will travel faster along the lower index axis (fast axis) than the higher-index axis (slow axis). In the simplest form, the polarization interferometer functions by taking light from a standard single-mode optical fiber and launches it into a length of PM fiber such that light travels down both the fast and slow axes. These two paths represent the two paths for the interferometer. Any applied load to the PM fiber will induce an optical phase change between the two paths due to the difference in the refractive index between the two axes. The interference

Phase Detection There exist a number of means for interrogating interferometric fiber optic strain sensors. Each of these techniques is focused on extracting the optical phase change from the cosine function given in (14.10). Some techniques use an active modulation scheme where the reference path is perturbed by a sinusoidal or ramp function. Other forms of modulation include varying the optical source wavelength (optical frequency modulation). These active modulation techniques have different attributes in terms of frequency response and strain range. The selection of the phase detection technique is dependent on the selected interferometric sensor and the sensing application requirements. Passive interrogation techniques for interferometric sensors are concerned with optically deriving a sin(Δφ) and cos(Δφ) function from the sensing system. Then, an arctangent operation can be used to obtain the phase change. Another form of phase detection is based on low-coherence interferometry, also called white-light interferometry. These techniques use a broadband optical source with a coherence length shorter than the optical path difference (OPD) of the interferometer. In order to achieve an interference signal, the optical output of the sensing interferometer is passed through a reference interferometer, also called a readout interferometer. The resulting OPD between the two Intensity

Strain

Time

Fig. 14.17 Phase ambiguity with respect to change in strain (dashed line)

Optical Fiber Strain Gages

interferometers is less than the coherence length of the optical source leading to an interference signal. The reference interferometer can then be tuned to the sensing interferometer as a means of interrogation. This method is advantageous because it allows the use of lower cost and longer-life broadband optical sources such as lightemitting diodes (LEDs). Phase Ambiguity One of the main difficulties with interferometric detection is the issue of phase ambiguity. As an increasing strain field is applied to the sensor, the intensity signal is a cosine function given by (14.10). With this

14.4 Scattering

type of response, differentiating between an increasing strain and a decreasing strain is difficult. As shown in Fig. 14.17, the intensity response looks the same for increasing and decreasing strain. The crossover point can be identified by the change in the intensity pattern. If this change occurs at a maximum or minimum of the intensity function the crossover becomes difficult to detect. Some interrogation techniques limit the intensity response (and thus the phase change) to a single linear portion of the cosine function. Other interrogation techniques must account for multiple fringes in the response function. This is similar to fringe counting as discussed in Chap. 25.

where Va is velocity of sound in the optical fiber and λ is the free-space operating wavelength. The Brillouin frequency shift is dependent on the density of the optical fiber, which is dependent on strain and temperature.

14.4.2 Strain Sensing Using Brillouin Scattering Measuring strain with Brillouin scattering requires the generation of a narrow-line-width short-duration pulse. As the pulse travels down the optical fiber, scattering occurs at the Brillouin frequency. The location along the optical fiber is determined by gating the optical pulse and measuring the time of flight of the returned scattered signal. The standard spatial resolution for these sensor systems is approximately 1 m. This level Intensity E = hν =

hc λ 3 – 5 orders of magnitude

Anti-Stokes

14.4.1 Brillouin Scattering Brillouin scattering is a result of the interaction between light propagating in the optical fiber and spontaneous sound waves within the optical fiber. Because the sound wave is moving, the scattered light is Doppler-shifted to a different frequency. The Brillouin frequency shift (ν B ) is determined by:

Temperature dependent

Pump wavelength

Rayleigh scattering Brillouin scattering

Fluorescence Stokes

Raman scattering

Wavelength

Fig. 14.18 Picture of various scattering effects with respect to the

νB = 2nVa /λ ,

(14.25)

pump wavelength

Part B 14.4

14.4 Scattering As light travels through optical fiber, scattering mechanisms displayed in Fig. 14.18 come into play that can be exploited for sensing purposes. In order to achieve measurable scattering effects, a relatively high-powered optical source is required. Rayleigh scattering results from elastic collision between the phonons and the optical fiber material. The scattered wavelength (which will be guided back towards the optical source) is equivalent to the pump or excitation wavelength. Rayleigh scattering is extensively used with pulsed laser sources and time-gated detectors to monitor long lengths of optical fiber for attenuation sources such as connections, optical splices, tight bend locations, and breaks in the optical fiber. Raman scattering involves the inelastic scattering effect of the optical energy interacting with molecular bonds. Raman scattering is used extensively for distributed temperature monitoring. The Raman scattering effect is immune to strain effects, making it ideal for these applications. Brillouin scattering results from reflections from a spontaneous sound wave propagating in the fiber [14.10]. The Brillouin scattering effect is sensitive to both strain and temperature variations.

359

360

Part B

Contact Methods

a) Brillouin gain 1.016 1.012

Data acquisition 1.008 1.004 1 0.996 –100 0

Part B 14.4

100 200 300 400 500 600 700 Position along the fibre (m)

b) Gain 1.02 1.01 1

c) Brillouin frequency shift (GHz) 12.83 12.82 12.81 12.8 12.79 12.78 12.77 12.76 12.75 12.74 –100 0

12.7 12.75 12.8 12.85

700 600 500 Distance (m) 400 300 200 100

12.9 0 Frequency (GHz)

Data analysis

100 200 300 400 500 600 700 Position along the fibre (m)

Fig. 14.19 Illustration of a typical measurement from a Brillouin scattering system [14.11] (Courtesy of OmniSensTM )

of spatial resolution is adequate for applications on long structures such as pipeline monitoring. In order to construct the Brillouin frequency response along the optical fiber, the optical pulse is scanned through the frequency range of interest. In this manner, each pulse provides a measurement of the scattered intensity along the optical fiber at a particular frequency, as depicted in Fig. 14.19. A three-dimensional plot is generated with respect to optical frequency and position along the optical fiber, with each pulse producing an intensity response at a single frequency. The Brillouin frequency shift is then computed from this plot by taking the max-

imum gain intensity at each position along the optical fiber. Since Brillouin scattering is sensitive to both strain and temperature, a means for performing temperature compensation is required. In most cases, a secondary optical fiber is deployed in a loose tube fashion such that no strain is coupled and temperature can be monitored along this fiber and compensated for in the strain measurement, as discussed in Sect. 14.2.5. In these cases, an alternative optical method such as Raman scattering may also be used to monitor the temperature.

Optical Fiber Strain Gages

14.5 Fiber Bragg Grating Sensors

361

14.5 Fiber Bragg Grating Sensors Because of these difficulties, sensing applications with fiber gratings were not realized until a novel manufacturing technique was developed in 1989 [14.15]. In the case of Bragg grating formation, the change of the refractive index of an optical fiber is induced by exposure to intense UV radiation (typical UV wavelengths are in a range from 150 to 248 nm). To create a periodic change in the refractive index, an interference pattern of UV radiation is produced such that it is focused onto the core region of the optical fiber. The refractive index of the optical fiber core changes where the intensity is brightest in the interference pattern to produce a periodic refractive index profile [14.12]. The manner in which the refractive index change is induced has been linked to multiple mechanisms over the past 15 years of research. The various mechanisms are dependent on the intensity of the UV source, the exposure time, and any pretreatment of the optical fiber. The resulting FBG from each mechanism displays slightly different properties related to the optical spectrum response, sensitivity to strain and temperature, and performance at elevated temperatures. The most common type of FBG sensor (type I) is a result of short exposure times to relatively low-intensity UV radiation to standard telecommunications optical fiber. There exist other FBG sensor types (type IIa, type II, chemical composition) that have been researched for sensing applications. The discussion in this section will be limited to the standard type I FBG sensor. The length of a single period for the grating structure is called the grating pitch, Λ, as shown in Fig. 14.20. The pitch of the grating is controlled durCladding

14.5.1 Fabrication Techniques Fiber Bragg grating sensors are unique compared to the strain sensors discussed thus far in this chapter. The sensors up to this point have been fabricated from a combination of standard optical fibers or making use of inherent properties of optical fibers. The FBG sensor is fabricated by altering the refractive index structure of the core of an optical fiber. The refractive index change is made possible in standard communications-grade optical fiber due to a phenomena known as photosensitivity discovered in 1978 by Hill [14.13]. Hill-type gratings are limited in practicality because they reflect the UV wavelength that was used to induce the refractive index change and are difficult to manufacture.

Λ

Core

n neff ncore z

Fig. 14.20 Schematic of an FBG sensor

Part B 14.5

Fiber Bragg gratings (FBGs) have become an integral part of the telecommunications hardware, used in applications such as add/drop filters, fiber lasers, and data multiplexing [14.12]. Since the discovery of the photosensitive effect in optical fiber [14.13] by which ultraviolet (UV) light is used to induce a permanent change in the refractive index of optical fiber, researchers have been discovering new applications for this unique optical phenomena. Because of the widespread use and development of this technology, several textbooks dedicated to FBGs and their applications have been published [14.2, 12, 14]. In recent years, the FBG sensor has become a popular choice for fiber optic sensor applications for many reasons. As will be seen in this section, the optical response of the FBG sensor is wavelength-encoded, allowing many FBG sensors to be serially multiplexed via wavelength division multiplexing techniques. FBG sensors may also be time division multiplexed, allowing approximately 100 FBG sensors to be monitored along a single fiber strand. There exist other multiplexing techniques for FBG sensors that allow thousands of FBG sensors to be serially multiplexed and monitored with a single instrumentation system. Many commercially available interrogation systems have also come to market, making these types of sensors and systems readily available and relatively less complex for the enduser. Unfortunately, many of these systems use different technologies for interrogation of the FBG sensors, leading to different sensor specifications as well as different requirements for the fabrication of the FBG sensor. An overview of the different interrogation technologies is presented in Sect. 14.5.5.

362

Part B

Contact Methods

Lloyd mirror Focused UV Focused UV

Optical fiber

Phase mask

Optical fiber

–1

Interference pattern

+1

Interference pattern Reflective surface

Fig. 14.21 Phase mask fabrication technique

Part B 14.5

ing the manufacturing process, and is typically on the order of ≈ 0.5 μm, while the amplitude of the index variation is only on the order of 0.01–0.1% of the original refractive index [14.12]. Many methods are used to create the periodic interference pattern (e.g., phase masks [14.16], Mach–Zehnder interferometers [14.15], and Lloyd mirrors [14.14]). Phase masks are corrugated silica optical components, as shown in Fig. 14.21. As laser radiation passes through the phase mask, the light is divided into different diffraction beams. The diffracted beams in the example are the +1 and −1 beams, which create an interference pattern that is focused on the optical fiber core. Other orders of diffraction are designed to be minimized in these passive optical devices. The interference pattern induces a periodic refractive index change along the exposed length of the optical fiber, thus creating the Bragg grating. Phase masks provide a stable, repeatable interference pattern and are used for high-volume production of FBG sensors at a common wavelength. The Mach–Zehnder interferometer technique uses a bulk optic Mach–Zehnder interferometer to create an interference pattern on the optical fiber. The optical arrangement for a basic Mach–Zehnder technique is shown in Fig. 14.22. Light from the laser emission is passed through focusing optics and is split via an optical beam splitter. The two divergent laser beams are Focused UV

Mirror

Interference pattern Beam splitter Mirror Optical fiber

Fig. 14.22 Mach–Zehnder interferometer technique

Fig. 14.23 Lloyd mirror technique

redirected via mirrors to combine at the optical fiber location, resulting in an interference pattern from the combination of these two optical paths. The focusing optics are designed to focus the laser energy at the optical fiber location. This technique allows for the fabrication of FBG sensors with different pitches through adjustment of the angle of incidence of the interference beams. The Lloyd mirror technique uses a Lloyd mirror to create an interference pattern on the optical fiber, as shown in Fig. 14.23. Light from the laser emission is passed through focusing optics and transmitted to the Lloyd mirror arrangement. The Lloyd mirror causes the input laser beam to split into two beams. These beams are recombined and focused at the optical fiber location, which creates an interference pattern and forms the FBG sensor.

14.5.2 Fiber Bragg Grating Optical Response Light traveling in an optical fiber can be classified by an optical modal parameter β, as discussed in Sect. 14.3 and given by [14.7] 2πn core (14.26) , λ where n core is the refractive index of the core of the optical fiber and kw is the wavenumber associated with the wavelength of light propagating in the optical fiber. The function of the Bragg grating is to transfer a forward-propagating mode (β1 ) into a backwardpropagating mode (−β1 ) (i. e., reflect the light) for a particular wavelength meeting the phase-matching criterion. The phase-matching condition is derived from coupled-mode theory and is given by β = nkw =

βi − β j =

2πm , Λ

(14.27)

Optical Fiber Strain Gages

14.5 Fiber Bragg Grating Sensors

363

Fig. 14.24 Schematic of an FBG sensor with reflected and transmitted spectra

Reflected spectrum

Incoming spectrum

Transmitted spectrum

Λ

λB = 2n eff Λ ,

(14.29)

(ΔT ), so that the Bragg wavelength shifts to higher or lower wavelengths in response to applied thermalmechanical fields. For most applications, the shift in the Bragg wavelength is considered a linear function of the thermal-mechanical load. The treatment of FBG sensors here will ignore the thermal effects, because the thermal effects can be modeled as an independent response of the Bragg grating. The shift in the Bragg wavelength due to an incremental change of length (ΔL) is given by [14.9]:

∂Λ ∂n eff (14.30) + n eff ΔL . ΔλB = 2 Λ ∂L ∂L Assuming that the strain field is uniform across the Bragg grating length (L), the term ∂Λ/∂L can be replaced with Λ/L. Likewise, the term ∂n eff /∂L can be replaced by Δn eff /ΔL in (14.30). The terms Λ and L are physical quantities that are determined by the interference pattern formed during fabrication and are known. The change in the effective refractive index (Δn eff ) can be related to the optical indicatrix, Δ[1/n 2eff ], as discussed in Sect. 14.3 [14.9]. Δn eff = −



 n 3eff 1 . Δ 2 2 n eff

(14.31)

where the subscript ‘B’ defines the wavelength as the Bragg wavelength. Equation (14.29) states that, for a given pitch (Λ) and average refractive index (n eff ), the wavelength λB will be reflected from the Bragg grating, as illustrated in Fig. 14.24.

In the case of small strains, the optical indicatrix is related to the strain on the optical fiber as:   1 Δ 2 = pij ε j , (14.32) n eff i

14.5.3 Strain Sensing Using FBG Sensors

where pij is the strain–optic tensor of the optical fiber and ε j is the contracted strain tensor. The directions associated with the indices in (14.32) are illustrated in Fig. 14.25. In the general case, the strain–optic tensor will have nine distinct elements, termed photoelastic constants. Fortunately, for a homogenous, isotropic material (such as optical fiber), the strain–optic tensor can be represented by two photoelastic constants p11 and

Bragg gratings operate as wavelength selective filters reflecting the Bragg wavelength λB which is related to the grating pitch Λ and the mean refractive index of the core, n eff , given by (14.29). Both the effective refractive index (n eff ) of the core and the grating pitch (Λ) vary with changes in strain (ε) and temperature

Part B 14.5

where m is an integer value representing the harmonic order of the grating [14.12]. From (14.27), multiple propagation modes can be used to satisfy the phasematching condition. Therefore, a single FBG sensor will reflect multiple wavelengths with respect to the order of the integer m. Most optical sources do not have a large enough bandwidth to excite multiple wavelength reflections from a single FBG sensor. Some research has been performed using multiple optical sources to excite more than the first-order Bragg condition as a means of providing a temperature and strain measurement with a single FBG sensor [14.17]. For all practical applications to date, only the first-order Bragg condition of m = 1 is employed for standard uniform Bragg gratings. For this case, the backward-propagating mode (−β1 ) is substituted into (14.27) for β j , and the following equation is derived π 2πn eff (14.28) β1 = = . Λ λ The index of refraction is now noted as an effective (or average) refractive index, n eff , for the Bragg grating, due to the periodic change across the length of the optical fiber. Solving (14.28) for the wavelength (λ) provides the Bragg wavelength equation

364

Part B

Contact Methods

2

1 3

Fig. 14.25 Indicial directions for an FBG optical fiber

Part B 14.5

p12 as [14.9]: ⎞ ⎛ p11 p12 p12 ⎟ ⎜ pij = ⎝ p12 p11 p12 ⎠ . p12 p12 p11

Transverse loads may also be modeled using this analysis by choosing the proper contracted strain tensor formulation. For example, the measurement of a uniform pressure field on the optical fiber can be modeled with the following contracted strain tensor ⎛ ⎞ εz ⎜ ⎟ (14.39) ε j = ⎝εr ⎠ . εr Using (14.39) in the above formulation leads to the following response function for the FBG sensor response:

(14.33)

For the case of a surface mounted FBG sensor, it has been shown that the contracted strain tensor may be written as [14.2, 9]: ⎛ ⎞ 1 ⎜ ⎟ (14.34) ε j = ⎝−ν ⎠ εz , −ν where ν is Poisson’s ratio for the optical fiber. Due to the symmetry of the optical fiber, the optical indicatrix is equivalent for i = 2 or 3. Incorporating the strain tensor into (14.32) leads to the following expression for a surface-mounted optical fiber sensor:   1 (14.35) Δ 2 = εz [ p12 − ν( p11 + p12 )] . n eff

n2 ΔλB = εz − eff [εz p12 + εr ( p11 + p12 )] . λB 2

(14.40)

For other strain states, such as diametric loading, the analysis of the FBG response becomes more complicated. This is due to the induced polarization effects and nonuniform loading on the optical fiber. In extreme diametric loading cases, the FBG sensor response will split due to the reflection from the different polarization axis and nonuniform strain on the grating structure [14.18]. Research is continuing in this field for the development of transversely sensitive FBG sensors with emphasis on using FBG sensors written into PM optical fiber to take advantage of the pre-existing polarization properties.

14.5.4 Serial Multiplexing

One of the main advantages of FBG sensors is the ability to measure multiple physical parameters. This ability combined with serial multiplexing of FBG sensors alAfter substituting (14.31) and (14.35) into (14.30), the lows for multiple parameters to be monitored not only result is by a single instrument, but also with all the data transmitted on a single piece of optical fiber [14.19]. This is 3 n eff εz [ p12 − ν( p11 + p12 )] + 2n eff Λεz . advantageous in applications where minimal intrusion ΔλB = −2Λ 2 into an environment is required. Many applications of (14.36) FBG sensors concern measuring strain and/or temperNormalizing (14.36) by the Bragg wavelength demon- ature. In these applications, many sensors (from fewer strates the dependence on the wavelength shift of the than ten to over a hundred) are multiplexed to provide FBG sensor on the refractive index, strain–optic coeffi- measurements across the structure. Examples of these applications are monitoring civil infrastructure [14.20], cients, and Poisson’s ratio for the optical fiber. naval/marine vessels [14.21, 22], and shape measure n 2eff ΔλB  ment of flexible structures [14.23]. = 1− [ p12 − ν( p11 + p12 )] εz . (14.37) The ability to serially multiplex FBG sensors is arλB 2 guably the most prominent advantage of this sensor The terms multiplying the strain in (14.37) are constant type. With over a decade of development, researchers over the strain range of the Bragg grating, and (14.37) have devised three main techniques for serial multiis often written in simplified form as plexing FBG sensors: wavelength-domain multiplexing (WDM), time division multiplexing (TDM), and optical ΔλB = Pe εz . (14.38) frequency-domain reflectometry (OFDR). λB

Optical Fiber Strain Gages

Wavelength response of FBG #2 Λ1

Λ2

Λ3

Thermomechanical load

Reflected spectrum

Fig. 14.26 Serial multiplexing of FBG sensors

from which sensor. The default is to place the lowerwavelength signal with the previous lower-wavelength sensor signal. If the lower-wavelength signal completely overwhelms the higher-wavelength signal, as is depicted in Fig. 14.27, then the instrumentation records the wrong wavelength data for each of these sensors after the overlap event. Knowledge of the potential measurement range for each sensor is required to prevent sensor overlap issues. When this is not provided, conservative estimates should be used to select sensor wavelength separations. In cases where the FBG sensors are expected to experience similar responses, such as thermal measurements, the wavelength spacing of the sensors may be more tightly spaced. Time Division Multiplexing TDM uses a time-of-flight measurement to discriminate the FBG sensors on a fiber. The basic TDM architecture is shown in Fig. 14.28. A light source generates a short pulse of light that propagates down the optical fiber to a series of FBG sensors. All the FBG sensors initially have the same wavelength and reflect only a small portion of the light at the Bragg wavelength. The light source has a somewhat wide spectrum that is centered about the unstrained Bragg wavelength of the FBG sensors. Each FBG sensor generates a return pulse when it reflects the light at its particular Bragg wavelength, which is dependent upon the strain/temperature state of the FBG. For the example shown in Fig. 14.28, there are five return pulses generated by the FBG sensors. These pulses propagate back along the fiber and are coupled to a high-speed photodetector, which measures when each pulse was detected. The FBG sensors must be separated by a minimum distance along the fiber to allow accurate measurement of the difference in arrival time of the reflected pulses, often by 1 m or more. An additional wavelength measurement system is needed to measure the Bragg wavelength of each sensor as it is detected. TDM systems have become more popular in

ε, T

λB1

λB2

λ

λB1 λB2

λ

ε, T FBG #1

FBG #2

λB2 λB1 λ

Fig. 14.27 Wavelength shift description and sensor overlap for WDM systems

365

Part B 14.5

Wavelength Division Multiplexing As shown in Fig. 14.26, WDM is accomplished by producing an optical fiber with a sequence of spatially separated gratings, each with different grating pitches, Λi = 1, 2, 3, . . .. The output of the multiplexed sensors is processed through wavelength selective instrumentation, and the reflected spectrum contains a series of peaks, each peak associated with a different Bragg wavelength given by λi = 2nΛi . As indicated in the figure, the measurement field at grating 2 is uniquely encoded as a perturbation of the Bragg wavelength λ2 . Note that this multiplexing scheme is completely based on the optical wavelength of the Bragg grating sensors. The upper limit to the number of gratings that can be addressed in this manner is a function of the optical source profile width and the expected strain range. WDM was the first form of multiplexing explored for the FBG sensor and is common in commercially available systems. A major concern for WDM FBG systems is sensor wavelength overlap. This occurs when two neighboring sensors in wavelength space experience loads that cause the reflected Bragg wavelengths from each sensor to approach each other and overlap. During the overlap event, the monitoring instrumentation cannot process both sensors independently and will record only a single senor. After the overlap event when both sensors can be resolved, the instrumentation cannot distinguish which of the reflected Bragg wavelengths is

14.5 Fiber Bragg Grating Sensors

366

Part B

Contact Methods

I (t)

Light source

FBG sensors t

S-1 S-2 S-3 S-4 S-5

3 dB coupler I (t)

t1

Detector 1

t2

t3

t4

t5

t

Time delayed signals from each of the sensors

Light dump

Fig. 14.28 Time division multiplexing technique

Part B 14.5

recent years due to advances in optical sources, data acquisition, and computing technology.

determine the Bragg wavelength of that particular sensor [14.24].

Optical Frequency-Domain Reflectometry OFDR is a more complex interrogation technique then the other systems discussed. The basic concept of an OFDR system is to create a response function from the serially multiplexed FBG sensors that spatially isolates the individual FBG sensor response in the frequency domain of the system response [14.24]. A schematic of a typical OFDR optical system layout is shown in Fig. 14.29. Light from a tunable laser source is coupled into an optical arrangement containing the FBG sensors and a reference interferometer. The reference interferometer is used to trigger the sampling on detector 1, thus ensuring that the response from the FBG sensors is sampled at a constant wavenumber interval. The FBG sensors are low reflectivity, typically at a common wavelength, and can be very closely spaced (on the order of 1 cm). Each sampled set of FBG data from detector 1 is initially processed via a Fourier transform. The frequency spectrum obtained from the Fourier transform displays a collection of peaks at the physical distance of each FBG along the sensing array. Each peak is then bandpass filtered in the frequency spectrum and an inverse Fourier transform is used to

14.5.5 Interrogation of FBG Sensors, Wavelength Detection As the popularity of using FBG sensors has grown in recent years, so has the number of techniques to perform interrogation. Of course, the method of interrogation used depends on the type of multiplexing, the sampling rate requirements, and the measurement resolution requirements. The existing technologies for interrogating FBG sensors include scanning lasers, tunable filters, linear optical filters, and spectroscopic techniques. The end-user must ensure the instrumentation system chosen meets the requirements for the application. In terms of sampling rate, the response of FBG sensors is very fast; unfortunately, all instrumentation systems are limited by the speed of the electronics, data acquisition, and data processing. This is especially true for serially multiplexed systems. In most cases, the sampling rate for individual sensors is not selectable. The instrumentation records data at one sampling rate for all sensors interrogated by the system. The end-user may design additional data-acquisition software to store the data into separate files with different

Detector 1 Coupler Tunable laser source

FBG sensors

Coupler R

Li

Coupler

R R

Detector 2

Lref

Reference interferometer

Fig. 14.29 Schematic of typical OFDR interrogation design

Optical Fiber Strain Gages

14.5.6 Other Grating Structures Although not commonly employed for the measurement of strain, there do exist other fiber grating structures that are founded upon the variation of refractive index in the core of the optical fiber. Two structures in particular are the chirped grating and the long-period grating.

The chirped grating is an FBG with a linear variation of the periodicity (Λ(z)). Without a uniform periodicity, the reflected spectrum of the grating structure becomes spread out. This device is commonly used as a linear wavelength filter. Although the chirped grating has similar sensitivities to strain as the FBG sensor, it is almost never used as such because of the difficulty in obtaining a wavelength peak detection value, as well as increased fabrication costs. A long-period grating (LPG) is similar to the Bragg grating such that the periodicity is uniform, but the periodicity is much larger, on the order of Λ = 1 mm. Examination of the phase-matching condition (14.27) for the Bragg grating permits interaction between allowable optical modes (β y i). For the case of an LPG, the forward-propagating optical mode in the core is transferred to cladding modes, which are attenuated from the optical fiber, leading to wavelength-dependent attenuation in the transmitted signal. Once the light gets beyond the LPG structure, the various modes are coupled back to the core region and the wavelength attenuated signal is transmitted through the optical fiber. The attenuation mechanisms for the cladding modes are highly dependent on the boundary condition of the cladding. Therefore, LPG sensors have been examined for potential use as chemical sensors with the presence of a chemical species inducing wavelength attenuation shifts [14.25].

14.6 Applications of Fiber Optic Sensors Although fiber optic strain sensors are not widely used in any industry, they have been tested in many applications. Often, these applications have attributes that make traditional sensing technologies ineffectual or difficult to implement. With advantages of long lead lines, embedding, and multiplexing, fiber optic sensors have proven their ability to perform measurements were traditional sensors cannot. The following sections provide a brief overview of some of the applications. References [14.2, 3, 14] provide additional applications. With new applications of fiber optic sensors occurring regularly, a quick internet search will provide information on the latest and greatest applications.

14.6.1 Marine Applications In a marine environment, fiber optic sensors have the advantage of not requiring extensive waterproof-

ing for short-duration tests. Also, many sensors can be multiplexed to achieve a high sensor count for large structures such as ships and submarines. FBG sensors have been used to monitor wave impacts and loads on surface ships [14.21, 22] and for American Bureau of Shipping (ABS) certification of a manned submarine [14.26].

14.6.2 Oil and Gas Applications Fiber optic sensing in the oil and gas industry has made great strides in recent years. Of particular use are distributed temperature sensing (DTS) systems based on Raman scattering [14.27, 28]. For structural monitoring, the area that has witnessed use of fiber optic strain sensors is in riser monitoring. Risers are long structural components used on offshore platforms for many applications including drilling, water injection, and col-

367

Part B 14.6

sampling rates. Alternatively, postprocessing the oversampled data through averaging or filtering techniques may provide improved results. Some WDM multiplexed instrumentation systems require the FBG sensors to have certain nominal wavelengths to fit within the instrument’s wavelength filtering windows. This limits the flexibility of the wavelength spacing discussed previously and limits the number of FBG sensors that can be serially multiplexed with these instrumentation systems. Other WDM instrumentation systems allow for the wavelength spacing of neighboring FBG sensors to be less than 1 nm, allowing for over 100 FBG sensors to be serially multiplexed, but issues of sensor wavelength overlap must be taken into consideration. For TDM and OFDR systems, the special design of the FBG sensors in terms of low reflectivity and physical spacing requirements limits the availability of commercial vendors to provide these sensing arrays. The end-user can typically only purchase FBG sensors that function with the systems from the system vendors themselves.

14.6 Applications of Fiber Optic Sensors

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Contact Methods

lection of the oil. Tidal currents are known to induce vibrations in these structures, and fiber optics sensors have recently been used to gain an understanding of the structural effects the vibrations have on the risers [14.29, 30].

14.6.3 Wind Power Applications

Part B 14.7

With the upswing in oil prices, more effort is being expended on research and development of alternative energy sources, including renewable energy sources such as wind power. Wind power turbines are large structures that make use of composite blades. Wind turbine manufacturers are examining ways in which to improve the efficiency of these devices and are turning to fiber optic sensing to provide load monitoring and control data [14.31]. Other wind turbine applications using fiber optic sensors include health monitoring of the wind turbine [14.31] and shape measurement of the wind turbine blade [14.32]. Attributes that make fiber optic sensors attractive for these applications include multiplexing, immunity to radiofrequency (RF) noise, and immunity to damage caused by the electrical charge from lightning strikes. Fiber optic sensors also have the ability to be embedded into these composite structures.

14.6.4 Civil Structural Monitoring The application area that has received the most interest in fiber optic strain sensing has been civil structure

monitoring. Using the attributes of long lead fibers and sensor multiplexing, monitoring the loads and structural health of these large structures has been an ideal application for fiber optic sensors. FBG sensors have been used to monitor bridge structural components including bridge decks [14.2], composite piles [14.20], and stay-cables [14.2]. In each of these cases, the FBG sensor has been embedded into the structural component. As the growth of FBG sensor applications continues, many more examples of FBG sensors being employed in civil structural monitoring will be realized. Michelson-interferometer-based systems have also been used for civil structural monitoring purposes including bridges [14.33], dams [14.34], and buildings [14.35]. The Michelson interferometer has a long sensor gage length, which is beneficial when measuring the deformation of these large structures. The reference arm of the Michelson can also be packaged alongside the sensing arm, allowing for straightforward temperature compensation of the measured signal. Brillouin scattering sensor systems are also finding applications in civil structures, including pipeline monitoring and dam monitoring [14.36]. Again, the ease of embedding these sensors directly into the structure is leveraged for these applications. Furthermore, the distributed sensing nature of Brillouin-based sensors allows for a strain measurement at approximately every meter along the optical fiber, making Brillouin scattering appealing for these applications.

14.7 Summary Optical fiber sensing is a growing field, full of potential. In general, fiber optic sensing is a viable technology for strain monitoring applications in many industries. Advancement of this technology is dependent on leveraging its many advantages over traditional sensors. Arguably the key aspect of fiber optic sensors is the ability to multiplex many sensors or obtain distributed measurements using a single strand of optical fiber. This allows for a vast number of sensing points with reduced cabling weight and minimal intrusion into the application environment. The ability to embed optical fiber into composite structures is also a primary driver for fiber optic sensing in the composite structures field. As discussed in this chapter, there are many different types of fiber optic strain sensors. Some of

these sensor types have their roots in classical optics, while others take advantage of the special properties of optical fiber. There exist single-point sensors (EFPI, FBG), cumulative strain sensors (MZ, Michelson), and distributed strain sensors (Brillouin). The selection of a strain measurement technique is highly dependent on the application requirements. Just as there are multiple types of fiber optic strain sensors, there are often multiple interrogation techniques for each of the sensor types. This leads to a very complex design space for the novice engineer tasked with selecting the appropriate sensing system. Careful consideration of measurement resolution, sampling rate, and number of sensors will assist in making an informed decision.

Optical Fiber Strain Gages

References

369

References 14.1

14.2 14.3 14.4 14.5

14.7 14.8 14.9

14.10 14.11 14.12 14.13

14.14

14.15

14.16

14.17

14.18

14.19

14.20

14.21

14.22

14.23

14.24

14.25

14.26

14.27

14.28

14.29

14.30 14.31 14.32

14.33

14.34

C. Baldwin, T. Salter, J. Niemczuk, P. Chen, J. Kiddy: Structural monitoring of composite marine piles using multiplexed fiber Bragg grating sensors: Infield applications, Proc. SPIE 4696, 82–91 (2002) A. Kersey, M.A. Davis, T.A. Berkoff, A.D. Dandridge, R.T. Jones, T.E. Tsai, G.B. Cogdell, G.W. Wang, G.B. Havsgaard, K. Pran, S. Knudsen: Transient load monitoring on a composite hull ship using distributed fiber optic Bragg grating sensors, Proc. SPIE 3042, 421–430 (1997) C. Baldwin, J. Kiddy, T. Salter, P. Chen, J. Niemczuk: Fiber optic structural health monitoring system: Rough sea trials testing of the RV Triton, MTS/IEEE Oceans 2002, 3, 1807–1814 (2002) C. Baldwin, T. Salter, J. Kiddy: Static shape measurements using a multiplexed fiber Bragg grating sensor system, Proc. SPIE 5384, 206–217 (2004) B.A. Childers, M.E. Froggatt, S.G. Allison, T.C. Moore Sr., D.A. Hare, C.F. Batten, D.C. Jegley: Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure, Proc. SPIE 4332, 133–142 (2001) Z. Zhang, J.S. Sirkis: Temperature-compensated long period grating chemical sensor, Techn. Dig. Ser. 16, 294–297 (1997) J. Kiddy, C. Baldwin, T. Salter: Certification of a submarine design using fiber Bragg grating sensors, Proc. SPIE 5388, 133–142 (2004) P. E. Sanders: Optical Sensors in the Oil and Gas Industry, Presentation for New England Fiberoptic Council (NEFC) (2004) P. J. Wright, W. Womack: Fiber-Optic Down-Hole Sensing: A Discussion on Applications and Enabling Wellhead Connection Technology, Proc. of Offshore Technology Conference, OTC 18121 (2006) D.V. Brower, F. Abbassian, C.G. Caballero: Realtime Fatigue Monitoring of Deepwater Risers Using Fiber Optic Sensors, Proc. of ETCE/OMAE2000 Joint Conference: Energy for the New Millennium, ASME OFT-4066 (2000) pp. 173–180 L. Sutherland: Joint Industry Project, DeepStar, www.insensys.com, Insensys Ltd. (2005) Insensys: Wind Energy, www.insensys.com (2006) J. Kiddy: Deflections Measurements of Wind Turbine Blades using Fiber Optic Sensors, WINDPOWER 2006, Poster (2006) A. Del Grosso, D. Inaudi: European Perspective on Monitoring-Based Maintenance, IABMAS ’04, International Association for Bridge Maintenance and Safety (Kyoto 2004) P. Kronenberg, N. Casanova, D. Inaudi, S. Vurpillot: Dam monitoring with fiber optics deformation sensors, Proc. SPIE 3043, 2–11 (1997)

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14.6

K.T.V. Grattan, B.T. Meggitt (Eds.): Optical Fiber Sensor Technology, Vol. 1-4 (Kluwer Academic, Boston 1995) R.M. Measures: Structural Monitoring with Fiber Optic Technology (Academic, San Diego 2001) E. Udd (Ed.): Fiber Optic Smart Structures (Wiley, New York 1995) Crystal Fibre, Birkerød, Denmark (www.crystalfibre.com) N.G. Skinner, J.L. Maida Jr.: Downhole fiber-optic sensing: The oilfield service provider’s perspective, SPIE Fiber Optic Sensor Technol. Appl. III, 5589, 206–220 (2004) B. Culshaw: Optical fiber sensor technologies: Opportunities and – perhaps – pitfalls, J. Lightwave Technol. 22(1), 39–50 (2004) A. Snyder, J. Love: Optical Waveguide Theory (Chapman Hall, New York 1983) C.D. Butter, G.B. Hocker: Fiber optics strain gauge, Appl. Opt. 17(18), 2867–2869 (1978) S.M. Melle, K. Lui, R.M. Measures: Practical fiberoptic Bragg grating strain gauge system, Appl. Opt. 32, 3601–3609 (1993) B. Hitz: Fiber sensor uses Raman and Brillouin scattering, Photonics Spectra 39(7), 110–112 (2005) M. Nikles: Omnisens, www.omnisens.ch R. Kashyap: Fiber Bragg Gratings (Academic, San Diego 1999) K.O. Hill, Y. Fujii, D.C. Johnson, B.S. Kawasaki: Photosensitivity in optical fiber waveguides: Application to reflection filter fabrication, Appl. Phys. Lett. 32, 647–649 (1978) A. Othonos, K. Kalli: Fiber Bragg Gratings, Fundamentals and Applications in Telecommunications and Sensing (Artech House, Boston 1999) G. Meltz, W.W. Morey, W.H. Glenn: Formation of Bragg gratings in optical fibers by a transverse holographic method, Opt. Lett. 14, 823–825 (1989) K.O. Hill, B. Malo, F. Bilodeau, D.C. Johnson, J. Albert: Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask, Appl. Phys. Lett. 62, 1035–1037 (1993) P. Sivanesan, J.S. Sirkis, Y. Murata, S.G. Buckley: Optimal wavelength pair selection and accuracy analysis of dual fiber grating sensors for simultaneously measuring strain and temperature, Opt. Eng. 41, 2456–2463 (2002) R.B. Wagreich, J.S. Sirkis: Distinguishing fiber Bragg grating strain effects, 12th International Conference on Optical Fiber Sensors, OSA, Vol. 16 (Technical Digest Series, Williamsburg 1997) pp. 20–23 C. Baldwin: Multi-parameter sensing using fiber Bragg grating sensors, Proc. of SPIE 6004, Fiber optic sensor technologies and applications IV, 60040A-1 (2005)

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14.35

B. Glisic, D. Inaudi, K.C. Hoong, J.M. Lau: Monitoring of building columns during construction, 5th Asia Pacific Structural Engineering and Construction Conference (APSEC) (2003) pp. 593–606

14.36

D. Inaudi, B. Glisic: Application of distributed fiber optic sensory for SHM, 2nd International Conference on Structural Health Monitoring of Intelligent Infrastructure (2005)

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371

Residual Stre 15. Residual Stress

Yuri F. Kudryavtsev

15.1 Importance of Residual Stress ................ 15.1.1 Definition of Residual Stresses ....... 15.1.2 Origin of Residual Stresses ............. 15.1.3 Residual Stress Management: Measurement, Fatigue Analysis, and Beneficial Redistribution ........ 15.2 Residual Stress Measurement................. 15.2.1 Destructive Techniques for Residual Stress Measurement .... 15.2.2 Nondestructive Techniques for Residual Stress Measurement .... 15.2.3 Ultrasonic Method for Residual Stress Measurement ..................... 15.3 Residual Stress in Fatigue Analysis ......... 15.4 Residual Stress Modification .................. 15.5 Summary ............................................. References ..................................................

371 372 372

373 373 373 375 377 381 383 386 386

residual stresses published by the Society of Experimental Mechanics (SEM) in 1996 and 2005.

15.1 Importance of Residual Stress Residual stress can significantly affect the engineering properties of materials and structural components, notably fatigue life, distortion, dimensional stability, corrosion resistance, and brittle fracture [15.1]. Such effects usually lead to considerable expenditure in repairs and restoration of parts, equipment, and structures. For this reason, residual stress analysis is a compulsory stage in the design of parts and structural elements and in the estimation of their reliability under real service conditions. Systematic studies had shown that, for instance, welding residual stresses might lead to a drastic reduction in the fatigue strength of welded elements. In multicycle fatigue (N > 106 cycles), the effect of residual stresses can be comparable to the effect of stress concentration [15.2]. Even more significant are the ef-

fects of residual stresses on the fatigue life of welded elements in the case of relieving harmful tensile residual stresses and introducing beneficial compressive residual stresses in the weld toe zones. The results of fatigue testing of welded specimens in the as-welded condition and after the application of ultrasonic peening shows that, in the case of non-load-carrying fillet welded joint in high-strength steel, redistribution of residual stresses resulted in approximately twofold increase in the limit stress range [15.1]. The residual stresses are therefore one of the main factors determining the engineering properties of materials, parts, and welded elements, and should be taken into account during the design and manufacturing of different products. Although certain progress has been achieved in the development of techniques for re-

Part B 15

In many cases residual stresses are one of the main factors determining the engineering properties of parts and structural components. This factor plays a significant role, for example, in fatigue of welded elements. The influence of residual stresses on the multicycle fatigue life of butt and fillet welds can be compared with the effects of stress concentration. The main stages of residual stress management are considered in this chapter with the emphasis on practical application of various destructive and nondestructive techniques for residual stress measurement in materials, parts, and welded elements. Some results of testing showing the role of residual stresses in fatigue processes as well as aspects and examples of ultrasonic stress-relieving are also considered in this chapter. The presented data on residual stresses are complimentary to the detailed review of various methods of residual stress analysis considered in two handbooks on

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It is the first level or macroscopic (type I) residual stress that is of interest to mechanical engineers and design offices and that is considered in this paper.

σres (MPa) 300 12

15.1.2 Origin of Residual Stresses x

200

12 100

0

–100

0

25

50

75

100 x (mm)

Part B 15.1

Fig. 15.1 Distribution of longitudinal (oriented along the weld) residual stresses near the fillet weld in a bridge span. x is the distance from the weld toe [15.3]

sidual stress management, considerable effort is still required to develop efficient and cost-effective methods of residual stress measurement and analysis as well as technologies for the beneficial redistribution of residual stresses.

15.1.1 Definition of Residual Stresses Residual stresses (RS) can be defined as those stresses that remain in a material or body after manufacture and material processing in the absence of external forces or thermal gradients. Residual stresses can also be produced by service loading, leading to inhomogeneous plastic deformation in the part or specimen. Residual stresses can be defined as either macro- or microstresses and both may be present in a component at any one time. Residual stresses can be classified as Type I: Macro residual stresses that develop in the body of a component on a scale larger than the grain size of the material Type II: Micro residual stresses that vary on the scale of an individual grain Type III: Micro residual stresses that exist within a grain, essentially as a result of the presence of dislocations and other crystalline defects

Residual stresses develop during most manufacturing processes involving material deformation, heat treatment, machining or processing operations that transform the shape or change the properties of a material. They arise from a number of sources and can be present in the unprocessed raw material, introduced during manufacturing or arise from in-service loading. The origins of residual stresses in a component may be classified as

• • •

differential plastic flow differential cooling rates phase transformations with volume changes etc.

For instance, the presence of tensile residual stresses in a part or structural element are generally harmful since they can contribute to, and are often the major cause of, fatigue failure and stress-corrosion cracking. Compressive residual stresses induced by different means in the (sub)surface layers of material are usually beneficial since they prevent origination and propagation of fatigue cracks, and increase wear and corrosion resistance. Examples of operations that produce harmful tensile stresses are welding, machining, grinding, and rod or wire drawing. Figure 15.1 shows a characteristic residual stress profile resulting from welding. The residual stresses were measured by an ultrasonic method in the main wall of a bridge span near the end of one of the welded vertical attachments [15.3]. In the vicinity of the weld the measured levels of harmful tensile residual stresses reached 240 MPa. Such high tensile residual stresses are the result of thermoplastic deformations during the welding process and are one of the main factors leading to the origination and propagation of fatigue cracks in welded elements. On the other hand, compressive residual stresses usually lead to performance benefits and can be introduced, for instance, by peening processes such as shot peening, hammer peening, laser peening, and ultrasonic peening [15.1]. Figure 15.2 shows characteristic distributions of beneficial compressive residual stress in the surface layers of material resulting from conventional and ultrasonic shot peening processes [15.4].

Residual Stress

15.1.3 Residual Stress Management: Measurement, Fatigue Analysis, and Beneficial Redistribution

15.2 Residual Stress Measurement

373

σr (MPa) 200 100

It is very important to consider the problem of residual stress as a complex problem including, at least, the stages of determination, analysis, and beneficial redistribution of residual stresses. The combined consideration of these stages of the residual stress analysis and modification gives rise to so-called the residual stress management (RSM) concept approach [15.5]. The RSM concept includes the following main stages

0 –100 –200 –300 – 400

Stage 1. Residual stress determination:

• •

Experimental studies Computation

Stage 3. Residual stress modification (if required):

• •

Changes in technology of manufacturing/assembly Application of stress-relieving techniques

The main stages of residual stress management are considered in this chapter with the emphasis on examples of practical application of various destructive and nondestructive techniques for residual stress measurement in materials, parts, and welded elements. Some results of testing showing the role of residual stresses in fatigue

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 0.9 Depth (mm)

Fig. 15.2 In-depth profile of residual stress in 2014-T6 aluminum

alloy produced by conventional (CS) and ultrasonic (US) shoot peening [15.4]

processes as well as beneficial modification of residual stresses directed mainly towards fatigue life improvement are also considered. New engineering tools such as a computerized ultrasonic system for residual stress measurement and a technology and corresponding compact system for ultrasonic hammer peening are also introduced. The data on residual stresses presented in this chapter are complimentary to the detailed consideration and comparison of different methods of residual stress analysis considered in [15.6] and [15.1].

15.2 Residual Stress Measurement Over the last few decades, various quantitative and qualitative methods for residual stress analysis have been developed [15.6]. In general, a distinction is usually made between destructive and nondestructive techniques for residual stress measurement.

15.2.1 Destructive Techniques for Residual Stress Measurement The first series of methods is based on destruction of the state of equilibrium of the residual stress after sectioning of the specimen, machining, layer removal or hole drilling. The redistribution of the internal forces leads to local strains, which are measured to evaluate the residual stress field. The residual stress is deduced from

the measured strain using the elastic theory through the use of an analytical approach or finite element calculations. The most usual destructive methods are:

• • • •

the hole-drilling method the ring core technique the bending deflection method the sectioning method

The application of these destructive, or so-called partially destructive, techniques is limited mostly to laboratory samples. Hole Drilling The hole-drilling method requires drilling a small hole, typically 1–4 mm in diameter, to a depth approximately

Part B 15.2

• •

–500

Measurement: destructive, nondestructive Computation

Stage 2. Analysis of the residual stress effects:

US CS

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Strain gauge rosette

Strain gauge rosette

Residual stress (MPa)

100 0 –100

–500 Hole

Ring core x-ray method Hole drilling method

–1000

Fig. 15.3 Schematic illustrations of the application of

hole-drilling and ring core methods for residual stress measurement (after [15.6])

Part B 15.2

equal to its diameter. A specialized three-element rosette, such as that shown in Fig. 15.3, measures the surface strain relief in the material around the outside of the hole. The ring core method is similar, except that a ring hole, typically with an internal diameter of 15–150 mm, is drilled instead of a hole. The measurements of relieved strain are then made on the surface of the material remaining inside the ring, as shown in Fig. 15.3. The typical depth of the ring core is 25–150% of its internal diameter. In both methods, residual stresses existing in the material before hole/ring drilling can be calculated from the measured relieved strains. The ring core method is quite sensitive compared with the hole-drilling method because it involves almost complete relief of the surface strains. It is also insensitive to any minor diameter errors or eccentricity of the annular hole with respect to the strain gages. However, the size of the annular hole in relatively large, causing much more damage than the hole-drilling method. Also, the results are much less localized because the hole/ring diameter defines the region of residual stress measurement. Another concern with the ring core method is the need to disconnect the strain gage wires to allow ring drilling to proceed. Also, any diameter errors or eccentricity of the hole with respect to the strain gages can introduce significant errors in the residual stress calculation. Despite some shortcomings, the hole-drilling technique remains a popular means of measuring residual stress. Recent developments include the introduction

0

Phase γ(λKαMn) Phase α(λKαMn)

Axial

Axial Tangential

0.1 0.2 0.3 0.4 0.5 0.6

Depth (mm)

Fig. 15.4 Results of residual stress measurement by incremental hole-drilling and x-ray diffraction methods in steel after shot peening (after [15.7])

of new rosette designs [15.8], the development of laser speckle interferometry [15.9], moiré interferometry [15.10], and holography [15.11] for carrying out the residual stress measurements. One result of the application of the hole-drilling technique is presented in Fig. 15.4. The residual stresses in this case were measured by using the incremental hole-drilling method in comparison with the application of x-ray diffraction method in a steel specimen after shot peening [15.7]. As can be seen, good correlation of results of residual stress measurement by the two different techniques was observed. Curvature and Layer Removal Layer-removal techniques are often used for measuring residual stress in samples with a simple geometry. The methods are generally quick and require only simple calculations to relate the curvature to the residual stresses. The curvature depends on the original stress distribution present in the layer that has been removed and on the elastic properties of the sample. By carrying out a series of curvature measurements after successive layer removals the distribution of stress in the original plate can then be deduced. Figure 15.5 presents the results of residual stress measurement using the layer-removal technique in material after shot peening. It is shown that, after shot peening, the magnitude and character of the distribution of residual stresses depends on the mechanical properties of the material.

Residual Stress

100

100

0

0 –100

–100

–200 455 HV

–300

–300

–400

– 400 Shot peening condition: Cast steel shot.dia. = 0.4 mm Speed of turbine: 2500 tr/min Time of treatment: 4 min Thickness of sample: 20 mm

–600 –700

0.5

280 HV 365 HV

5454 aluminum alloy AISI 304 grade steel AISI 1070 grade steel

–500

620 HV

– 500 – 600 – 700 0.5

1

1 Depth (mm)

Depth (mm)

15.2.2 Nondestructive Techniques for Residual Stress Measurement The second set of methods for residual stress measurement are based on the relationship between physical and crystallographic parameters and the residual stress and do not requires the destruction of the part or structural element; they can therefore be used for field measurements. The most well-developed nondestructive methods are: Residual stress (MPa) 100 0 –100 –200 –300 x-direction y-direction

–400 –500 0

0.2

0.4

0.6 0.8 1 1.2 Depth from surface (mm)

Fig. 15.6 Residual stress distribution in A572 Gr.50 steel after shot peening measured by the x-ray diffraction method combined with layer removal [15.12]

•

• •

X-ray and neutron diffraction methods. These methods are based on the use of the lattice spacing as the strain gauge. They allow study and separation of the three kinds of residual stresses. Currently, the x-ray method is the most widely used nondestructive technique for residual stress measurements. Ultrasonic techniques. These techniques are based on variations in the velocity of ultrasonic wave propagation in materials under the action of mechanical stresses. Magnetic methods. These methods rely on the interaction between magnetization and elastic strain in ferromagnetic materials. Various magnetic properties can be studied, such as permeability, magnetostriction, hysteresis, and Barkhausen noise.

X-Ray Method The x-ray method is a nondestructive technique for the measurement of residual stresses on the surface of materials. It can also be combined with some form of layer-removal technique so that a stress profile can be generated, but then the method becomes destructive. One of the major disadvantages of the x-ray method is the limitation imposed on the sample geometry for residual stress measurement. The geometry has to be such that an x-ray can both hit the measurement area and still be diffracted to the detector without hitting any obstructions. Portable diffractometers that can be taken out into the field for measurements of structures

Part B 15.2

Fig. 15.5 Results of residual stress measurement by the layer-removal method in shot-peened samples, illustrating the effect of the mechanical properties of materials [15.6]

–600

375

Residual stress (MPa)

Residual stress (MPa)

–200

15.2 Residual Stress Measurement

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Part B 15.2

such as pipelines, welds, and bridges are now available [15.13]. The speed of measurement depends on a number of factors, including the type of material being examined, the x-ray source, and the degree of accuracy required. With careful selection of the x-ray source and test set-up speed of measurement can be minimized. New detector technology has also greatly reduced the measurement time. x-ray diffraction has a spatial resolution of 1–2 mm down to tens of microns and a penetration depth of around 10–30 μm, depending on the material and source. Figure 15.6 shows the results of residual stress measurement by the x-ray diffraction method combined with layer removal in A572 Gr.50 steel after shoot peening [15.12]. Measurements were made at the surface and at 11 nominal depths until the residual stress decayed to zero. Stresses were obtained for both the x- and y-directions. The maximum compressive stress was −448 MPa in the x-direction and −514 in the ydirection. The depth of beneficial compressive residual stresses in this case was approximately 0.8 mm. As can be seen form Figs. 15.2 and 15.4–15.6, the depth of the beneficial compressive residual stresses after conventional and ultrasonic shoot peening is typically 0.3–0.8 mm. Deeper beneficial compressive residual stresses can be induced by using the ultrasonic impact technique or ultrasonic hammer peening (UP). The UP technique is based on the combined effect of high-frequency impacts and the induction of ultrasonic energy in the treated material [15.1]. Figure 15.7 shows Residual stress (MPa) 100

the components of residual stresses parallel and perpendicular to the direction of treatment [15.12]. In this case the x-ray diffraction method combined with layer removal was also used. The maximum compressive stress in the direction of treatment was −427 MPa, In the perpendicular direction the maximum compressive stress was −302 MPa. The depth of the compressive residual stresses induced by UP was about 1.5 mm. Neutron Diffraction Method The neutron diffraction method relies on elastic deformations within a polycrystalline material that cause changes in the spacing of the lattice planes from their stress-free condition. The advantage of the neutron diffraction method in comparison with the x-ray technique is its larger penetration depth. The neutron diffraction technique enables the measurement of residual stress at near-surface depths of around 0.2 mm down to bulk measurements of up to 100 mm in aluminum or 25 mm in steel [15.6]. With high spatial resolution, the neutron diffraction method can provide complete three-dimensional maps of the residual stresses in material. However, compared to other diffraction techniques such as x-ray diffraction, the relative cost of application of neutron diffraction method is much higher, mainly because of the equipment cost [15.14]. The neutron diffraction method was applied for the measurement of residual stresses in A572 Gr.50 steel after ultrasonic impact treatment [15.12]. The three components of residual stresses, parallel (x) and perpendicular (y) to the direction of treatment as well as Residual stress (MPa) 300

0

200

–100

100

–200

0

–300

–100 Parallel Perpendicular

–400 –500 –600

–200 –300

Sxx (parallel) Syy (perpend.) Szz (deep)

– 400 –500 0

0.2

0.4

0.6

0.8

1 1.2 1.4 1.6 1.8 Depth from surface (mm)

Fig. 15.7 Distribution of residual stresses as a function of

the distance from the treated surface of material received by using of the x-ray diffraction method combined with layer removal [15.12]

– 600

0

1

2

3

4 5 6 7 Depth from surface (mm)

Fig. 15.8 Distribution of residual stresses as a function of

the distance from the treated surface of material received by using of the neutron diffraction method [15.12]

Residual Stress

a) ΔC/C x 10–3

b) ΔC/C x 10–3

2

2

1

1

0

0

0

–1

–1

–2 –200

C sx3 C sx2 CL –100

0

100

–200 σ (MPa)

–2 –200

–100

377

c) ΔC/C x 10–3

2

1

–1

15.2 Residual Stress Measurement

0

100

–200 σ (MPa)

–2 –200

–100

0

100

–200 σ (MPa)

Fig. 15.9a–c Changes in the ultrasonic longitudinal (CL ) and shear wave velocities with orthogonal polarization (CSX3 ; CSX2 ) depending on the mechanical stress σ in (a) steel A, (b) steel B, (c) and aluminum alloy [15.15]

sive external loads in steel and aluminum alloys are presented in Fig. 15.9 [15.15]. As can be seen from Fig. 15.9, the intensity and character of these changes can be different, depending on the material properties. Different configurations of ultrasonic equipment can be used for residual stress measurements. In each case, waves are launched by a transmitting transducer, propagate through a region of the material, and are detected by a receiving transducer, as shown in Fig. 15.10 [15.6]. The technique in which the same transducer is used for excitation and receiving of ultrasonic waves is often called the pulse-echo method (Fig. 15.10a). This method is effective for the analysis of residual stresses in the interior of the material. In a)

b)

15.2.3 Ultrasonic Method for Residual Stress Measurement One of the promising directions in the development of nondestructive techniques for residual stress measurement is the application of ultrasound. Ultrasonic stress measurement techniques are based on the acousticelasticity effect, according to which the velocity of elastic wave propagation in solids is dependent on the mechanical stress [15.16, 17]. The relationships between the changes of the velocities of longitudinal ultrasonic waves and shear waves with orthogonal polarization under the action of tensile and compres-

c)

Fig. 15.10a–c Schematic view of ultrasonic measurement configurations: (a) through-thickness pulse-echo, (b) through-thickness pitch-catch, and (c) surface pitch-

catch

Part B 15.2

in the depth direction (z), are shown in Fig. 15.8. It can be seen that the depth of the beneficial compressive residual stresses is 1.6–1.7 mm for all three components. The maximum surface compressive residual stresses was −458 MPa in the parallel to the treatment direction and −416 MPa in the perpendicular direction. There are various destructive and nondestructive methods to detect and quantify the residual stresses described in the technical literature. However, new industrial problems, new geometrical and material complexities related to them, combined with a general need for fast and economical residual stress measurements, create a strong demand for new and effective techniques and devices. The ideal technology must be reliable and user-friendly, i. e., it should not require guessing and intuition from the engineer/technician and it must be computerized for quick analysis. The demand for sophisticated systems is increasing dramatically.

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Contact Methods

Part B 15.2

this case the through-thickness average of the residual stresses is measured. In the configuration shown in Fig. 15.10c, the residual stress in a (sub)surface layer is determined. The depth of this layer is related to the ultrasonic wavelength, often exceeding a few millimeters, and hence is much greater than that obtained by x-ray method. Other advantages of the ultrasonic technique are the facts that the instrumentation is convenient to use, quick to set up, portable, inexpensive, and free of radiation hazards. In the technique proposed in [15.15], the velocities of longitudinal ultrasonic wave and shear waves with orthogonal polarization are measured at a considered point to determine the uni- and biaxial residual stresses. The bulk waves in this approach are used to determine the stresses averaged over the thickness of the investigated elements. Surface waves are used to determine the uni- and biaxial stresses at the surface of the material. The mechanical properties of the material are represented by the proportionality coefficients, which can be calculated or determined experimentally under external loading of a sample of the considered material. In general, the change in the ultrasonic wave velocity in structural materials under mechanical stress amounts to only tenths of a percentage point. Therefore the equipment for practical application of ultrasonic technique for residual stress measurement should be of high resolution, reliable, and fully computerized. Ultrasonic Equipment and Software for Residual Stress Measurement The ultrasonic computerized complex (UCC) for residual stress analysis was developed recently based on an improved ultrasonic methodology [15.15]. The UCC includes a measurement unit with supporting software and a laptop with an advanced database and expert sys-

Fig. 15.11 The ultrasonic computerized complex for the

measurement of residual and applied stresses

tem (ES) for the analysis of the influence of residual stresses on the fatigue life of welded components. The developed device with gages/transducers for ultrasonic residual stress measurement is presented in Fig. 15.11. The UCC allows the determination of uni- and biaxial applied and residual stresses for a wide range of materials and structures. In addition, the developed ES can be used for the calculation of the effect of the measured residual stresses on the fatigue life of welded elements, depending on the mechanical properties of the materials, the type of welded element, the parameters of cyclic loading, and other factors. The developed equipment enables the determination of the magnitudes and signs of the uni- and biaxial residual and applied stresses for a wide range of materials as well as stress, strain, and force in fasteners of various sizes. The sensors, using quartz plates measuring from 3 × 3 mm to 10 × 10 mm as ultrasonic transducers, are attached to the object under investigation by special clamping straps (Fig. 15.11) and/or electromagnets. The main technical characteristics of the measurement unit are:

• • • • • • •

stress can be measured in materials with thickness 2–150 mm; the error in stress determination (from an external load) is 5–10 MPa; the error in residual stress determination is 0.1 σy (yield strength) MPa; stress, strain, and force measurement in fasteners (pins) 25–1000 mm long; independent power supply (accumulator battery 12 V); the overall dimensions of the measurement device are 300 × 200 × 150 mm3 ; the weight of the unit with sensors is 6 kg.

The supporting software allows the control of the measurement process, storage of the measured and other data, and calculation and plotting of the distribution of residual stresses. The software also allows easy connection with standard personal computers (PCs). An example of the residual stress measurement data, using the developed software, is shown in Fig. 15.12. The software allows the comparison of different sets of residual stress measurement data and transfer of selected data for further fatigue analysis. In Fig. 15.12, the left side of the screen displays information on the measured ultrasonic wave velocities as well as other technical information on the sample. The right-hand side of the screen displays the distribution of calculated residual stresses.

Residual Stress

15.2 Residual Stress Measurement

379

Fig. 15.12 Screenshot showing the distribution of residual stresses in a low-carbon steel plate after local heating [15.15]

σ (MPa) 200

the compression zone, located between the edge of the plate and the center of the heating zone, the longitudinal component of residual stress reaches −140 MPa. Examples of Residual Stress Measurements Using the Ultrasonic Method One of the main advantages of the developed technique and equipment is the possibility to measure the residual and applied stresses in samples and real structure elements. Such measurements have been performed for a wide range of materials, parts, and structures. A few examples of the practical application of the developed technique and equipment for residual stress measurement based on the ultrasonic technique are presented below. Specimen for Fatigue Testing. The residual stresses

100

0 Local heating

Fatigue crack

–100 0

20

40

60

80 L (mm)

Fig. 15.13 Distribution of residual stresses induced by local

heating in a specimen made of an aluminum alloy with a fatigue crack. L is the distance from the center of specimen [15.15]

were measured in a 500 × 160 × 3 mm3 specimen made of an aluminum alloy (σy = 256 MPa, σu = 471 MPa) with a fatigue crack. The residual stresses were induced by local heating at a distance of 30 mm from the center of the specimen. As can be seen from Fig. 15.13, in the heating zone, both components of the residual stress are tensile. In the compression zones, the longitudinal component of residual stress reaches −130 MPa. Compound Pipes and Pipes with Surfacing. Another example of measuring the residual stresses by the ultrasonic method is associated with compound pipes. Compound pipes are used in various applications and

Part B 15.2

In the example of residual stress measurement presented in Fig. 15.12, a plate made from lowcarbon steel, with a yield strength of 296 MPa, was heated locally, with the focal point of heating located approximately 50 mm from the left side of the plate. The distribution of both components of residual stresses in the specimen as a result of this local heating are shown on the right-hand side of Fig. 15.12. As can be seen, in the heating zone, both residual stress components are tensile and reach the yield strength of the considered material. In

380

Part B

Contact Methods

σ (MPa) 200

σ (MPa) 200

150

σx

100

σy

50

0

0 y

–50 –200 –100

x

1000

–150 –400

0

10

20

–

30 δ (mm)

+

Part B 15.2

Fig. 15.14 Residual stress distribution in a compound ring with the following dimensions [15.15]. (inner ring D1 = 160 mm and D2 = 180 mm; outer ring D1 = 180 mm and D2 = 220 mm; D1 inner diameter, D2 outer diameter, width of ring 16 mm)

a) σ (MPa)

b) σ (MPa)

160

160

80

80

0

0

–80

0

5

–

10 15 δ (mm)

+

–80

0

5

–

10

15 10 δ (mm)

+

Fig. 15.15a,b Residual stress distribution in rings with inner surfacing [15.15]: (a) ring with D1 = 150 mm and D2 = 180 mm; (b) ring with D1 = 180 mm and D2 = 220 mm. (D1 inner diameter, D2 outer diameter, width of ring 16 mm)

500

–200 –250

36

0

100

200

300

400

500 x (mm)

Fig. 15.16 Distribution of longitudinal (along the weld) and transverse components of residual stresses along a butt weld toe [15.18]

are made by fitting a pipe with a certain outer diameter inside a pipe with approximately the same inner diameter under pressure. For residual stress measurement, rings were cut off from a number of compound pipes of different diameters. The width of the rings was 16 mm. Residual stresses were measured across the prepared cross-sections in three different locations at 120◦ to each other with subsequent averaging of the measurement results. Depending on the differences between the inner diameter D1 of the outer pipe and the outer diameter D2 of the inner pipe, the measurements were made at three to five points along the radius. The distribution of residual stresses as measured across the wall thickness of the compound pipe is presented in Fig. 15.14. The results of residual stress measurement using the ultrasonic method in rings cut off from a pipes with inner surfacing are presented in Fig. 15.15. Measurement of Residual Stresses in Welded Elements. The residual stresses were measured in a 1000 ×

500 × 36 mm3 specimen, representing a butt-welded element of a wind tunnel. The distribution of biaxial residual stresses was investigated in the x- (along the weld) and y-directions after welding and in the process of cyclic loading of a specimen [15.18]. Figure 15.16 shows the distribution of longitudinal (along the weld) and transverse components of residual stresses along the weld toe. Both components of residual stress reached

Residual Stress

15.3 Residual Stress in Fatigue Analysis

381

b) σ (MPa)

a)

160

x2 σ22

x3 σ33

80

II 70

x1 σ11

III I

I

0

II III 900

σ22 σ11

–80

140

–160

c) σ (MPa)

0

20

40

60

80

100

120 140 x (mm)

d) x (mm) 75

120

Part B 15.3

σ22 σ11

80 40

35 σ11 σ33

0 –40 –80

0

20

40

60

80

100

0 –300

120 140 x (mm)

–200

–100

0

100

200 300 σ (MPa)

Fig. 15.17 (a) Welded specimen and (b) distribution of the residual stresses along the butt weld (I–I), (c) perpendicular to the weld (II–II), and (d) through the thickness near the weld (III–III) [15.5]

their maximum levels in the central part of a specimen: longitudinal – 195 MPa, transverse – 110 MPa. The ultrasonic method was also applied for residual stress measurement in a 900 × 140 × 70 mm3 specimen of low-alloyed steel, representing the butt weld of

a structure [15.5]. The distribution of residual stress components in the x3 (along the weld) and x2 (perpendicular to the weld) directions as well as through the thickness of the specimen near the weld (x1 ) are presented in Fig. 15.17.

15.3 Residual Stress in Fatigue Analysis Despite the fact that the residual stresses have a significant effect on the strength and reliability of parts and welded elements, their influence is not sufficiently reflected in corresponding codes and regulations. This is mainly because the influence of residual stresses on the fatigue life of parts and structural elements depends greatly not only on the level or residual stresses, but also on the mechanical properties of the materials used, the

type of welded joints, the parameters of cyclic loading, and other factors [15.1, 2]. Presently elaborate, timeand labor-consuming fatigue tests of large-scale specimens are required for this type of analysis. Generally, in modern standards and codes for fatigue design, for instance, of welded elements the presented data correspond to the fatigue strength of real welded joints including the effects of welding tech-

382

Part B

Contact Methods

Part B 15.3

nology, type of welded element, and welding residual stresses [15.19]. Nevertheless, in many cases there is a need to consider the influence of welding residual stresses on the fatigue life of structural components in greater detail. These cases include the use of the results of fatigue testing of relatively small welded specimens without high tensile residual stresses, the analysis of the effects of factors such as overloading, spectra loading, and the application of improvement treatments. A few examples of the analysis of the effect of residual stresses on the fatigue life of welded elements based on fatigue testing and computation are described below. It is known that the tensile residual stresses induced by welding can lead to drastic reduction of the fatigue strength of welded elements [15.2]. Figure 15.18 illustrates the results of one of these studies in which butt joints in low-carbon steel were tested using a symmetric loading cycle (stress ratio R = −1). There were three types of welded specimens. Specimens of the first type were cut from a large welded plate. Measurements of the residual stresses revealed that in this case the specimens after cutting had a minimum level of residual stresses. Additional longitudinal weld beads along both sides in specimens of the second type created tensile residual stresses close to the yield strength of the base material in the central part of these specimens. These beads did not change the stress concentration of the considered butt weld in the direction of loading. In the specimens of the third type the longitudinal beads were deposited and then the specimens were bisected and re-

joined by butt-welding. Due to the small length of this butt weld the residual stresses in these specimens were very small (approximately the same as in the specimens of the first type) [15.2]. Tests showed that the fatigue strength of the specimens of the first and third types (without residual stresses) was practically the same as the limit stress range 240 MPa at N = 2 × 106 cycles. The fatigue limit of specimens with high tensile residual stresses (second type) was only 150 MPa. In all specimens the fatigue cracks originated near the transverse butt joint. The reduction of the fatigue strength in this case can only be explained by the effect of welding residual stresses. These experimental studies also showed that, at the level of maximum cycle stresses close to the yield strength of the base material, the fatigue life of specimens with and without high tensile residual stresses was practically identical. With the decrease of the stress range there is a corresponding increase in the influence of the welding residual stresses on the fatigue life of the welded joint. In the multicycle region (N > 106 cycles) the effect of residual stresses can be compared with the effect of stress concentration [15.2]. The effect of residual stresses on the fatigue life of welded elements can be more significant in the case of the relief of harmful tensile residual stresses and the introduction of beneficial compressive residual stresses in the weld toe zones. Beneficial compressive residual stresses at a level close to the yield strength of material are introduced in the weld toe zones by the ultrasonic Δσ (MPa)

Δσ (MPa)

3

450 400 350

400 350

300

300

250

250

200

1

2

200 150

150 log N = –3.183753 x log σ + 13.31484 105

106

2 3

N cycles

1

105

106

N cycles

Fig. 15.18 Fatigue curves of a butt-welded joint in low-

Fig. 15.19 Fatigue curves of non-load carrying fillet

carbon steel [15.2]: 1 – without residual stresses; 2 and 3 – with high tensile residual stresses (fatigue testing and computation)

welded joint in high-strength steel [15.20]: 1 – in the aswelded condition; 2, 3 – after the application of ultrasonic hammer peening (fatigue testing and computation)

Residual Stress

Δσ (MPa) 300 250 6 5

200 2

150 4 3

100 1

105

106

N cycles

Fig. 15.20 Fatigue curves of transverse loaded butt weld at

peening process. The results of fatigue testing of welded specimens in the as-welded condition and after the application of ultrasonic hammer peening (UP) are presented on Fig. 15.19. The fatigue curve of the welded element in the as-welded condition (with high tensile residual stresses) was also used as the initial fatigue data for the computation of the effect of the UP. In the case of a non-load-carrying fillet welded joint in high-strength steel (σ y = 864 MPa, σu = 897 MPa), the redistribution of residual stresses resulted in an approximately twofold increase in the limit stress range and

over tenfold increase in the fatigue life of the welded elements [15.20]. The results of the computation of the effect of residual stresses correlates well with the results of fatigue testing if the relaxation of residual stress is considered during the cyclic loading of the welded elements, taking into account the effect of the stress concentration created by the shape of weld (Figs. 15.18 and 15.19). A significant increase in the fatigue strength of welded elements can be achieved by the redistribution of high tensile residual stresses [15.5]. The calculated fatigue curves for a transverse-loaded butt weld with different levels of initial residual stresses at R = 0 are shown in Fig. 15.20. The fatigue curve of the welded element will be located between lines 1 and 2 in the case of partial relief of harmful tensile residual stresses (lines 3 and 4). The decrease of the tensile residual stresses from an initial high level to 100 MPa causes, in this case, an increase of the limit stress range at N = 2 × 106 cycles from 100 MPa to 126 MPa. The relief of the residual stresses in a welded element to the level of 100 MPa can be achieved, for example, by heat treatment or overloading of this welded element at a level of external stresses equal to 0.52 σy . As a result, this fatigue class 100 welded element becomes fatigue class 125 [15.19]. After modification of the welding residual stresses, the considered welded element will have enhanced fatigue performance and, in principle, could be used instead of a transverse-loaded butt weld ground flush to plate (no. 211) or longitudinal weld (nos. 312 and 313) [15.19]. Introduction of compressive residual stresses in the weld toe zone can increase the fatigue strength of welded elements to an even greater extend (lines 5 and 6 in Fig. 15.20).

15.4 Residual Stress Modification In many cases, beneficial redistribution of residual stresses can drastically improve the engineering properties of parts and welded elements. Detrimental tensile residual stresses can be removed and beneficial compressive residual stresses introduced by the application of heat treatment, overloading, hammer peening, shot peening, laser peening, and low-plasticity burnishing. A new and promising process for the effective redistribution of residual stresses is ultrasonic (hammer) peening (UP) [15.1, 21–23]. The development of the ultrasonic peening (UP) technology was a logical continuation of work directed towards the investigation and

further development of known techniques for surface plastic deformation such as hammer peening, needle peening, and conventional and ultrasonic shot peening. UP is a very effective and fast technique for the relief of harmful tensile residual stresses and the introduction of beneficial compressive residual stresses in the surface layers of parts and welded elements. Figure 15.21 shows the results of the measurement of residual stresses in a part produced by the electric discharge machining (EDM) process. Application of the x-ray technique showed that UP provided the relief of harmful tensile residual stresses and induced compressive residual

383

Part B 15.4

R = 0 [15.5]: 1 – with high tensile residual stresses; 2, 3, 4, 5, and 6 – with residual stresses of 0, 200, 100, −100, and −200 MPa, respectively

15.4 Residual Stress Modification

384

Part B

Contact Methods

Residual stress (MPa) 600

UP treated surface

Zone C

0 0.01– 0.4 mm

400

–

Application of UP

Zone A

200 1–1.5 mm

0

Typical distribution of residual stresses after UP

+

–200

Zone B

–400 –600 –800

15 mm and more

0

20

40

60

80

100 120 Treatment time (s)

Fig. 15.21 Relief of harmful tensile residual stresses induced by EDM, and the introduction of beneficial compressive residual stresses in the surface layers of a material by UP [15.24]

Part B 15.4

stresses equal to the yield strength of material in the surface layers of the considered part [15.24]. UP has been successfully applied to increase the fatigue life of parts and welded elements, eliminate distortions caused by welding and other technological processes, relief residual stress, and increase the hardness of materials. Fatigue testing of welded specimens has shown that UP is the most efficient improvement treatment as compared with traditional techniques such as grinding, TIG (tungsten inert gas)-dressing, heat treatment, hammer peening, shot peening etc. [15.2]. The improvement in fatigue performance is achieved mainly by the introduction of compressive residual stresses into the surface layers of metals and alloys, the decrease in stress concentration in weld toe zones, and the enhancement of the mechanical properties of the surface layer of the material. A schematic view of

Distance from surface

Fig. 15.22 Schematic view of the cross section of a mater-

ial/part improved by ultrasonic peening [15.21]

the cross section of a material/part improved by UP is shown in Fig. 15.22 and a description of the benefits of UP is presented in Table 15.1. A compact system for ultrasonic peening (UP) of materials, parts, and welded elements is shown in Fig. 15.23. The new optimized UP system (total weight 6 kg) includes: 1. A hand tool, based on a piezoelectric transducer. The weight of the tool is 2.8 kg and it is convenient for use. A number of working head types have been designed for different industrial applications. 2. Ultrasonic generator. The weight of the generator is 3.2 kg with a power consumption of only 400 W. Output frequency is ≈ 22 kHz. 3. A laptop with a software package for remote control and optimum application of ultrasonic peening.

Table 15.1 Zones of material/part improved by ultrasonic peening

Zone

Description of zone

A

Zone of plastic deformation and compressive residual stresses Zone of relaxation of welding residual stresses Zone of nanocrystallization (produced under certain conditions)

B C

Penetration (distance from surface) (mm) 1–1.5 mm 15 mm and more 0.01–0.1 mm

Improved characteristics Fatigue, corrosion, wear, distortion Distortion, crack propagation Corrosion, wear, fatigue at elevated temperature

Residual Stress

15.4 Residual Stress Modification

385

Δσ (MPa) 400 350 300 250 200 150 9 7 5

100

3

1

Fig. 15.23 Application of the ultrasonic peening system for beneficial redistribution of residual stresses and fatigue-life improvement of a tubular welded joint [15.23]

•

• • •

Determination of the maximum possible increase in the fatigue life of the welded elements by UP, depending on the mechanical properties of the material, the type of welded element, the parameters of cyclic loading, and other factors Determination of the optimum technological parameters for UP (maximum possible effect with minimum labor and power consumption) for each considered welded element Quality monitoring of the UP process Final fatigue assessment of the welded elements or structures after UP, based on detailed inspection of the UP-treated zones and computation

The developed software allows the assessment of the influence of the residual stress redistribution caused by the UP process on the service life of the welded elements without having to perform time- and laborconsuming fatigue tests and to compare the results of these calculations with the effectiveness of other improvement treatments such as heat treatment, vibration treatment, and overloading. The computation results presented in Fig. 15.24 show the effect of the application of UP on the fatigue life of welded joints in steels of different strength. The data for fatigue testing of non-load-carrying fillet weld specimens in the as-welded condition (with high tensile residual stresses) were used as initial fatigue data for

106

N cycles

Fig. 15.24 Fatigue curves of a non-load-carrying fillet welded joint: 1 – in the as-welded condition for all types of steel; 3, 5, 7, and 9 – after the application of UP to steels 1–4 (see text)

calculating the effect of UP. These results are in agreement with the existing understanding that the fatigue strength of a certain welded element in steels of different strength in the as-welded condition is represented by a single fatigue curve [15.2, 19]. Four types of steels were considered for fatigue analysis: steel 1 (σy = 270 MPa, σu = 410 MPa), steel 2 (σy = 370 MPa, σu = 470 MPa), steel 3 (σy = 615 MPa, σu = 747 MPa), and steel 4 (σy = 864 MPa, σu = 897 MPa). Line 1 in Fig. 15.24 is the single fatigue curve of the considered welded joint for all types of steel in the as-welded condition, determined experimentally. Lines 3, 5, 7, and 9 are the calculated fatigue curves for the welded joint after the application of UP for steels 1–4, respectively. As can be seen from Fig. 15.24, the stronger the mechanical properties of the material, the higher the fatigue strength of the welded joints after the application of UP. The increase in the limit stress range at N = 2 × 106 cycles after UP for a welded joint in steels 1–4 is 42%, 64%, 83%, and 112%, respectively. These results show a strong tendency for increasing fatigue strength of welded connections after the application of UP with increasing mechanical properties of the material used. The developed computerized complex for UP was successfully applied in different applications for increasing of the fatigue life of welded elements, eliminating distortions caused by welding and other

Part B 15.4

For optimum UP application, the maximum possible increase in the fatigue life of the welded elements with minimum labor and power consumption is desired. The main functions of the developed software are:

105

386

Part B

Contact Methods

technological processes, relieving residual stress, increasing the hardness of the surface of materials, and surface nanocrystallization. Areas/industries where UP

was applied successfully include railway and highway bridges, construction equipment, shipbuilding, mining, automotive, and aerospace.

15.5 Summary

Part B 15

1. Residual stresses play an important role in the operating performance of materials, parts, and structural elements. Their effect on the engineering properties of materials such as fatigue and fracture, corrosion resistance, and dimensional stability can be considerable. Residual stresses should therefore be taken into account during the design, fatigue assessment, and manufacturing of parts and structural elements. 2. Various destructive and nondestructive techniques can be efficiently used for the measurement of residual stresses in laboratory and field conditions in many applications for a wide range of materials. In many cases the residual stress analysis was successfully applied to increase the reliability and service life of parts and welded elements in the construction industry, shipbuilding, railway and highway bridges, nuclear reactors, aerospace industry, oil and gas engineering, and in other areas during manufacturing, in service inspection, and the repair of real structures. 3. Certain progress has been achieved during the past few years in the improvement of traditional tech-

niques and the development of new methods for residual stress measurement. A number of new engineering tools for residual stress management such as the described ultrasonic computerized complex for residual stress measurement, technology and equipment for ultrasonic hammer penning, and software for the analysis of the effect of residual stresses on the fatigue life of welded elements and structures have recently been developed and verified for various applications. 4. The beneficial redistribution of residual stresses is an efficient way of improving the engineering properties of parts and structural elements. The application of ultrasonic hammer peening causes a remarkable improvement in the fatigue strength of parts and welded elements in various materials: the stronger the treated materials, the greater the efficiency of the application of ultrasonic peening. This allows more effective use of the advantages of high-strength materials in parts and welded elements that are subjected to fatigue loading.

References 15.1 15.2

15.3

15.4

15.5

J. Lu (Ed.): Handbook on Residual Stress, Vol. 1 (SEM, Bethel 2005) p. 417 V. Trufyakov, P. Mikheev, Y. Kudryavtsev: Fatigue Strength of Welded Structures (Harwood Academic, London 1995) p. 100 Y. Kudryavtsev, J. Kleiman, O. Gushcha: Ultrasonic measurement of residual stresses in welded railway bridge, Structural Materials Technology: An NDT Conference (Technomic Publishing Co. Inc., Atlantic City 2000) pp. 213–218 J. Lu, P. Peyre, C. Oman Nonga, A. Benamar, J. Flavenot: Residual stress and mechanical surface treatments, current trends and future prospects, Proceedings of the 4th International Congress on Residual Stresses (ICRS4) (SEM, 1994) pp. 1154–1163 Y. Kudryavtsev, J. Kleiman: Residual stress management: Measurement, fatigue analysis and beneficial redistribution, X Int. Congress and Exposition on Experimental and Applied Mechanics (Costa Mesa, 2004) pp. 1–8

15.6 15.7

15.8

15.9

15.10

J. Lu (Ed.): Handbook of Measurement of Residual Stresses (SEM, Bethel 1996) p. 238 J. Lu, J.F. Flavenot: Application of the incremental hole drilling method for the measurement of residual stress distribution in shot-peened components. In: Shot Peening, Science Technology Application (DGM, Frankfurt am Main 1987) pp. 279– 288 G. Montay, A. Cherouat, J. Lu: The hole drilling technique applied on complex shapes, SEM Annual Conference and Exposition: Experimental Mechanics in Emerging Technologies (Portland, 2001) pp. 670–673 M. Pechersky, E. Estochen, C. Vikram: Improved measurement of low residual stresses by speckle correlation interferometry and local heat treating, IX Int. Congress on Experimental Mechanics (Orlando, 2000) pp. 800–804 Z. Wu, J. Lu: Residual stress by moiré interferometry and incremental hole drilling. In: Experimental

Residual Stress

15.11

15.12

15.13

15.14

15.15

15.17

15.18

15.19

15.20

15.21

15.22

15.23

15.24

velopment of fracture toughness requirement for weld joints in steel structures for arctic service. VTT-MET. B-89. Espoo. (1985), pp 62– 76 Recommendations for Fatigue Design of Welded Joints and Components. IIW Doc. XIII-1965-03/XV1127-03. International Institute of Welding. (2003) p. 147 Y. Kudryavtsev, J. Kleiman, G. Prokopenko, V. Trufiakov, P. Mikheev: Ultrasonic peening of weldments: Experimental studies and computation, IX Int. Congress on Experimental Mechanics. Orlando (2000) pp. 504–507 Y. Kudryavtsev, J. Kleiman, L. Lobanov, V. Knysh, G. Prokopenko: Fatigue Life Improvement of Welded Elements by Ultrasonic Peening. International Institute of Welding. IIW Document XIII-2010-04. (2004), p. 20 Y. Kudryavtsev, J. Kleiman, A. Lugovskoy, L. Lobanov, V. Knysh, O. Voitenko, G. Prokopenko: Rehabilitation and Repair of Welded Elements and Structures by Ultrasonic Peening International Institute of Welding. IIW Document XIII-2076-05. (2005), p. 13 Y. Kudryavtsev, J. Kleiman, A. Lugovskoy, G. Prokopenko: Fatigue Life Improvement of Tubular Welded Joints by Ultrasonic Peening International Institute of Welding. IIW Document XIII-2117-06. (2006), p. 24 Y. Kudryavtsev, J. Kleiman, G. Prokopenko, V. Knysh, L. Gimbrede: Effect of ultrasonic peening on microhardness and residual stress in materials and welded elements, SEM Int. Congress and Exposition on Experimental and Applied Mechanics, Costa Mesa (2004) pp. 1–11

387

Part B 15

15.16

Mechanics, ed. by Allison (Balkema, Rotterdam 1998) pp. 1319–1324 A. Makino, D. Nelson: Measurement of biaxial residual stresses using the holographic hole drilling technique, Proc. 1993 SEM Spring Conference. Experimental Mechanics (1993) pp. 482–491 X. Cheng, J. Fisher, H. Prask, T. Gnaupel-Heroid, B. Yen, S. Roy: Residual stress modification by post-weld treatment and its beneficial effect on fatigue strength of welded structures, Int. J. Fatigue 25, 1259–1269 (2003) J. Lu, D. Retraint: A review of recent developments and applications in the field of X-ray diffraction for residual stress studies, J. Strain Anal. 33(2), 127–136 (1998) A. Allen, M. Hutchings, C. Windsor: Neutron diffraction methods for the study of residual stress fields, Adv. Phys. 34, 445–473 (1985) Y. Kudryavtsev, J. Kleiman, O. Gushcha, V. Smilenko, V. Brodovy: Ultrasonic technique and device for residual stress measurement, X Int. Congress and Exposition on Experimental and Applied Mechanics. Costa Mesa (2004) pp. 1–7 F. Belahcene, J. Lu: Study of Residual Stress Induced in Welded Steel by Surface Longitudinal Ultrasonic Method, Proceedings of the SEM Annual Conference on Theoretical, Experimental and Computational Mechanics. Cincinnati (1999) pp. 331–334 T. Leon-Salamanca, D.E. Bray: Residual stress measurements in steel plates and welds using critically refracted (LCR) waves, Res. Nondestructive Eval. 7(4), 169–184 (1996) Y. Kudryavtsev: Application of the ultrasonic method for residual stress measurement. De-

References

389

Nanoindenta 16. Nanoindentation: Localized Probes of Mechanical Behavior of Materials

David F. Bahr, Dylan J. Morris

16.1 Hardness Testing: Macroscopic Beginnings ........................ 16.1.1 Spherical Impression Tests: Brinell and Meyers ....................... 16.1.2 Measurements of Depth to Extract Rockwell Hardness ......... 16.1.3 Pyramidal Geometries for Smaller Scales: Vickers Hardness 16.2 Extraction of Basic Materials Properties from Instrumented Indentation ............. 16.2.1 General Behavior of Depth Sensing Indentation ........ 16.2.2 Area Functions ............................. 16.2.3 Assessment of Properties During the Entire Loading Sequence

389 390 390 391 392 392 394 395

16.3 Plastic Deformation at Indentations ....... 396 16.3.1 The Spherical Cavity Model ............ 397 16.3.2 Analysis of Slip Around Indentations ............................... 398 16.4 Measurement of Fracture Using Indentation ................................ 399 16.4.1 Fracture Around Vickers Impressions................................. 399 16.4.2 Fracture Observations During Instrumented Indentation .. 400 16.5 Probing Small Volumes to Determine Fundamental Deformation Mechanisms.. 402 16.6 Summary ............................................. 404 References .................................................. 404

16.1 Hardness Testing: Macroscopic Beginnings Out of the many ways in which to measure the mechanical properties of a material, one of the most common is the hardness test. While more basic materials properties can be determined from testing materials in uniaxial tension and compression (along with shear testing), the

hardness test remains useful in both industrial and laboratory settings for several reasons. First, the testing method is very quick and easy, and allows many measurements on a given sample. Secondly, the equipment for the test methods and required sample preparation

Part B 16

This chapter focuses on mechanical probes of small volumes of materials using contact based testing methods performed on instruments designed to measure mechanical properties, rather than those which are inherently developed for scanning probe microscopy. Section 16.1 consists of basic information used in the measurement of localized mechanical properties and provides a brief review of engineering hardness testing, followed in Sect. 16.2 by a discussion of the basis for instrumented indentation testing to determine elastic and plastic properties of the material being tested: the common modulus and hardness techniques. More advanced methods for using localized probes beyond determining basic elastic and plastic properties, including developments in the analysis of the deformation zone around indentations and the resulting information that can be garnered from this information, are covered in Sect. 16.3. Methodologies that are used to estimate other mechanical properties such as fracture, are covered in Sect. 16.4, which focuses on bringing instrumented indentation testing into realms similar to current macroscopic mechanical testing. Finally, the chapter concludes in Sect. 16.5 with methods that use nanoscale indentation testing to determine more fundamental properties, such as the onset of dislocation activity, size-dependent mechanisms of deformation, and activation volumes.

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Contact Methods

Part B 16.1

is relatively inexpensive, making the tests attractive for small laboratories with limited budgets. Finally, and most importantly, the tests can be made on the actual parts and is sensitive to lateral position. For large parts, a small indentation is usually nondestructive, and allows for quality control checks of each part. Sectioning the sample allows testing of parts which have varying properties through the thickness of the part, such as case hardened gears. While hardness may not be as pure a measurement of a material’s properties as uniaxial testing, the advantages of these methods ensure that the test methods will continue to be used in both industrial and research settings for many applications. Traditional hardness tests have been used for decades to measure the resistance of a material to deformation. In the late 18th and early 19th century several attempts were made to rank the deformation of materials by scratching one material with another. If the scratching material marked the tested material, the scratching material was considered harder than the scratched material. While this was useful for comparative purposes (and is still referred to in the Mohs hardness scale), there remained the problem of quantifying hardness. In the late 19th century work by Hertz [16.1] and Auerbach [16.2] brought about early examples of static indentation testing. In these cases, the samples (either balls or flat plates) were probed with a ball of a given material, and the deformation was measured by considering the contact area between the ball and sample.

16.1.1 Spherical Impression Tests: Brinell and Meyers It was not until the early 1900s that Brinell [16.3] presented a standard method of evaluating hardness, based on applying a fixed load to a hard spherical indenter tip into a flat plate. After 15–30 s, the load was removed and the diameter of the impression was measured using optical microscopy. The Brinell hardness number (BHN) is defined as 

BHN = π D2

2P ,   d 2 1− 1− D

(16.1)

where P is the applied load, D the diameter of the indenter ball, and d is the chordal diameter of the residual impression. Note that this method evaluates the load applied to the surface area of the residual impression. However, the Brinell test has been shown to be affected

by both the applied load and diameter of the ball used for the indenter. The idea that hardness should be a materials property, and not dependent on the test method, led to the observations that the parameter which remains constant for large indentations appears to be the mean pressure, defined as the load divided by the projected contact area of the surface. Meyer suggested in 1908 [16.4] that the hardness should therefore be defined as 4P , (16.2) H= πd 2 where H is the hardness. For a confirmation of the work of Meyer (and a report of the ideas proposed in English) the reader is referred to the study of Hoyt [16.5] in 1924. This type of hardness measurement is alternately referred to as the mean pressure of an indentation, and denoted by either p0 or H in the literature. Tabor describes in detail one of the direct benefits of utilizing spherical indentation for determining the hardness of a material[16.6]. As the indentation of a spherical indentation progresses, the angle of contact of the indentation changes. From extensive experiments in the 1920s to 1950s, it was found that the effective strain of the indentation εi imposed by a spherical tip could be approximated by d (16.3) . D Similarly, for materials such as copper and steel, H is approximately 2.8 times the yield stress. This allowed the creation of indentation stress–strain curves; by applying different loads to spherical tips and measuring the impression diameters one found that the hardness of a material which work hardens increased with increasing indentation area for spherical tips. εi ≈ 0.2

16.1.2 Measurements of Depth to Extract Rockwell Hardness Both the Brinell and Meyer hardness methods require each indentation to be observed with an optical microscope to measure the diameter of the indentation. This makes the testing slow and requires a skilled operator to carry out the test, with the result that the method is more appropriate to a laboratory setting than industrial settings. The Rockwell hardness test, designed originally by Stanley P. Rockwell, focused on measuring the depth an indenter penetrated into a sample under load. As the depth would correspond to an area of contact if the geometry of the indenter was know, it was thought that measuring the depth would provide an

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

accurate and automatic measurement of hardness. For a detailed description of the history of the development of hardness testing, the reader is referred to the work of Lysaght [16.7]. In standardizing the Rockwell test methods, it became common to use a conical diamond indenter tip which was ground to an included angle of 120◦ when testing very hard materials. This indenter tip, referred to as a Brale indenter, is the basis for the common Rockwell C, A, and D scale tests. During a Rockwell test, the sample is loaded to a preload of 10 kg to alleviate effects of surface roughness. After this preload is applied, a larger load is applied, referred to as the major load. The depth measured during the test is the change in distance between the penetration of the tip at preload and after the application of the major load. Effectively, this is the first time that depth sensing was instrumented during indentation testing, and eventually leads to the development of instrumented indentation methods.

16.1.3 Pyramidal Geometries for Smaller Scales: Vickers Hardness

where DPH is the diamond pyramid hardness number, P is the load applied in grams, and now d is the length of the diagonals in mm (usually the average of the two

diagonals). This is analogous to the Brinell test, in that the DPH is the ratio of the load to the surface area of the indentation. The main advantages of this method is that there is a continuous scale between very soft and very hard materials, and that the DPH is constant over a wide range of loads until very low loads (less than 50–100 g) are reached. However, the test still required each indentation to be measured optically to determine the hardness. The DPH can be converted to a mean pressure by simply using (16.2), by substituting the projected area for the ratio. The DPH tests were then adapted to loads of less than 1000 g, which are now commonly referred to as microindentation techniques. This allowed testing of very small areas around the indentation. In addition, as the tests began to gain use in probing cross sections of case hardened materials, the lateral resolution was altered by adding the Knoop indenter tip, which is a four-sided diamond with an aspect ratio of 7:1 between the diagonals [16.6]. The small areas deformed by the DPH and Knoop methods allowed testing of thin coatings and of different microstructural features, such as areas of ferrite and martensite in a steel alloy. Other indenter geometries have been developed, in particularly the Berkovich tip, which is a three-sided pyramid with the same projected contact area as the Vickers tip [16.9]. This results in a face angle between an edge and the opposing face of approximately 65.3◦ . The driving force towards using the Berkovich geometry over conical or Vickers pyramids is that fabricating diamond tips with extremely small root radii is challenging. As three planes intersect at a point, experimentally it is easier to fabricate the Berkovich geometry. One should be aware that even the most finely fabricated diamond tips do not reach the pyramidal geometries at their apex, and commonly are approximated by an effective tip radius [16.10]. Experimental measurements of commercially available tips [16.11] show that the tip radius of Berkovich indenters is commonly less than 200 nm. If a simple geometrical model of an axisymmetric indenter such as a cone radiating from a spherical cap is applied, one finds that the depth at which the indenter tip transitions from a spherical region of radius R to the conical aspect of the tip, h tip occurs at h tip = 1 − sin θ R

(16.5)

for a conical tip with an included half angle of θ. As there are many analytical mechanics solutions developed for the contact of axisymmetric solids, one

391

Part B 16.1

While depth sensing indentation methods were being developed, a change in imaging-based indention methods was also occurring. The development of the Vickers (or diamond pyramid hardness) indentation test, and determination of the angles used during pyramidal indentations [16.8], followed the basic method of the Brinell test, but replaced the steel ball with a diamond pyramid ground to an included angle between faces of 136◦ . It is interesting to note that the angle chosen was based on the Brinell test. It had become common to make indentations with the Brinell method to residual indent diameters corresponding to 0.25–0.5 of the ball diameter, and measurements of BHN as a function of applied load showed an effective constant hardness region between d/D ratios of 0.25–0.5. The average of these is 0.375, and if a four-sided pyramid is formed around a spherical cap such that the ratio of the chordal diameter to ball diameter is 0.375, the angle must be 136◦ . After applying a load, the indent is imaged, and the diagonals of the indentation are measured. The Vickers hardness is then defined as  ◦ 2P sin 136 2 (16.4) DPH = d2

16.1 Hardness Testing: Macroscopic Beginnings

392

Part B

Contact Methods

common technique is to apply an equivalent included half angle for a cone that would provide the same relation between depth of contact and the projected contact area of the solid. For both the Vickers and Berkovich geometries, the ideal equivalent conical included half angle is approximately 70.3◦ . For a tip radius of 200 nm, the effective depth of transition is 12 nm. A microhardness testing machine was modified to continuously monitor the load and depth during an indentation by Loubet et al. [16.12] among others [16.13]. This led to the development of what is commonly

referred to as nanoindentation, though the term instrumented indentation is more accurate. Any procedure which is capable of monitoring the load and depth during an indentation can be referred to as instrumented indentation, continuous indentation or nanoindentation, however inappropriate the scale may be when discussing indentations which penetrate depths of microns and contact areas over 10 μm2 . The following section details the processes by which one experimentally determines the properties of a material using these instrumented indentation techniques.

16.2 Extraction of Basic Materials Properties from Instrumented Indentation

Part B 16.2

There are many recent reviews and resources which describe the general methods of determining the elastic and plastic properties of materials using nanoindentation. This section will provide a brief review of the developments in this area; readers that wish for a more extensive treatment of the technique or specific aspects are referred to several reviews [16.14–18]. In particular, this section will address the calculation of elastic properties from an indentation load–depth curve, the methods to determine hardness from the same test, methods which have improved the technique, and address some of the limitations of the overall methodology of nanoindentation.

16.2.1 General Behavior of Depth Sensing Indentation When a hard tip is impressed into a solid, the resulting load to the sample and resulting penetration of the tip relative to the initial surface can be recorded during the process. This is typically referred to as the load– depth curve for a particular tip–solid system, and is shown schematically in Fig. 16.1. The load–depth curve has several important features which will be used in the following discussion. The maximum load Pmax , the maximum penetration depth h max , and the final depth h f will be required to describe the indentation process. In addition, the load–depth relationship can be determined throughout the loading process, and the stiffness is given by dP/ dh. As shown by Doerner and Nix [16.19], and further developed by Oliver and Pharr [16.20], an analysis of the unloading portion of the load–depth curve allows the modulus and hardness of the material to be determined; the following treatment is based on

that work. This type of analysis is based on the work of Sneddon for the indentation of an elastic half space with a rigid shape or column [16.21]. While Sneddon originally used the analysis for the impression of an axisymmetric punch, the current nanoindentation approach is often used for relatively shallow indenters of pyramidal geometries such as the Berkovich geometry. Under load, one assumes that the profile of the indentation is given by Fig. 16.2, where the contact radius (if a conical indentation were being used) would be defined by a, and the maximum penetration is given by h max . Contact mechanics since the time of Hertz has recognized that the elastic deformation of two solids can be Load P Creep Pmax S = dP/dh

Loading

Unloading hf hmax Displacement h

Fig. 16.1 Schematic load–depth curve for the penetration of a indentation tip into a flat solid

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

described by composite terms. Extensive treatments of contact mechanics for the advanced reader can be found in Johnson’s text [16.22]. For the elastic compression of two solids in contact, the compliances add in series, and it is convenient to define a reduced elastic modulus E ∗ 1 − νi2 1 − νs2 1 = + , E∗ Ei Es

(16.6)

√ 2 dP = β √ E∗ A , dh π

Initial surface

(16.7)

a

Indenter

At Pmax hc

hmax

Surface profile

After unloading

Initial surface

hf

Fig. 16.2 Schematic of the surface of a solid while in con-

tact with an indentation tip and after removing the loaded tip

where A is the projected contact area of the solid and β is a constant that attempts to account for the differences between the axisymmetric contact models and the experimental variations in using pyramidal geometries as well as possible variations due to plastic deformation, and the fact that the contact area is beyond the small-strain conditions assumed in many contact mechanics problems. This correction factor β is still a subject of debate and recent research and reported values range from the commonly references King’s 1.034 [16.23] to values of over 1.1, as reviewed by Oliver and Pharr [16.17] and Fischer-Cripps [16.24]. A recent study [16.25] describes these effects in detail. The currently accepted recommendation is that using β = 1.034 [16.17] or 1.05 [16.25] will allow reasonably accurate values of elastic modulus to be determined, given the other possible inaccuracies in the calibration process. There remain two system characteristics which must be calibrated if an accurate assessment of the elastic modulus and hardness of the sample are to be determined. Of course, the load–depth curve provides measurements of P and h, but not A, which is needed to calculate both the elastic properties and hardness of the solid. First, one must estimate the contact depth from the load–depth curve. The common method of assessing this requires the initial unloading slope, dP/ dh, to be determined. If the indenter were a flat punch, then the elastic unloading would be linear in load–depth space. This was the initial assumption made by Doener and Nix [16.19], that the initial unloading slope was linear. However, if the contact was the elastic contact of a blunt axisymmetric cone in the manner of Sneddon [16.26], the relationship would be parabolic with depth. If the contact were that of a sphere of radius R, rather than a flat punch or cone, the original Hertzian elastic contact solution could be used, in which the relationship would be to the 1.5 power. The loading conditions for an axisymmetric elastic contact between an indenter and a flat surface, such that a contact radius of a can be described as P = 2aE ∗ h for a flat punch indenter of constant radius a, 2E ∗ 2 P= h tan α π for a conical indenter of included half angle α, 4 P = E ∗ Rh 3 3 for a spherical indenter of radius R. (16.8)

393

Part B 16.2

where E and v are the elastic modulus and Poisson’s ratio respectively, and the subscripts ‘i’ and ‘s’ refer to the indenter and sample, respectively. For the conventional indentation technique, diamond is selected for the indenter tip, and so 1141 GPa is commonly used for E i , while vi is 0.07. During nanoindentation, one standard assumption is that both elastic and plastic deformation occur during loading, beginning at the lowest measurable loads. Section 16.5 will describe cases in which this is not a logical assumption, but for many tests this assumption will be valid. In this case, and with a shallow indenter (i. e., one with an included angle similar to that of the Berkovich geometry), the unloading of the indentation will be used to determine the elastic properties of the solid, since the common assumption is that the unloading is purely described by the elastic response of the system. In many metallic and ceramic systems this is indeed experimentally observed (i. e., after partially unloading, subsequent reloading will follow the load–depth curve until the previous maximum load is achieved). If this is the case, then the unloading of the system can be described by

Extraction of Basic Materials Properties

394

Part B

Contact Methods

Therefore, if the unloading slope, dP/ dh, is to be determined, it would at first glance appear to be possible to select the appropriate relationship, perform a leastsquares fit on experimental data, and determine the elastic modulus of the material. However, due to plastic deformation, none of these methods are exactly accurate. In practice the easiest manner found to determine the unloading slope is to perform a least-squares fit to the unloading data using a power-law function which can be described by P = A(h − h f )m ,

(16.9)

Part B 16.2

where m will be between 1 and 2, and h f can be estimated from the load–depth curve. This form is convenient in that the slope of this form will never be negative during the unloading segment. In practice various portions of the unloading segment are selected for fitting due to efforts to minimize drift or noise in the system; hold periods are often inserted in the loading schedule. Any curve fitting procedure should ignore regions of the unloading curve which contain extended holding segments. With the proper analysis of the unloading stiffness, dP/ dh at the maximum depth and load, the contact depth (Figs. 16.1 and 16.2) can be determined. For an axisymmetric indenter the common method is to determine the contact depth h c as hc = hf − ε

Pmax , dP/ dh

The frame compliance, commonly referred to in the indentation literature as Cf , relates the displacement in the system at a given applied load, wherein the system consists of the springs which suspend the indenter tip, the shaft or screw which holds the indenter tip, and the mounting mechanism for the sample. The displacement carried by the total system will be dependent on each given sample due to mounting, but this is often accounted for by determining the compliance for a typically stiff sample and then assuming that this frame compliance is constant for other similarly mounted samples. The total load–displacement relationship can be described by     dh dh = + Cf . (16.11) dP measured dP sample In practice, if the unloading stiffness is measured at a variety of contact depths for a variety of stiff materials (sometimes fused silica, sapphire, and tungsten are recommended), and once these values are inverted this measured compliance is plotted as a function of inverse contact depth, the intercept is Cf . This also allows the definition of the contact stiffness of the sample, rather than of the system. The sample stiffness, S, is just ( dP/ dh)sample and will be used to determine the elastic modulus from a given indentation.

16.2.2 Area Functions (16.10)

where ε is a constant equal to 0.72 for an axisymmetric indenter. Commonly ε is chosen to be 0.75, which describes the data well for pyramidal indenter geometries. One last parameter must be determined experimentally to accurately determine the stiffness at unloading: the frame compliance of the system itself. When any mechanical test is performed, there will be a portion of the displacement carried by the elastic deformation of the loading frame. In bulk mechanical testing this is well documented; if one wishes to utilize crosshead displacement and load data the compliance of the system is subtracted at a given load, assuming a linear spring constant. In the case of bulk testing, the suggested method to carry out mechanical testing is to utilize strain gauges and extensometers rather than relying on the cross head displacement, because the displacement given by the frame may be a substantial part of the overall displacement. However, during nanoindentation testing there is no direct analog to the extensometer, and therefore the frame compliance must be accounted for during the test.

With all these parameters now defined, the final step is to determine the area function of the tip. The initial point of analysis for this method relied upon the contact area, not the depth of penetration, and therefore, in a manner reminiscent of the work of Rockwell, the depth of penetration can be related to an area for a given tip geometry. Many authors have developed a series of techniques to relate the penetration depth to the projected contact area of the tip with the sample. Two popular methods are described in the literature; one chooses an empirical functional dependence of the projected contact area to contact depth, and the other models the tip as a spherical cap on an effectively conical contact. In both cases, the calibration begins by performing indentations to various depths in materials of known elastic modulus (hence both are sometimes referred to as constant-modulus methods). The traditional materials chosen include fused silica and single-crystal aluminum; however other materials such as sapphire and tungsten are sometimes used as calibration standards. Rearranging (16.7), it is possible to

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

describe A=

π 4



S β E∗

2 (16.12)

and if E ∗ is constant, then A can be determined for any given S at a contact depth h c . A can either be described empirically [16.20] as A = C1 h 2c + C2 h c + C3 h 0.5 c + C4 h 0.25 + C51 h 0.125 +... , c c

(16.13)

16.2.3 Assessment of Properties During the Entire Loading Sequence Analysis of the unloading slope provides information from an individual point during an indentation, and as much more data during an instrumented indentation tests is collected, there has been an effort to develop methods to extract more information about the properties of materials from the entire load–depth curve. These fall into three categories, those which approach the loading curve from an analytical standpoint, those which impose a small oscillation upon the loading curve

395

to create a series of unloading segments, and those which utilize finite element solutions to solve inverse problems that provide more information about materials properties including strain hardening coefficients and rate sensitivity of the test. The first two portions, as they are still being used to primarily calculate E and H, will be discussed here, while the third topic will be covered in Sect. 16.3. The first method to extract materials properties from the entire loading segment focuses on the fact that for a given material–tip combination the loading curve tends to describe the balance between elastic and plastic deformation, referred to by Page as the mechanical fingerprint of the material. Hainsworth and coworkers [16.27] developed a novel method initially proposed by Loubet et al. [16.28] in which the shape of the loading curve instead of the unloading curve is considered. This model is particularly applicable for evaluating materials where unloading curves exhibit a considerable amount of elastic recovery. For instance, some coated materials, such as CNx on silicon, show a well-behaved loading curve but a highly curved unloading segment. The relationship between loading displacement and applied load was described by [16.27] P = Kmh2 ,

(16.15)

where P is the applied load, h is the corresponding displacement, and K m is an empirical constant which is a function of Young’s modulus, hardness as well as indenter geometry. For a Berkovich tip, K m is expressed as

  −2 E H . (16.16) K m = E 0.194 + 0.930 H E Therefore, if either E or H is known, the other may be calculated by curve fitting the experimental loading curve using the combination of (16.16) and (16.15). A second method to determine the properties of materials during indentation builds upon the unloading aspect of the test. If a sample is repeatedly loaded and unloaded to greater depths, one effectively determines the unloading slope at different contact depths. As the initial unloading slope can be determined from a partially unloaded segment, the ability to sample in one position the properties as a function of depth is particularly appealing in two cases: firstly where either the material is layered or has some positional variations in properties (or size-dependent properties) and secondly when a spherical indentation is used for the experiment. The instrumented indentation of a material with a sphere would allow sampling at different contact-area-

Part B 16.2

where in this case C1 is often chosen to fit the ideal tip shape (24.5 for a Berkovich tip). Subsequent constants are empirically selected to provide a best fit to the elastic modulus from the measured data. The other method assumes that the tip is described using a spherical cap on an effective cone of included half angle α with a tip radius of R [16.10]   h c C1 2 + A= C1 C2 πh 2c + 4Rπh c + 4R2 π cot2 α , = (16.14) cot2 α where the constants C1 and C2 are related to α and R, respectively. The other methodology used for tip calibration relies upon actual measurement of the tip as a function of position using scanning probe microscopy (or some other quantitative profilometer method) [16.11], and is less common due to the difficulty in performing these experiments; most users will likely utilize the constant-modulus methods. Having defined a method to determine the projected contact area as a function of contact depth, the hardness H as determined in (16.2) is easily determined. The hardness measured by nanoindentation is expected to deviate from the hardness measured by post-indentation inspection if the surface does not remain flat during the entire test.

Extraction of Basic Materials Properties

396

Part B

Contact Methods

to-depth ratios, which is the same effect as measuring hardness at different d/D ratios (see the previous section). Field and Swain demonstrated this for nanoindentation using a cyclic loading–partial unloading sequence [16.29]. By this time nanoindentation testing was becoming more standardized, and they noted that spherical indentations would probe different strains, while conical or pyramidal indentation would probe a fixed strain value; effectively (just like (16.3) but under load) a (16.17) εi ≈ 0.2 R for spheres while for pyramids and cones εi ≈ 0.2 tan α ,

(16.18)

Part B 16.3

allowing them to calculate hardness as a function of penetration depth. This paper also deals with the prediction of the load–depth graphs knowing materials properties such as modulus and yield strength. The last method of determining materials properties during nanoindentation is commonly referred to as the continuous stiffness method, though other terms are utilized by a variety of commercial manufacturers of systems. In all cases, the process relies upon an imposed oscillation in the force (or displacement) applied to the tip. This is carried out at a specific frequency. The technique imposes a sinusoidal forcing function at a fixed frequency ω, and uses this signal to calculate the contact stiffness from

POS  −1 2

S−1 + Cf + K s − mω2 + ω2 D2

h(ω) = (16.19)

or from the phase difference between the displacement and force signals from ωD , (16.20) tan(φ) =  −1 −1 S + Cf + K s − mω2 where Cf is the compliance of the load frame, K S is the stiffness of the column support springs, D is the damping coefficient, POS is the magnitude of the forcing oscillation, h(ω) is the magnitude of the resulting

displacement oscillation, φ is the phase angle between the force and displacement signals, and m is the mass. Material properties can then be calculated at each of these tiny unloading sequences. Further extension of this work by Joslin and Oliver [16.30] led to the realization that, as both H and E ∗ use the area of contact and applied load, at any point during the loading cycle if the stiffness were measured using a lock-in amplifier the quantity P 4 H = 2 (16.21) 2 S π E∗ (neglecting correction factors such as β) provides a measurement of the resistance to deformation which does not require the use of an area function. For assessing the properties of materials which may be expected to have depth-dependent hardnesses (for instance, in ionimplanted solids) this technique promised the ability to assess these properties without the need to calibrate tips to extremely shallow depths. Of course, if one knows the area function of the tip through a calibration technique, then it is possible to determine the modulus directly if the stiffness is known at all points during loading. The continuous stiffness method and its variations are now common methods of assessing materials properties as a function of depth during a single indentation. The obvious benefits of this method include the ability to sample small volumes (i. e., probe laterally distinct features) which may not be able to be the subject of many tests, to perform tip calibrations and property measurements in thin-film systems rapidly, and to provide extremely sensitive methods of system constants. For instance, a new method of determining frame compliance uses this technique[16.17]; when fused quartz is indented using the continuous stiffness method, P/S2 is a constant 0.0015 GPa−1 . This method is quite sensitive to a system’s frame compliance, and obviously not to the area function. Therefore, users of indentation systems can rapidly calibrate the frame compliance to account for the variations in the systems with time and ensure more accurate area functions for use in materials properties assessment.

16.3 Plastic Deformation at Indentations The importance of imaging and examining indentations after they are made is often underestimated. While there certainly is valuable data contained in load–depth curves, there is also much information contained in the

plastically deformed region around the indentation, be it in localized probes of bulk materials or in thin-film coated systems [16.31]. Specifically, information about the dislocation mechanisms responsible for deformation

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

can be obtained by examining atomic force microscopy (AFM) images of the slip steps which develop on the surface.

16.3.1 The Spherical Cavity Model

Θ

a w

hf c

Fig. 16.3 Definition of variables for indentations. Note that

the included angle Θ is only used here to represent the half angle of the indentation. After the indenter tip has been removed from the sample, the remaining angle in the residual cavity will increase due to the elastic recovery of the indentation

included angle of the indenter is 2θ, the applied load is P, and the extent of plastic deformation or plastic zone radius is c. For this discussion the plastic zone will be defined in the manner of Samuels and Mulhearn [16.33] given as the radius of the edge of the vertical displacements of the surface. Lockett [16.36] used a model of indentation which relied on slip lines for included indenter angles between 105◦ and 160◦ . These calculations predict an h/a ratio for a 105◦ indenter of 0.185, for a 120◦ indenter of 0.137, and for a 140◦ indenter of 0.085. Dugdale [16.35] measured the pile up as a function of indenter angle normalized by 2c, providing experimental data for steel, aluminum, and copper. When converted to w/a, these values fall between approximately 0.2 and 0.1 for cone angles between 105◦ and 160◦ . The trend for Dugdale’s experimental data is that the ratio c/a decreases as the indenter angle increases, as do the calculations of Lockett. Mulhearn [16.37] proposed that the maximum pile up depended on the indenter angle, and as a first approximation the surface of the sample being indented would remain planar during indentation. Upon removing the load, elastic recovery would occur, and as an upper bound the maximum pile up could be approximated as 1/(6 tan θ). Bower et al.’s model can be modified into a similar ratio, and has a 1/ tan θ dependence, just as with the model presented by Mulhearn. In addition, Bower’s model suggests that the w/a ratio for a given indenter should be constant, independent of the applied load. The extent of plastic deformation around an indentation, often assumed to be linked to the lateral dimensions of the out-of-plane deformation around the indentation, is not the only way in which plastic zones around indentations can be defined. In addition to the upheaval around the indenter, there is a region centered on the indenter tip which experiences radial compression from the indenter tip. The ratio of the plastic zone size to the contact radius, c/a, will quantify the extent of plastic deformation. According to Johnson’s conical spherical cavity model [16.32] of an elastic–plastic material, the plastic zone is determined by  ∗   1 E tan β 2 1 − 2ν 3 c , = + a 6σ y (1 − ν) 3 1 − ν

(16.22)

where α is the angle between the face of the cone and the indented surface, 90 − θ and σy is the yield strength of an elastic–plastic material. When the ratio E ∗ tan α/σy is greater than 40, Johnson suggests that

397

Part B 16.3

The plastic deformation caused by an indentation has two components, radial and tangential. The spherical cavity model which has been developed extensively by Johnson relies on this mode of deformation [16.32]. However, the spherical cavity model does not address the amount of upheaval around an indentation, and instead focuses on the radial expansion of the plastic zone. It has been shown [16.33] that materials which exhibit perfectly plastic behavior tend to pile up around an indenter, while annealed materials which have high work hardening exponents tend to sink in around an indenter. This behavior has been modeled using finite element methods by Bower et al. [16.34]. Previous work on pile up around indentations has shown the amount of pile up to vary with the indenter angle for a given load, with sharper indenters providing more pile up [16.35]. This discussion will only consider cones between the angles of 105◦ and 137◦ , where the spherical cavity model has been used to successfully determine plastic zones around indentations. Cones sharper than this may begin to approach cutting mechanisms modeled by slip line fields, and may be inappropriate for using a spherical cavity model. Figure 16.3 defines the variables which will be used in this discussion. The maximum pile up is referred to as w, the indentation depth relative to the nominal surface is h final , the contact radius of the indenter is a, the

16.3 Plastic Deformation at Indentations

398

Part B

Contact Methods

the analysis of elastic–plastic indentation is not valid, and that the rigid–plastic case has been reached. Once the fully plastic state has been reached, the ratio c/a becomes about 2.33. Another model of the plastic zone size [16.38] based upon Johnson’s analysis suggests that the plastic zone boundary c is determined by  3P . (16.23) c= 2πσ y With this model, and as noted by Tabor, the hardness is approximately three times the yield strength of a fully plastic indentation and the ratio c/a should be 2.12 (similar to the suggested values of Johnson). In either case, this provides a mechanistic reason for the conventional rules of indentation which require lateral spacing of at least five times the indenter diameter to ensure no influence of a previous indentation interferes with the hardness measurement. Experimental measurements of pile up around macroscopic indentations [16.39] have shown the pile up models described in this section to be reasonable, and both on the macroscopic and nanoscale [16.40] the plastic zone size approximation for the yield strength is reasonable.

Part B 16.3

16.3.2 Analysis of Slip Around Indentations From what is known about the dislocation mechanisms taking place beneath an indenter tip during indentation experiments, it seems a valid assumption that dislocation cross-slip must take place for out-of-plane deformation, or material pile up, on free surfaces around indentations to occur [16.41–49]. Several studies on the structure of residual impressions have been reported in the literature. Kadijk et al. [16.50] has performed indentations in MnZn ferrite single crystals using spherical tips and identified slip systems responsible for the patterns which resulted in the residual depression. Using a combination of controlled etch pitting, chemomechanical polishing, and AFM, Gaillard et al. [16.46] described dislocation structures beneath indentations in MgO single crystals. Other work has been performed on body-centered cubic (BCC) materials to identify changes in surface topography with crystal orientation [16.48] as well as transmission electron microscopy (TEM) images of the subsurface dislocation structure [16.49, 51]. Many of these results have noted that the extent of plastic deformation may be dependent on the overall size of the indentation, as well as exceeding the c/a ratio of 2.33 for nanoscale testing of relatively defect-free materials.

The face-centered cubic (FCC) crystal structure contains only four unique slip planes. Chang et al. [16.52] describe a method for determining the surface orientation of a particular grain in an FCC material by measuring the angles of the slip step lines on the surface around indentations and calculating the orientation from the combination of angles. With the current availability of orientation imaging microscopy (OIM), it is possible to carry out a similar process in reverse and use the known orientation of a grain to determine the slip plane responsible for each slip step. Each slip plane can be indexed with respect to a reference direction taken from the OIM. Several studies of surface topography have been performed [16.53, 54] using this technique, and effects such as the include angle of the tip are shown to influence the extent of the plastic zone around materials. Figure 16.4 demonstrates the extent of surface deformation in a stainless-steel alloy indented with a spherical tip with a tip radius of approximately 10 μm, and a 90◦ conical indenter with a tip radius of approximately 1 μm. For the second indentation the extent of plastic deformation surpasses the expected c/a ratio of 2.33. These results are not unique to surface measurements; dislocation arrays identified by etching after indentation in MgO have demonstrated similar effects [16.44]. Clearly the extent of deformation in the lateral direction which is transferred to the surface is influenced by the effective strain of the indentation. Of particular interest in recent studies using dislocation etch pitting in conjunction with nanoinden-

20 μm

20 μm

Fig. 16.4 Slip steps from only the positively inclined slip planes around blunt tip indentations in a grain with surface orientation of (1 2 30). When the effective strain is increased by using a sharp tip, steps form from both the positively and negatively inclined slip planes

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

tation experiments [16.55] is that the lattice friction stress for individual screw and edge dislocation components can be measured; Gaillard and co-workers

16.4 Measurement of Fracture Using Indentation

399

have determined lattice friction stresses of 65 MPa for edge dislocations and 86 MPa for screw dislocations in MgO.

16.4 Measurement of Fracture Using Indentation If nanoindentation is to prove as useful as bulk mechanical tests, properties other than hardness (which is poorly defined in a mechanics sense) and elastic modulus must be considered. Some of these properties, such as yield strength and strain hardening, were covered in the previous section. Two other main areas which provide data to be utilized in assessing a material’s response to an applied load are time-dependent deformation and mechanisms of fracture. In the case of fracture measurements observation of the surface of the material will generally be required.

16.4.1 Fracture Around Vickers Impressions

where c is the crack radius, P is the peak indentation load, and χr is an dimensionless constant dependent on the specific indenter–material system that can be found by applying indentations to various maximum loads and measuring crack lengths. If the material system has a constant toughness, then a fitting procedure can be used to determine the dimensionless constant.

15 μm

Fig. 16.5 Vickers indentation in silicon, showing radical

cracks

Part B 16.4

The indentation fracture technique is widely used to characterize the mechanical behavior of brittle materials due to its simplicity and economy in data collection [16.56]. The fracture patterns strongly depend on loading conditions, which fall into two basic categories: blunt or sharp contact. While blunt indenters (for instance, spherical tips) are associated with a Hertzian elastic stress field, sharp indenters (for instance Vickers or Knoop) lead to an elastic–plastic field underneath the contact region [16.57]. In the later case, radial cracks may be generated from the corners of the contact impression, and median cracks propagate parallel to the load axis beneath the plastic deformation zone in the form of circular segments truncated by the material surface. At higher peak loads, lateral cracks may also be formed in the manner of a saucer-shaped surface centered near the impression base [16.57]. As the driving force for lateral crack growth is rather complex compared to the centerloaded symmetric radial/median stress field, for the sake of simplicity, only radial/median crack system is considered in terms of indentation fracture mechanics herein. Lawn and coworkers developed a model to quantitatively represent the relationship of the radial crack size and the fracture toughness for a ceramic material [16.58]. The key idea was to divide the indentation stress field into elastic and residual components where the reversible elastic field enhances median extension

during the loading half cycle and the irreversible plastic field primarily provides the driving force for radial crack evolution in the unloading stage. In a later study, they further the discussion by considering the effect of a uniform biaxial applied field on the crack evolution [16.59, 60] based on experimental observation that an imposed uniform stress field could alter the fracture behavior by either increasing or decreasing the crack size [16.61, 62]. In general, the toughness of a material is related to the extent of radial cracks that emanate from the corners of an indentation, such as those shown in Fig. 16.5. An extensive review of the modes of fracture can be found in the literature [16.57]. For the commonly reported fracture tests using Vickers indentation, the stress intensity factor due to the residual impression is  0.5 E P P = χr 1.5 , (16.24) K r = ζr H c1.5 c

400

Part B

Contact Methods

tests have arrested, the assumption that K r is equal to the fracture toughness is used to provide a measurement of the toughness of the system.

Load P (mN) 600 500

16.4.2 Fracture Observations During Instrumented Indentation

400 300 Pop-in 200 100 0

hf

0

1000

2000

3000 4000 5000 Displacement h (nm)

Fig. 16.6 Nanoindentation in fused silica using a 42◦ in-

denter

Of course, if a different material is tested one must determine the constant again and ζr is often used instead, where ζr is given as 0.016 [16.61] or sometimes 0.022 [16.63]. Since the cracks measured during these

Part B 16.4

a) Load P (mN)

There are drawbacks to using static indentation for fracture. First, many materials which are brittle do not exhibit fracture during nanoindentation using conventional indentation probes. Secondly, imaging the indentation can become problematic if very small features or indentations are made in the sample. Therefore, it would be convenient to determine the occurrence of fracture during an instrumented indentation. Some authors have chosen to identify fracture mechanisms using acoustic emission sensors during nanoindentation [16.64–66], but this does require specialized equipment and analysis to separate fracture from plasticity events. Morris et al. [16.67] have utilized sharper indenter probes to determine the effects of included angle on the resulting load–displacement curves, as shown in the typical result in Fig. 16.6. More sensitive measurements of these pop-in events are found if one measures the stiffness continually during the ex-

b) Load P (mN)

600

600

500

500 52°

400

400 65°

300

65° 52° 42° 35°

300 42°

200

200 35°

100 0

100 0

100

200

300

400

500

0

0

Displacement h (nm)

c)

100

200

300

400

500

Displacement h (nm)

d)

Fig. 16.7a–d Unloading curves and

10 μm

respective images of residual indentation impressions showing cracking an systems ((a) and (b)), while uniform unloading curves occur in materials which undergo only plastic deformation ((c) and (d))

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

16.4 Measurement of Fracture Using Indentation

401

Load (μN) 3000 Lower than the excursion load Higher than the excursion load fim thickness: 92 nm

2500

2000

1500

Cracks in oxide film 1000

1000 nm 500

Fig. 16.8 AFM image of circumferential cracks around indentations into anodized titanium. The large ring corresponds to a through-thickness crack, while the smaller impression in the center is the contact area defined by the indenter tip [16.68]

0

20

40

60

80

100

120

140

Depth (nm)

Fig. 16.9 Indentation into a 92 nm-thick TiO2 film grown anodically on a Ti substrate. The distinct excursion in depth, in this case corresponds to a through-thickness fracture in the film [16.68]

to produce cracks in the film [16.75–80]. Malzbender and de With [16.75] demonstrated that the dissipated energy was related to the fracture toughness of the coating and interface by performing simple integrations of the loading and unloading portions from indentation to determine the amount of energy needed to damage the film. Other studies have suggested that the fracture energy could be estimated as the energy consumed during the first circumferential crack during the load drop or plateau on the load–displacement curve [16.74, 76, 78, 81]. A recent study has demonstrated one technique which relies upon examination of the load–depth curve, knowing the general behavior of hard film–soft substrate systems [16.68, 82]. In this case, the load–depth curve of a coated system, shown in Fig. 16.9 for a 90 nm TiO2 on Ti, also demonstrates a pop-in or excursion in the load–depth curve. These pop-in events, present only because the indentation testing system is a load-controlled device, are in this case indicative of film fracture because permanent deformation is observed prior to the pop-in. The plastic zone c from Sect. 16.3 is used to estimate the radius of the fracture, and the toughness of the film can be estimated from the energy difference between that which goes into the indentation of the film–substrate system and indentations to similar depths for the substrate only.

Part B 16.4

periment. While it may be tempting to identify the presence of a pop-in to a fracture event, these authors demonstrate conditions under which no pop-in event occurs and yet the resulting indentation impression instead shows clear evidence of cracking. Probes of increasing acuity (moving from the conventional Berkovich geometry to sharper cube-corner-type systems) are more likely to generate fracture. A comparison of the unloading slopes of probes of various acutities demonstrated that samples which exhibit indentationinduced fracture demonstrate differences in unloading (while having the same final displacement), while materials which only exhibit plastic deformation do not show differences in unloading curves, as shown in Fig. 16.7. Therefore, the authors suggest utilizing a shallow (Berkovich) and acute (cube corner or similar) indenter pair, and that comparing the ratio of h max /h f for the different tips will enable the user to identify materials which are susceptible to indentation-induced fracture. Another case for fracture measurements particularly suited to continuous indentation is the application of hard films on compliant or soft substrates, which results in circumferential fracture around the indentation, as shown in Fig. 16.8. Thin films have been shown to fracture during indentation processes at critical loads and depths [16.69]. Some studies have analyzed these fracture events by calculating an applied radial stress at fracture [16.68, 70, 71], an applied stress intensity at fracture [16.71–74], and the amount of energy required

0

402

Part B

Contact Methods

16.5 Probing Small Volumes to Determine Fundamental Deformation Mechanisms

Part B 16.5

It has been well documented that small volumes of materials regularly exhibit mechanical strengths greatly in excess of more macroscopic volumes. Observations in whiskers of metals were among the first studies to demonstrate the ability of metals to sustain stresses approaching the theoretical shear strength of the material. The often described indentation size effect in metals [16.83–86] has been the subject of many studies with nanoindentation due to the ability to neglect elastic recovery (i. e., the indentation hardness is measured under load) and to measure very small sizes. In ceramics this effect has been explained by variations in elastic recovery during the indentation [16.87]. Results on nanocrystalline metals have continued that trend, until the literature has reached a point where models suggest that grain sizes too small to support the nucleation and growth of stable dislocation loops will likely deform via other mechanisms. Similarly, nanostructured metallic laminates have been demonstrated to exhibit particularly unique deformation behavior. Other recent experiments in which a flat punch in a nanoindenter has been used to carry out compression tests on machined structures fabricated via focus ion beam machining have demonstrated that just having a smaller volume of material, with the concomitant increase in surface area and decrease in sample size, may indeed impact plasticity in ways not immediately obvious through scaling arguments [16.88, 89]. Recent work has also demonstrated the ability to quantify the stress required to cause the onset of plasticity dislocations in relatively dislocation-free solids [16.90–94]. This method is only possible using small-scale mechanical testing; large-volume mechanical testing will generally measure the motion of pre-existing dislocations under applied stresses. Nanoindentation couples well with small-scale mechanical modeling, as it approaches volumes which can be simulated using molecular dynamics [16.95, 96] and the embedded atom technique [16.97]. In materials with low dislocation densities, the sudden onset of plasticity occurs at stresses approaching the theoretical strength of a material. Recent studies have proposed two primary models of these effects: homogenous nucleation and thermally activated processes. Gane and Bowden were the first to observe the excursion phenomena on an electropolished surface of gold [16.98]. A fine tip was pressed on the gold surface while observing the deformation in a scan-

ning electron microscope. No permanent penetration was observed initially, but at some point during the process a sudden jump in displacement occurred, which corresponded to the onset of permanent deformation. Similar observations of nonuniform loading during indentation were made by Pethica and Tabor [16.99], who applied a load to bring a clean tungsten tip into contact with a nickel (111) surface in an ultrahighvacuum environment while monitoring the electrical resistance between the tip and surface. The nickel had been annealed, sputtered to remove contaminants, and then annealed again to remove surface damage from the sputtering process. When an oxide was grown on the nickel to thicknesses of greater than 50 Å, loading was largely elastic while the electrical resistance remained extremely high, with only minimal evidence of plastic deformation. However, past some critical load the resistance between the tip and sample dropped dramatically, with continued loading suggestive of largely plastic deformation. With the advent of instrumented indentation testing it was possible to monitor the penetration of an indenter tip into a sample. The pop-in or excursion in depth during indentation was observed by many researchers during this period, and is represented in Fig. 16.10. Load (mN) 6 5 Elastic loading prediction

4 3 2 1 0

0

50

100

150

200 Depth (nm)

Fig. 16.10 Nanoindentation into electropolished tungsten,

with an initial elastic loading followed by plastic deformation. Elastic loading prediction by [16.26]

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

Some researchers have utilized experimental methods that are either depth controlled [16.100,101] or a system in which the overall displacement (and not penetration) was held constant [16.91, 102]. Post-indentation microscopy demonstrated that, after these discontinuous events during loading in single-crystal ceramics, dislocation structures were present [16.49, 55]. Using the Hertzian elastic model, 4 (16.25) P = E ∗ Rδ3 , 3 the maximum shear stress under the indenter tip τmax is related to the mean pressure pm and is given by  1 6(E ∗ )2 3 1/3 P (16.26) τmax = 0.31 π 3 R2 at a position underneath the indenter tip at a depth of 0.48a, where a is the contact area, and is determined by solving  ∗2 1/3 6E 3P = P 1/3 . (16.27) 2 2πa π 3 R2

Recently, Schuh and co-workers [16.106] carried out indentation experiments that probed the onset of plastic deformation at elevated temperatures to determine an effective activation energy and volume for the initiation of plasticity. Using a method describing the probability of an excursion occurring underneath the indenter tip they use an Arrhenius model for dislocation nucleation. They suggest that the activation energy is similar to that of vacancy motion, and that the activation volume is on the order of a cubic Burgers vector. This suggests that point defects have a direct relationship with dislocation nucleation. Another recent study of incipient plasticity in Ni3 Al [16.107] suggests that the growth of a Frank dislocation source at subcritical stresses controls plasticity at lower stresses during indentation, and that self-diffusion along the indenter– sample surface dominates the onset of plasticity in these cases of nanoindentation. Bahr and Vasquez [16.108] have shown that, for solid solutions of Ni and Cu, the shear stress under the indenter tip at the point of dislocation nucleation can be correlated to changes in the elastic modulus of the material. Using a diffusion couple of Ni and Cu to create a region of material with a range of compositions ranging from pure Ni to pure Cu, it was shown that the changes in shear stress required to nucleate dislocations are on the order of the changes in elastic modulus, between 1/30 and 1/20 of the shear modulus of this alloy. The implication of these data is that overall dislocation nucleation during nanoindentation is strongly related to shear modulus in this system, but does not preclude the diffusion mechanism suggested by Ngan and Po [16.107]. Another example of these methods is nanoindentation being used to probe the effects of hydrogen on dislocation nucleation and motion in a stainless steel [16.109]. These pop-in events were used to demonstrate that, while dissolved hydrogen lowers the indentation load at which dislocations are nucleated, this was likely a result of hydrogen decreasing the shear modulus and not directly related to hydrogen creating or modifying nucleation sites. The pop-ins occurred more slowly in the presence of hydrogen, most likely due to hydrogen inhibiting the mobility of fast-moving dislocations as they are rapidly emitted from a source. Finally, a second excursion occurs after charging with hydrogen, but is uncommon in uncharged material. When coupled to analysis of slip steps around the indentations that show the existence of increased slip planarity, this suggests that dislocations emitted during the first excursion are inhibited in their ability to cross-slip and there-

403

Part B 16.5

The shear stress field does not drop off that rapidly, and there is a region under the indenter tip extending to approximately 2a which has an applied shear stress of approximately 0.25 pm . Many research groups have noted that this stress approaches the theoretical shear stress in a crystal. The similarity between the applied shear stress and shear modulus is close to the classical models of homogeneous dislocation nucleation espoused by the early theoretical work on dislocation mechanisms. For instance, Cotrell noted that the classical theoretical shear strength of a material is often approximated by μ/30 to μ/100 [16.103], where μ is the shear modulus. A new set of experiments utilizing in situ nanoindentation in a transmission electron microscope has been used to support this model of homogeneous dislocation nucleation. Minor et al. [16.92] observed dislocations to pop out underneath an indenter tip after some initial elastic loading during contact between the tip and aluminum sample. This direct observation is strong evidence that dislocations nucleate underneath the indenter tip. However, the most recent studies from in situ microscopy have shown that even the very initial contact can generate dislocations that nucleate and propagate to grain boundaries, after which an elastic loading behavior is again demonstrated followed by a second strain burst [16.104]. Multiple yield points during nanoindentation are often referred to as staircase yielding [16.93, 101, 105].

Probing Small Volumes

404

Part B

Contact Methods

fore cannot accommodate a fully evolved plastic zone. A second nucleation event is then needed on a differ-

ent slip system to accommodate the plasticity from the indenter.

16.6 Summary This chapter has provided a summary of the basic methodologies for assessing the mechanical properties of materials on a small scale via instrumented indentation, often more commonly referred to as nanoindentation. The first two sections described the history of indentation testing and the concept of hardness, beginning with the quantification carried out by Brinnel. This aimed to provide a background for the reader, who may feel at times that many of the conventions in nanoindentation appear to be based on somewhat arbitrary rules. Moving to instrumented techniques, the development of contact mechanics approaches for determining elastic and plastic properties of solids via small-scale indentations was outlined, including the analysis of the unloading slope and the imposed oscillation

techniques currently available on commercial instruments. The later sections of the chapter focused on more advanced uses of small-scale indentations. The deformation surrounding an indentation provides a researcher with more information than just the elastic properties; sophisticated analysis methods allow for determination of more complex mechanical properties such as strain hardening. Indentation-induced fracture, while still a subject of debate in the literature, is often used in brittle materials, so this topic is covered herein to provide the reader with a current status of the methods. Finally, the last section covers more fundamental dislocation behavior (such as nucleation or multiplication issues) that is often examined by very low-load indentations.

Part B 16

References 16.1

16.2 16.3

16.4 16.5 16.6 16.7 16.8

16.9

16.10

16.11

H. Hertz: Miscellaneous Papers. English translation by D.E. Jones, G.E. Schott (MacMillan, New York, 1896) pp. 163-183 F. Auerbach: Smithsonian Report for 1891 (Government Printing Office, Washington 1893) pp. 207–236 J.A. Brinnnel: II.Cong. Int. Methodes d’Essai, J. Iron Steel Inst. 59, 243 (1901), translated to English by A. Wahlberg E. Meyer: Untersuchungen über Härteprüfung und Härte, Z. Ver. Dtsch. Ing. 52, 645–654 (1908) S.L. Hoyt: The ball indentation hardness test, Trans. Am. Soc. Steel Treating, 6, 396 (1924) D. Tabor: The Hardness of Metals (Oxford Univ. Press, Oxford 1951) V.E. Lysaght: Indentation Hardness Testing (Reinhold Pub., New York 1949) R.L. Smith, G.E. Sandland: Some notes on the use of a diamond pyramid for hardness testing, J. Iron. Steel Inst. 3, 285–301 (1925) E.S. Berkovich: Three faceted diamond pyramid for micro hardness testing, Industrial Diamond Rev. 11, 129–133 (1951) J. Thurn, R.F. Cook: Simplified area function for sharp indenter tips in depth-sensing indentation, J. Mater. Res. 17, 1143–1146 (2002) M.R. Vanlandingham, T.F. Juliano, M.J. Hagon: Measuring tip shape for instrumented indentation

16.12

16.13

16.14 16.15

16.16

16.17

16.18

16.19

using atomic force microscopy, Measurement Sci. Tech. 16, 2173–2185 (2005) J.L. Loubet, J.M. Georges, O. Marchesini, G. Meille: Vickers indentation curves of magnesium oxide (MgO), J. Tribology, 106, 43–48 (1984) D. Neweyt, M.A. Wilkins, H.M. Pollock: An ultralow-load penetration hardness tester, J. Phys. E: Sci. Instrum., 15, 119–122 (1982) A.C. Fischer-Cripps: Nanoindentation, 2nd edn. (Springer, Berlin Heidelberg 2004) B. Bhushan (Ed.): Springer Handbook of Nanotechnology, 2nd edn. (Springer, Berlin Heidelberg 2004) X. Li, B. Bhushan: A review of nanoindentation continuous stiffness measurement technique and its applications, Mater.Char. 48, 11–36 (2002) W.C. Oliver, G.M. Pharr: Review: Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology, J. Mater. Res. 19, 3–20 (2004) M.R. VanLandingham: Review of instrumented indentation, J. Res. Natl. Inst. Stand. Technol. 108, 249–265 (2003) M.F. Doerner, W.D. Nix: A method for interpreting the data from depth-sensing indentation instruments, J. Mater. Res. 1, 601–609 (1986)

Nanoindentation: Localized Probes of Mechanical Behavior of Materials

16.20

16.21

16.22 16.23

16.24

16.25

16.26 16.27

16.28

16.30

16.31

16.32 16.33

16.34

16.35 16.36

16.37

16.38

16.39

16.40

16.41

16.42

16.43

16.44

16.45

16.46

16.47

16.48

16.49

16.50

16.51

S. Harvey, H. Huang, S. Vennkataraman, W.W. Gerberich: Microscopy and microindentation mechanics of single crystal Fe-3 wt.% Si: Part I. Atomic force microscopy of a small indentation, J. Mater. Res. 8, 1291–1299 (1993) D.F. Bahr, W.W. Gerberich: Pile up and plastic zone size around indentations, Metall. Mater. Trans. A 27A, 3793–3800 (1996) D. Kramer, H. Huang, M. Kriese, J. Robach, J. Nelson, A. Wright, D. Bahr, W.W. Gerberich: Yield strength predictions from the plastic zone around nanocontacts, Acta Mater. 47, 333–343 (1998) Y. Gaillard, C. Tromas, J. Woirgard: Pop-in phenomenon in MgO and LiF: observation of dislocation structures, Phil. Mag. Leters 83, 553–561 (2003) E. Carrasco, M.A. Gonzalez, O. Rodriguez de la Fuente, J.M. Rojo: Analysis at atomic level of dislocation emission and motion around nanoindentations in gold, Surf. Sci. 572, 467–475 (2004) M. Oden, H. Ljungcrantz, L. Hultman: Characterization of the induced plastic zone in a single crystal TiN(001) film by nanoindentation and transmission electron microscopy, J. Mater. Res. 12, 2134–2142 (1997) C. Tromas, J. C. Girard, V. Audurier, J. Woirgard, Study of the low stress plasticity in single-crystal MgO by nanoindentation and atomic force microscopy, J. Mater. Sci. 34, 5337–5342 (1999) W. Zielinski, H. Huang, W.W. Gerberich: Microscopy and microindentation mechanics of single crystal Fe-3 wt.%Si: Part II. TEM of the indentation plastic zone, J. Mater. Res. 8, 1300–1310 (1993) Y. Gaillard, C. Tromas, J. Woirgard: Study of the dislocation structure involved in a nanoindentation test by atomic force microscopy and controlled chemical etching, Acta Mater. 51, 1059– 1065 (2003) P. Peralta, R. Ledoux, R. Dickerson, M. Hakik, P. Dickerson: Characterization of surface deformation around Vickers indents in monocrystalline materials, Metall. Mater. Trans. A 35A, 2247–2255 (2004) N.A. Stelmashenko, M.G. Walls, L.M. Brown, Y.U.V. Milman: Microindentations on W and Mo oriented single crystals: An STM study, Acta Metal. Mater. 41, 2855–2865 (1993) T.F. Page, W.C. Oliver, C.J. McHargue: Deformation behavior of ceramic crystals subjected to very low load (nano)indentations, J. Mater. Res. 7, 450–473 (1992) S.E. Kadijk, A. Broese van Groenou: Cross-slip patterns by sphere indentations on single crystal MnZn ferrite, Acta Metall. 37, 2625–2634 (1989) W. Zielinski, H. Huang, S. Venkataraman, W.W. Gerberich: Dislocation distribution under a microindentation into an iron-silicon single crystal, Phil. Mag. A 72, 1221–1237 (1995)

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W.W. Gerberich, N.I. Tymiak, J.C. Grunlan, M.F. Horstemeyer, M.I. Baskes: Interpretations of indentation size effects, J. Appl. Mech. 69, 433–442 (2002) N. Gane, F.P. Bowden: Microdeformation of solids, J. Appl. Phys. 39, 1432–1435 (1968) J.B. Pethica, D. Tabor: Contact of characterized metal surfaces at very low loads: Deformation and adhesion, Surf. Sci. 89, 182–190 (1979) T.A. Michalske, J.E. Houston: Dislocation nucleation at nano-scale mechanical contacts, Acta Mater. 46, 391–396 (1998) J.D. Kiely, J.E. Houston: Nanomechanical properties of Au(111), (001), and (110) surfaces, Phys. Rev. B 57, 12588–12594 (1998) D.F. Bahr, D.E. Wilson, D.A. Crowson: Energy considerations regarding yield points during indentation, J. Mater. Res. 14, 2269–2275 (1999) A.H. Cotrell: Dislocations and Plastic Flow in Crystals (Oxford Press, Oxford 1953) pp. 11–12 A.M. Minor, S.A. Syed Asif, Z. Shan, E.A. Stach, E. Cyrankowski, T.J. Wyrobek, O.L. Warren: A new view of the onset of plasticity during the nanoindentation of aluminium, Nature Mater. 5, 697–702 (2006) S.G. Corcoran, R.J. Colton, E.T. Lilleodden, W.W. Gerberich: Anomalous plastic deformation at surfaces: Nanoindentation of gold single crystals, Phys. Rev. B 55, R16057–16060 (1997) C.A. Schuh, J.K. Mason, A.C. Lund: Quantitative insight into dislocation nucleation from high-temperature nanoindentation experiments, Nature Mater. 4, 617–621 (2005) A.H.W. Ngan, P.C. Wo: Delayed plasticity in nanoindentation of annealed crystals, Phil. Mag. 86, 1287–1304 (2006) D.F. Bahr, G. Vasquez: The effects of solid solution impurities on dislocation nucleation during nanoindentation, J. Mater. Res. 20, 1947–1951 (2005) K.A. Nibur, D.F. Bahr, B.P. Somerday: Hydrogen effects on dislocation activity in austenitic stainless steel, Acta Materialia 54, 2677–2684 (2006)

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Atomic Force

17. Atomic Force Microscopy in Solid Mechanics

Ioannis Chasiotis

Scanning probe microscopes (SPMs) have become a key instrument for the application and sensing of nanonewton forces in small material volumes and the measurement of in-plane and out-ofplane strains with nanometer resolution. The versatility of SPMs lies with the ability to specify the nature of tip-material interaction forces in order to probe relevant nanoscale phenomena or control the position of individual atoms and molecules that is not possible by other highresolution imaging instruments. As a result, new capabilities in nanoscale experimentation and mechanics of materials at the nanometer scale have emerged along with new challenges and opportunities for further developments. It is the objective of this chapter to introduce the new as well as the advanced SPM user to the underlying operating principles, the advantages, and the limitations of SPMs.

17.2 Instrumentation for Atomic Force Microscopy................... 17.2.1 AFM Cantilever and Tip.................. 17.2.2 Calibration of Cantilever Stiffness ... 17.2.3 Tip Imaging Artifacts ..................... 17.2.4 Piezoelectric Actuator ................... 17.2.5 PZT Actuator Nonlinearities ............ 17.3 Imaging Modes by an Atomic Force Microscope .............. 17.3.1 Contact AFM................................. 17.3.2 Non-Contact and Intermittent Contact AFM ........ 17.3.3 Phase Imaging ............................. 17.3.4 Atomic Resolution by an AFM .........

412 414 415 417 419 420 423 423 425 430 431

17.4 Quantitative Measurements in Solid Mechanics with an AFM ............. 432 17.4.1 Force-Displacement Curves............ 432 17.4.2 Full Field Strain Measurements by an AFM ................................... 434 17.5 Closing Remarks ................................... 438 17.6 Bibliography ........................................ 439 References .................................................. 440

The term scanning probe microscopy (SPM) engulfs methods that utilize force interactions or tunneling current flow between a probe and a surface to construct a mapping of the geometric and material properties of the sample surface. The two most common methods are scanning tunneling microscopy (STM) [17.1] and atomic force microscopy (AFM) [17.2] that use a sharp tip to measure the tunneling current and the tip–sample force interactions, respectively. The latter are either short range (quantum mechanical, electrostatic) or long range (van der Waals), such as magnetic and electrostatic forces. Because the distance dependence of these forces is very strong, Angstrom-scale tip–sample separations can be detected and be translated into equivalent imaging resolution. Furthermore, local force interac-

tions can be used for spectroscopic analyses: the spatial distribution of various forces may be employed to construct an image of the surface and potentially its spatial composition. Along the same lines, optical information can be obtained by using near-field scanning optical microscopy (NSOM) [17.3, 4] that is built according to conventional AFM instrumentation. The spatial resolution depends on the nature and magnitude of the forces describing the tip-surface interaction, and the geometry and size of the probe. A first step to increase the spatial, in-plane, resolution is to control the tip diameter that varies between 10–15 nm, and, in part, determines the interaction volume of the sample material. SPM has evolved significantly since its birth [17.1] because it provides capabilities that are complemen-

Part B 17

17.1 Tip–Sample Force Interactions in Scanning Force Microscopy ................ 411

410

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Contact Methods

tary to existing electron microscopy methods. SPMs do not require vacuum environment, are minimally intrusive to organic and biological samples, and support force spectroscopy. On the other hand, scanning electron microscopes (SEM) are versatile instruments that provide three-dimensional images with large depth of field and in short image acquisition times. However, only in-plane dimensional measurements can be acquired by an SEM, which demands the precise knowledge of the sample viewing angle. Furthermore, high resolution SEM requires conducting surfaces and transmission electron microscopes (TEM) operate at very high voltages with samples that must be 500 Å or thinner, conditions that are not necessary for the operation of an AFM. The first SPM method to be reported was scanning tunneling microscopy (STM) [17.1]. In order to construct surface images, the electrical interaction between a sharp metallic probe and a conductive surface that are spaced apart by a few Angstroms is utilized, Fig. 17.1. Upon application of bias voltage Vb larger than the work function of the material, a tunneling current flows between the tip and the sample that is given by [17.6, 7] 1

It ∝ exp(−Aφ 2 d) ,

(17.1)

Part B 17

where d is the distance between the tip and the sample, φ is the average barrier height between the two electrodes, and A = 1.025 eV−1/2 Å−1 [17.8]. It is on the order of a few nA and d is 5–10 Å. It is a strong function of d, and, thus, small spatial changes in It are used as the feedback quantity so that the instrument repositions the tip to maintain a constant distance d from the sample surface. The operation and resolution of an STM hinge upon the electrical conductivity of the sample and the tip. Typically, STM tips are fabricated by

Tip

V

99% current 90% current

Sample

Fig. 17.1 Interaction surface between the tip of a scanning tunneling microscope and a conducting surface [17.5]

electrochemical etching of tungsten wires that must be freshly made to avoid extended surface oxidation. If not conductive, the sample surface is coated with a few nanometers of gold, which, however, may cover the fine surface features. The requirement for electrically conducting surfaces to produce a tunneling current was overcome by atomic force microscopy (AFM) [17.2]. AFM resolves the force interaction between a sample and a sharp tip attached to a compliant cantilever to generate spatial force mappings. If the nature and spatial dependence of the forces remain the same in the entire surface, then the force map can be converted into the true surface topography. Electrostatic or magnetic forces have longer persistence lengths compared to interatomic or van der Waals forces generating strong interactions at distances greater than a few tens of nanometers. The distinction between forces according to persistence lengths led to variants of AFM, such as magnetic force microscopy (MFM) [17.9] and electrostatic force microscopy (EFM), while the exploitation of other quantities, such as temperature, thermal conductivity, and light intensity led to scanning thermal microscopy (SThM) [17.10] and NSOM, respectively. In a different application of SPM, a compliant AFM cantilever is used to construct a semi-quantitative map of the lateral tip-surface force by measuring the torsional deflection of the cantilever. This method is commonly referred to as lateral force AFM (LFM) [17.11] and it is applied to generate surface adhesion or friction force maps [17.12]. An accurate calibration of the torsional constant [17.13] and deconvolution of adhesive forces from imaging artifacts, to be discussed in detail in this chapter, provide a good estimate for the magnitude of surface adhesion and friction. In addition to sensing and recording force interactions, an AFM probe is used for actuation and manipulation purposes both in contact [17.14] and in non-contact with a surface [17.15] in order to position particles, nanofibers, and nanotubes [17.16] to a desired location. Individual atoms are also possible to manipulate on atomically flat surfaces by an STM operated in vacuum [17.17, 18]. The ability of precise tip positioning allowed for small-scale surface patterning via nanolithography either by mechanically abrading a soft surface by a hard tip [17.19], or by local oxidation of a surface by the application of voltage between the tip and the sample [17.20] and subsequent etching of the resulting oxide. A more elaborate approach calls for the deposition of organic and inorganic molecules and atoms on a surface via tip immersion in a buffer of or-

Atomic Force Microscopy in Solid Mechanics

ganic molecules of interest and subsequent deposition of these molecules in a predetermined pattern benefiting from capillary transport processes [17.21]. This method, termed dip-pen lithography, has a resolution of approximately 50 nm, only limited by surface tension. Arrays of AFM cantilevers provide scalability to this method for surface read-and-write operations, as demonstrated by the millipede memory [17.22]. The force sensing capability of an AFM cantilever has been used in force spectroscopy [17.23] with applications in biology [17.24] and material property measurement such as nanoindentation [17.25]. Imaging in liquid environments [17.26] allows for electrochemical studies [17.27] with resolution that is not feasible by other techniques. However, AFM cantilevers are not limited to tip-surface interactions. The small dimensions and high resonance frequency of a micron-sized cantilever allows for sensing of molecules that are attracted and bound to its surface, which in turn permits the quantitative identification of chemical species [17.28]. The specificity of these methods is still subject to research. This chapter introduces the underlying operating principles and instrumentation for atomic force

17.1 Tip–Sample Force Interactions in Scanning Force Microscopy

411

microscopy (AFM) to provide the background for knowledgeable use of this class of instruments. The two main approaches in imaging by an AFM, contact and non-contact (NC-AFM) modes, are presented in conjunction with the accuracy that long and shortrange interatomic forces are resolved, which ultimately dictates the resolution of an AFM. The cantilever dynamics in NC-AFM are modeled as a damped forced oscillator and its amplitude and frequency characteristics are related to the functions performed during NC-AFM imaging. AFM metrology heavily depends on the determination of the cantilever stiffness and the deconvolution of true image features from cantilever tip artifacts and the nonlinearities of the piezoelectric actuator. Established calibration and deconvolution methods and their limitations are discussed from theoretical and practical viewpoints. Furthermore, two quantitative applications of AFM in experimental mechanics are presented: (a) The application and determination of pN and nN forces by an AFM probe (force spectroscopy) and the associated force-displacement curves, and (b) the measurement of in-plane strain and displacement fields with nanometer spatial resolution by AFM imaging and Digital Image Correlation analysis.

17.1 Tip–Sample Force Interactions in Scanning Force Microscopy

F=−

C , rn

n≥7.

(17.2)

The negative sign denotes attractive forces and the power n dictates the decay length, i. e. whether the force interaction is short or long range, and C is a constant. In the absence of electrostatic charges, the forces are distinguished in two categories according to the power n: the long and short-range forces. Short range forces scale with n = 9–16 while long-range forces scale with n = 7. Different modes of AFM imaging benefit from long or short range forces to construct topographic maps of surfaces. Short-range forces are the result of interatomic interactions and are important only at distances smaller than d = 1 nm. For AFM purposes such forces are strongly repulsive as a result of electronic repulsion and Pauli’s exclusion principle. Short-range forces can

also be attractive when they result in chemical binding but these forces are not beneficial in AFM imaging. Although physical contact is not established until d = 0.165 nm [17.29], at a few Angstrom distances two bodies are considered for all practical purposes to be in contact because the amplitude of repulsive forces is sufficient to induce surface deformations. The AFM mode of operation that employs short-range forces with possible permanent surface deformation is termed contact AFM. Long-range forces are described by smaller exponents n and thus are significant at larger separations compared to short-range forces. For n = 7 these forces are classified as van der Waals forces. The latter are due to permanent dipole interactions (Keesom or orientation forces), permanent and induced dipole interactions (Debye or induction forces), and inductive interactions, (London or dispersion forces). An approximate ratio of Keesom:Debye:London forces is 200:20:1 [17.29]. Their magnitude is a function of material polarizability and the dielectric constant of the medium. At distances larger than 20 nm van der Waals forces scale with n = 8

Part B 17.1

An AFM harnesses the interaction forces between a tip and a surface that are a strong function of distance. The force between two molecules or atoms is given by [17.29]

412

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Contact Methods

because of retardation effects [17.29]. It is notable that dispersion van der Waals forces are present even in the absence of dipoles. Their energy is equivalent to 1–1.5 kB T where kB is the Boltzman constant and T is the absolute temperature, i. e. they are temperature dependent. Dispersion forces can be sensed by an AFM tip if they exceed the background noise equal to kB T that is generated by thermal vibrations due to Brownian motion. Since thermal vibrations scale with absolute temperature, their effects can be overcome if AFM images are acquired at low absolute temperatures, which is the only approach to obtain atomic resolution by non-contact AFM (NC-AFM), which is the AFM mode of operation that employs long-range forces. Van der Waals forces between atoms and molecules are rather week, but additivity does generally apply so that the dependence of force interaction between the finite size AFM tip and the sample surface scales more favorably with distance compared to the case of two atoms or molecules. Specifically, probes with tip radius larger than 0.5 nm can be considered as finite size and the ensuing interactions are those between particles than individual atoms. For a conical AFM tip the force exerted to a sample is −HR tan2 θ , (17.3) d2 where H is the Hamaker constant, d is the tip-sample separation, θ is the half cone angle, and R is the radius of the AFM tip [17.30]. Long and short-range forces dominate very far and very close to a sample surface but they are in competition at intermediate distances. There are several models that describe the force interaction between a probe and a surface. The Lennard–Jones potential for instance, Fig. 17.2, provides a good description of the physical interactions that take place at various distances. The F=−

Potential energy Intermittent contact Repulsive force Contact

Distance

Non-contact Attractive force

Fig. 17.2 Correlation between nature of forces and AFM

imaging method as a function of tip–sample distance following a Lennard–Jones potential [17.5]

Part B 17.2

main two modes of AFM operation, contact versus noncontact, are possible at a range of forces that maintain different sign, i. e. repulsive and attractive forces, respectively. Due to the competition between attractive and repulsive forces at intermediate distances, Fig. 17.2, the same force magnitude is possible at two different separation distances. Interpretation of AFM raw data requires a homogeneous surface that is clean from impurities so that the interaction potential remains spatially the same. Impurities introduce image artifacts as they include additional forces between an AFM probe and the surrounding medium such as capillary, hydrophobic, and adhesive forces. Convolution of these forces results in imaging artifacts that are enmeshed with the true surface features. In this regard, tips with low reactivity and very clean sample surfaces are critical.

17.2 Instrumentation for Atomic Force Microscopy Before proceeding with the description of image formation by an AFM it is important to describe its basic components. Because of general similarities, Fig. 17.3 incorporates the basic components of an AFM and an STM. The main components of an SPM are the piezoelectric scanner, the probe that is a current or force reading device, an analog-to-digital converter that converts the analog signal recorded by the probe to a digital signal for processing and use by the control electron-

ics, a computer and control electronics, and a recording medium. The basic difference between an STM and an AFM is in the nature of the tip-surface interaction. In the STM this signal is a current, recorded by an ammeter and used by the control electronics to keep the tip at constant distance from the surface. In an AFM the deflection of a cantilever reflects the force exerted to the sample. In the majority of AFM instruments this deflection is computed from the signature

Atomic Force Microscopy in Solid Mechanics

of a reflected laser beam on a four quadrant photodetector (photodiode). Although the cantilever deflections are very small to be detected by naked eye, they can be measured very accurately because the source of the laser beam and the photodetector are positioned at an angle and far from the cantilever so that small changes in cantilever deflection result in large changes in the deflection arc that is recorded by the photodetector. Other methods to obtain cantilever deflections involve measurement of (a) changes in capacitance with respect to a reference electrode, (b) the tunneling current between the cantilever and a conductive tip placed on its rear side, and (c) interferometric measurements [17.8]. For details about different detection methods the reader is prompted to [17.24]. Contact AFM imaging is conducted in DC mode. The instrument continuously measures the deflection of the cantilever due to short-range forces at each surface location and uses this information to construct an image. In NC-AFM imaging the collective van der Waals forces, described for instance by (17.3), are weak to be reliably detected in DC (continuous) mode. Instead, AC measurements are significantly more sensitive and are employed to detect long-range forces that, in general, are three orders of magnitude weaker than short-range

17.2 Instrumentation for Atomic Force Microscopy

413

forces. As will be described later, changes in phase and amplitude of a modulated cantilever interacting with the specimen surface are computed to construct an image by NC-AFM. Compared to optical instrumentation, AFM imaging is not instantaneous because the data are collected pointwise. An SPM collects data by rastering a probe over a stationary surface (or by rastering the surface under a laterally stationary probe) along a straight line in x-direction (called the fast scanning direction). The motion in the fast scanning direction lasts 1 s or less, and the instrument collects a large number of equally spaced surface height readings, typically between 128 and 1024 in each line. Subsequently, the cantilever moves to the next scan line in y direction (called the slow scanning direction), repeats a fast scan of one line along the x direction and continues until it rasters the entire image area. Figure 17.4 shows a scanning sequence for AFM imaging. In Fig. 17.4 the fast scanning direction alternates between left-to-right and right-to-left. Due to considerations that stem from piezoelectric scanner nonlinearities, the true spacing between imaging points may not be constant, and the fast scanning direction should be fixed for each image, e.g. sampling from leftto-right only, and be the same as that during instrument

Control electronics

AFM PC

Piezoelectric scanner

Four quadrant photodetector

Part B 17.2

STM

Laser

Feedback loop maintains constant tunneling current AFM cantilever

Laser

STM tip Sample

Probe Ammeter

Fig. 17.3 Basic instrumentation layout of an STM and an AFM

Piezoelectric scanner

Sample Sample

414

Part B

Contact Methods

Fast scan PZT Slow scan

Fig. 17.4 Rastering data collection by an AFM cantilever

calibration. Note that the fast and slow scanning directions can be swapped.

17.2.1 AFM Cantilever and Tip

Part B 17.2

An AFM probe consists of a cantilever with an attached sharp tip at its free end. Cantilevers are manufactured by surface and bulk micromachining methods using single crystal silicon, Si3 N4 , and SiO2 [17.31]. To improve the signal-to-noise ratio of the photodetector, cantilevers are coated with Au or Al to increase reflectivity. To reduce its wear resistance the sharp tip is coated with Si3 N4 or diamond. The cantilever length typically varies from 100–200 μm, its width from 30–50 μm, and its thickness from 0.5–2.0 μm. For this range of cantilever geometries and materials the spring constant ranges between 0.01 N/m and 50 N/m, or more, while the resonance frequencies lie in the range of 30–400 kHz. At the free end of the cantilever a 4–10 μm long conical or pyramidal tip with an average radius of curvature of 10–20 nm is attached. The cantilever serves as the force sensor while the amplitude of the applied force is dictated by the radius of the sharp tip, the tip and sample materials, their distance, and the intermediate medium. In recent years cantilevers with integrated piezoelectric elements in the form of bimorphs have been employed to accomplish fast scanning by piezoelectric sensing and control of the cantilever deflection [17.32– 34]. When these cantilevers are combined with active damping by controlling the effective Q-factor of the cantilever, the transient response times are reduced and fast topographic measurements are achieved that increase the imaging speed by one order of magnitude.

There are two typical types of cantilevers: the rectangular beam (I-beam) and the triangular (V-shaped) cantilevers, as seen in Figs. 17.5 and 17.6. Both cantilevers can be manufactured with any desirable spring constant and resonance frequency. The main difference is their torsional stiffness. V-shaped cantilevers have significantly larger torsional and lateral stability compared to I-beam, which is an advantage in imaging samples with strong adhesive forces, deep steps (trenches), or imaging in contact AFM. I-beam cantilevers are more appropriate for frictional studies and measurement of in-plane forces because they allow for significant torsional deflections. The cantilever spring constant k must be known to determine the tip-sample force during imaging or force spectroscopy. The spring constant can be found analytically/numerically or be measured experimentally. For the dimensions shown in Fig. 17.5, the spring constant of an I-beam cantilever is [17.35] Etc3 w , (17.4) 4L 3 where tc is the cantilever thickness, and E is the Young’s modulus of the cantilever material. The torsional spring constant of an I-beam cantilever can be calculated by considering an applied lateral force at the end of the tip [17.36] kc =

ktorsion = a)

Gbtc3 , 3Lh 2

(17.5)

Normal z

L

Longitudinal x Lateral y

tc h

b) Longitudinal x Lateral y

w/sin α

L1

Normal z w 2α

w

d B

b

A

tc

h

Fig. 17.5 (a) I-beam and (b) V-shaped cantilever and geometry parameters for the calculation of normal and torsional stiffness [17.13]

Atomic Force Microscopy in Solid Mechanics

a)

 w 3L 1 (1 + ν) cot α , − d + L 2 Ewtc3 cos α sin α

L2 =

L 1 tan α + (w − d sin α)(1 − ν) cos α , 2 − (1 − ν) cos2 α (17.8b)

where ν is the Poisson’s ratio of the cantilever material. The torsional stiffness is  Etc3 1 w L 1 cos a ktorsion = log + 3(1 + ν) tan a d sin a w   2 6L 1 w sin a + 3w (1 + ν) cos2 a −1 × 1− . 8L 21 + 3w2 (1 + ν) cos2 a (17.9)

For L 1 > w (17.9) becomes  Etc3 1 w log ktorsion = 3(1 + ν) tan α d sin a L 1 cos a 3 sin 2a −1 + . (17.10) − w 8 The lateral stiffness at the tip position is calculated from the bending deflection of the cantilever assuming a force is applied at the end of the tip  ktorsion Etc3 1 w = log klateral = 2 2 d sin a h 3(1 + ν)h tan α −1 L 1 cos a 3 sin 2a + . (17.11) − w 8

17.2.2 Calibration of Cantilever Stiffness The cantilever stiffnesses calculated by (17.4)–(17.11) are accurate within certain assumptions and the accuracy in determining the cantilever dimensions and the elastic material properties. Alternatively, a cantilever can be calibrated experimentally with an accuracy of 10–40% depending on the specific method. The most common calibration methods are described here. Calibration of k by Using Resonance Frequency (Cleveland Method) The resonance frequency of a cantilever is a function of its effective mass and spring constant and may be measured accurately by the photodetector. If the dimensions

Part B 17.2

where h = h t + tc /2, h t is the length of the tip, b and L are given in Figs. 17.5a and 17.5b, and G is the shear modulus of the cantilever material. The spring constant of a V-shaped cantilever requires a more elaborate calculation. A closed form solution for its approximate bending stiffness based on a parallel beam calculation is [17.37]  3  Etc w w3 cos α 1 + kc = 3  2L 31 2 L 1 tan α + cosw α −1  3 L1 × (3 cos α − 2) , (17.6) L1 − d where L 1 is given in Fig. 17.5b and α is the included half angle. A thorough analysis in [17.13] provides more accurate values for the normal and torsional stiffnesses of a V-shaped cantilever. For the cantilever dimensions in Fig. 17.5b the normal stiffness is   −1 w , (17.7) −d k c = Δ1 + Δ2 + ϑ sin α where  2 w 3 − 2d Δ1 = 3 Etc tan α sin α   w 2 − d 2 log +1 d sin α  L 21 2L 1 + 3(w cot α Δ2 = Ewtc3 cos2 α cos α  (17.8a) − d cos α − L 2 sin α) ,

ϑ=

415

Fig. 17.6 (a) I-beam and (b) V-shaped cantilevers. The bright spots at the free ends of the cantilevers are their tips

b)

where

17.2 Instrumentation for Atomic Force Microscopy

416

Part B

Contact Methods

and density of the cantilever are known, the effective mass of an AFM cantilever with rectangular cross section is m ∗ ≈ 0.24m c , where m c is the mass of a tipless cantilever. The addition of an auxiliary end-mass M modifies the resonance frequency as [17.38]

k 1 ω ⇒ M = k(2πν)−2 − m ∗ = ν= 2π 2π M + m ∗ (17.12)

If individual microspheres of known mass are added at the end of the cantilever and the new resonance frequencies are measured, the effective mass and spring constant of the cantilever are obtained from the M versus (2πν)−2 plot. Uncertainties in the calculation of k stem from the precision in locating the mass, M, at the free end of the cantilever and the accuracy in knowing M. This method is also called the Cleveland method because of Cleveland et al. [17.38]. Contrary to using (17.4) to (17.11) to determine k, this method does not require knowledge of the Young’s modulus, which is an advantage when the cantilever is fabricated as a composite structure that includes anisotropic or inhomogeneous materials. Furthermore, this method can be applied to cantilevers of any geometry whose precise dimensions are not known as opposed to the application of (17.4)–(17.11). According to [17.38] the accuracy of this method is 10–15%.

Part B 17.2

Calibration of k by Using the Cantilever Thermal Fluctuations An AFM cantilever in free space and in the absence of a driving force is subjected to random thermal vibrations. The energy associated with this motion, which results in undesirable noise in AFM images, is related to the cantilever natural frequency and the average total energy, kinetic and dynamic, that is equal to kB T . The total energy is

1 p2 + m ∗ ω20 q 2 , (17.13) 2m ∗ 2 where q is the displacement of the cantilever, p is its momentum, and ω0 is the natural frequency. The mean value of each quadratic term in the Hamiltonian is equal to half the thermal energy  1 1 (17.14) mω20 q 2 = kB T . 2 2 H=

Ambient excitations have low frequency content and act as background noise. The mean-square amplitude of the cantilever’s thermal fluctuations is recorded after the ambient noise is filtered in the frequency domain.

Then, the area under the remaining peak of the power spectrum represents the mean square of the thermal fluctuations χ 2  that is used to calculate the spring constant as [17.39] kB T (17.15) kc = 2 . χ  This method is mostly effective for compliant cantilevers, with large thermally induced fluctuation amplitudes and for limited ambient vibrations. The accuracy of this method is 5–20% [17.40]. Determination of k Using a Pre-Calibrated Cantilever If a cantilever (or any probe attached to an apparatus that measures forces and displacements) is precisely calibrated, it can be used as reference to determine the spring constant of an AFM cantilever. The spring constant of the unknown cantilever is calculated by measuring the deflections of both cantilevers simultaneously by using a heterodyne interferometer and the AFM photodetector [17.41]. It is not necessary to know the dimensions of the AFM cantilever. The relative spring constants of the two probes determine the accuracy of this method: If the reference cantilever is much stiffer, the deflections of the cantilever with unknown spring constant is obtained with small relative uncertainty but the deflection of the reference cantilever is small, and as a result, the relative uncertainty in the applied force is large. Conversely, if the reference cantilever is compliant, the determination of the applied force is quite precise but the deflection of the cantilever with unknown stiffness is not measured accurately. Thus, the reference and the unknown cantilevers should have similar spring constants. Another major uncertainty originates in the location of the applied force on the unknown cantilever as its deflection scales with the third power of its length. In general, the accuracy of this method is 15–40% [17.40]. Calibration of AFM Cantilevers for Force Measurements in a Fluid The calibration of spring constants of cantilevers to be immersed in a liquid should be conducted in the same medium. The photodetector is not immersed into the liquid and, thus, the trajectory of the reflected laser beam intersects two different media, i. e. liquid and air, with different refractive indices [17.42]. A calibration of the cantilever in the same media is key when force-displacement measurements are the objective. One approach is to recalibrate the photodetector using a force-displacement curve (see Sect. 17.4.1) ob-

Atomic Force Microscopy in Solid Mechanics

tained on an infinitely stiff surface immersed in the same medium and at the same depth as the target surface so that the paths of the incident and the reflected laser beam are the same as those in the actual experiment. Then, a correction can be applied to the force-displacement curve to deduce a slope of −1 in the contact regime. This correction factor may be used in subsequent experiments in the same medium and for laser trajectories of the same length in the two media.

17.2.3 Tip Imaging Artifacts While the cantilever stiffness and geometry determine the force resolution, the tip geometry dictates the interaction volume and the resulting stresses to the sample. The tip geometry and radius are important because they

17.2 Instrumentation for Atomic Force Microscopy

417

determine the shape and dimensions of the surface features recorded in an AFM image [17.43]. Image features that resemble the tip geometry instead of the real surface are called imaging artifacts. The tip radius of curvature determines the size and shape of the in-plane features that may be resolved in an AFM image. The shape of the tip (conical, parabolic, pyramidal, or spherical) determines the geometry of three-dimensional surface features. This can be better understood with the aid of Fig. 17.7 that shows an AFM image of the surface of a compact disc (CD) where the digital information is stored in the line trenches. The cross-sectional image in Fig. 17.7 shows that the relatively straight trench walls are imaged by the cantilever tip as conical. Depending on the actual depth of the trenches, their opening, and the tip cone angle, it is pos-

(mm/div)

40

Part B 17.2

0

4

8 Height profile (μm)

(nm) 100 60

20 0

4

8 Height profile (μm)

Fig. 17.7 AFM image of the surface of a compact disc and a cross-section, as recorded with a conical AFM tip. The

geometry of the acute trenches can be used to calculate the tip cone angle

418

Part B

Contact Methods

sible that the profiles in Fig. 17.7 are truncated because the AFM probe could not reach the trench base. Thus, samples of this type could be used for inverse imaging and calibration of an AFM tip. Two general conditions are necessary for accurate AFM imaging of surface roughness. If a sinusoidal surface profile is assumed, y = A sin(2πx/λ) with roughness amplitude A and wavelength λ the slope of the tip (half angle) should be smaller than the maximum slope in the surface:

 2π A 2πx dy

dy 2π A = = cos ⇒ . dx λ λ dx λ max

(17.16)

Furthermore, the tip radius R should be smaller than the local radius of curvature everywhere in the surface: Rtip ≤ −

1 d2 y

=

dx 2

λ2  2πx  ⇒ Rtip ≤ 4π 2 A sin λ 1

4π 2 A λ2

(17.17)

a)

Scan direction

AFM tip

Part B 17.2

AFM image

b)

Scan direction

AFM tip

AFM image

Fig. 17.8a,b Determination of tip angle and radius of curvature by using (a) a nanopillar and (b) a sharp surface

step [17.44, 45]

Thus, in order to determine the limits in recording surface features one must know the tip geometry and radius of curvature. The exact tip geometry may be determined by imaging known structures. The latter must have lateral dimensions smaller than the tip radius and be more acute than the tip cone angle [17.44–46]. Such samples may contain sharp vertical pillars and surface steps, or trenches with sharp vertical walls as seen in Fig. 17.8a and 17.8b. To understand the construction of the AFM images in these figures one must consider the cantilever deflection due to vertical and lateral forces and the associated vertical motion of the PZT actuator in response to changes in cantilever deflection. A deflection of the cantilever is interpreted as a relative change in the surface height even if this is caused by lateral forces applied on the probe resulting in vertical deflection due to cantilever torsion. The nanopillar with vertical walls in Fig. 17.8a comes in lateral contact with the AFM probe before the probe axis overlaps with the vertical axis of the nanopillar. Because the x and y coordinates of surface features recorded by the AFM are those of the axis of symmetry of the cantilever tip, the path followed by the PZT to maintain constant cantilever deflection generates the image of a reverse AFM probe. When the nanopillar width is considerably larger than the tip radius, thus forming a vertical step, the resulting AFM image may be used to determine the tip cone angle and radius of curvature, Fig. 17.8b. In this case, the slope of the step sidewalls in the AFM image is the half cone angle, while the radius of curvature of the rounded corner of the step is that of the probe tip [17.44]. The examples in Fig. 17.8 point to the realization that the dimensions of surface features in an AFM image are overestimated and potentially have distorted geometry because of tip shape convolution. The true dimensions of features with known geometry may be determined by tip shape deconvolution. This procedure requires knowledge of the cantilever tip geometry and its radius of curvature. In its simplest form, deconvolution is applied to axisymmetric (cylindrical) and spherical surface features. The deconvolution function for a cylinder or a sphere with apparent width W that are imaged by the conical tip of half cone angle φ shown in Fig. 17.9 is [17.47]  Rcone −1 W = 2Rsphere cos (φ) 1 + Rsphere   Rcone + 1− (17.18) sin(φ) , Rsphere

Atomic Force Microscopy in Solid Mechanics

ϕ Rcone

Rp Rsphere

Rsphere

Fig. 17.9 Parabolic and conical AFM tips imaging a nanosphere with dimensions similar to the probe radius [17.47]. The profiles of the nanospheres imaged by each tip are shown in the second row. Note that the two spheres as imaged by the two different probes have different side profiles (constant slope versus parabolic)

(PZT) powder. The resulting material is comprised of domains with specific electric dipole moments. The application of an electric potential aligns the dipoles producing a net deformation of the piezoelectric actuator. Typical PZT scanners may image surface areas that vary from 10 × 10 nm2 to 100 × 100 μm2 with the z-range being 1/10-th of the lateral scan. The precision is 0.1 Å, or better, which is determined by the analog-to-digital converter in the AFM controller. An SPM actuator is either a tube or a stack. Tubular actuators are more suitable when the cantilever is rastered over the sample surface while PZT stacks are more appropriate when the sample is rastered under a laterally stationary cantilever that is attached to an actuator moving only vertically. Imaging resolution is a function of the electronics bandwidth and the signal-to-noise ratio. PZT scanners with smaller scan range provide better imaging resolution. Figure 17.10 shows the configuration of a PZT tube constructed for use as an AFM actuator. It is a hollow PZT cylinder with four attached electrodes that split it into four quadrants. An additional electrode is attached to the tube to generate motion in the z direction. The tube interior and exterior are silver-plated so that each quadrant is a PZT element sandwiched between two electrodes. When voltage is applied to the electrodes, the thickness of the PZT quadrant increases (decreases) and consequently the length of the quad-

–x

+x

–y

+y

z z

17.2.4 Piezoelectric Actuator While the vertical coordinates in an AFM image are based on the deflection of the cantilever, the lateral coordinates are determined by the position of the piezoelectric actuator. The piezoelectric actuator is fabricated by compaction of polycrystalline Lead Zirconate Titanate

419

x

y

Fig. 17.10 Combination of a split PZT tube for in-plane

motion and a continuous PZT tube for vertical motion

Part B 17.2

while if the same object is imaged by a parabolic tip its apparent width is  √ Rp 1 , (17.19) W = 2Rsphere 1 − a 1 + √ a Rsphere   where Rp 1 Rp a= 4+ 8 Rsphere Rsphere  2  Rp Rp − +8 (17.20) . Rsphere Rsphere The abovementioned equations assume that the tip is perfectly vertical to the sample surface. In reality, the cantilever is always tilted forward by 10◦ or more, which may result in asymmetric features. Furthermore, an I-beam cantilever has small torsional stiffness and the probe can be tilted resulting in additional asymmetries. Corrections similar to those in (17.18)–(17.20) for tilted cantilevers can be found in [17.47]. The spatial resolution of AFM images is also determined by the number of data points collected. As discussed later in this chapter, the true resolution in determining the dimensions of nanometer-scale surface features is limited by the smallest interaction force that can be measured by the probe.

17.2 Instrumentation for Atomic Force Microscopy

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Part B

Contact Methods

rant decreases (increases). Opposite quadrants provide motion in +x and −x directions and in +y, and −y directions respectively, by relative expansion and contraction. Thus, monotonic increase in voltage applied to the +x quadrant coupled with simultaneous decrease in voltage applied to the −x quadrant results in motion in the fast scanning direction. Reversal of this operation results in rastering motion of the cantilever, Fig. 17.4. A step increase in the voltage applied to the +y quadrant and a step decrease in the voltage applied to the −y quadrant after completion of one scan line in the x direction result in motion in the slow scanning direction. The lateral displacement on the x–y plane, produced by bending of the piezoelectric tube due to the transverse piezoelectric effect, is given by [17.48] √

l2 (17.21) V, π Rt where d13 is the transverse piezoelectric coefficient, l is the length of the piezoelectric tube, t is the tube thickness, R is the mean radius of the tube, and V is the voltage applied to each electrode. This actuation also results in axial tube deformation produced by an applied voltage to an inner tube electrode [17.49, 50] Δx =

2d13

Part B 17.2

l (17.22) Δl = d13 V . t Tube extension is accomplished if the same voltage is applied to all four quadrants that extend or contract by the same amount providing motion in the z direction. Alternatively, a dedicated tube may be used and the voltage applied to the z electrode provides the longitudinal scanner extension/contraction. The two tubes can be attached and operated simultaneously, Fig. 17.10. In Displacement (μm)

Voltage (V)

Fig. 17.11 Hysteretic behavior of a PZT scanner. The

dashed line is the linear calibration and the curved segments are the extension and contraction paths. The arrows point to the deviation from linearity

this configuration the lateral and axial motions of the tube are not as strongly coupled as in the former case. The voltage applied to the PZT scanner varies in the range ±200 V. A function of applied voltage per unit length of extension is obtained by a calibration standard with known periodic surface features. The accuracy in length measurements and positioning of the AFM cantilever on a surface relies on this calibration. Typical calibration standards have a checkerboard pattern comprised of steps constructed by photolithography, thus allowing for precise periodic dimensions, see Fig. 17.12 for instance. The calibration standard is imaged and the AFM measurements are compared to the real dimensions of the standard so that a correction factor is computed. A simple linear relationship between the applied voltage and the PZT motion is often assumed [17.49]. However, the PZT scanner is subject to intrinsic nonlinearities and as a result a simple correction factor is not sufficient. Instead, scanner calibration can be conducted at different length scales or by using a non-linear voltage-extension calibration function.

17.2.5 PZT Actuator Nonlinearities The intrinsic nonlinearities of a piezoelectric tube (hysteresis, cross-coupling, creep) are discussed in detail because they are usually not accounted for in the aforementioned calibration. Scanner nonlinearities [17.51–53] induce imaging artifacts that are convoluted with surface features and are detectable only by imaging surfaces with well-characterized periodic features. Hysteresis The paths of extension (trace) and contraction (retrace) of a PZT scanner are different due to a hysteretic relationship between the applied voltage and the change in its dimensions as seen in Fig. 17.11. The deviation from linearity depends on the scanner geometry and material and cannot be fully predicted. The deviation from linearity is maximized in its mid-range reaching as much as 15%, which is significant when, a linear conversion between applied voltage and tube extension is assumed. Thus, the exact same surface features appear to be different in two AFM images collected in the forward (trace) and the returning (retrace) direction. Moreover, the same surface features appear to have different physical dimensions if images are collected with the scanner fully contracted or half-extended. It is evident that the effects of hysteresis are pronounced in

Atomic Force Microscopy in Solid Mechanics

(μm/div)

17.2 Instrumentation for Atomic Force Microscopy

421

(nm) 120

0.2

80 40 0

40

80

Height profile (μm)

(nm) 250 150 50 0 0

40

80

40

80

Height profile (μm)

Height profile (μm)

Fig. 17.12 Effect of cross-coupling on a large AFM image of a calibration standard. The cross-coupling bow height is 100 nm in the 100 μm line scan

Cross-Coupling According to Fig. 17.10, a PZT tube rasters the specimen surface by lateral bending to generate the x–y motion. This motion is produced by extending one side

of the tube and contracting the opposite. In doing so the tube also extends towards the sample surface although the absolute surface height is constant. The cantilever is in feedback, which forces the PZT to retract the cantilever tip to avoid coming in contact with the surface. As a result, the z-scanner retracts and the AFM records an artificial concave surface that is the effect of cross-coupling in lateral and axial tube motions. The calibration sample in Fig. 17.12 and the two line profiles help to illustrate the scanner cross-coupling. The first height profile corresponds to a flat section. Although the true surface has no curvature, cross-coupling results in approximately 100 nm height error in a 100 μm line scan. Similarly, the cross-section of the periodic steps contains the same bow background height. Crosscoupling can be minimized by recording small size images or by calibration of the AFM scanner. A flat surface such as a glass-slide is imaged and the bow profile is subtracted from all subsequent AFM images. The use of separate tubes for the lateral and vertical motion of the cantilever can alleviate cross-coupling. Creep PZT scanner creep is pronounced when it moves from rest to a new position with sudden application of voltage. It results in distortion in the beginning of image recording, which is reduced with time. Creep is caused because the PZT actuator continues to extend or contract after an initial response to a voltage step. The

Part B 17.2

large AFM images. This distortion of in-plane features may be limited by recording only the trace or the retrace data. Alternatively, the precise hysteresis curve could be obtained by a calibration standard and subsequent software correction. Hysteresis affects z-height measurements especially when sharp corners are present. For instance, the two identical sides of a step (plateau) appear to be of different height in an AFM image. Since the voltage-displacement paths during scanner extension and contraction are different, a different voltage is required to contract the scanner to its reference position as the tip moves away from the step. The application of the same voltage difference for tip ascent and decent results in different contraction and extension. Thus, additional ΔV is required for the cantilever tip to reach the bottom of the step while it is in feedback. When the total Σ(ΔV ) is computed to calculate the step height, the step appears to be asymmetric. The effect of hysteresis is pronounced in large vertical features and is often convoluted with scanner creep and the effects of high controller gain. As a first approximation, the true step dimensions may be corrected with the plot in Fig. 17.11.

422

Part B

Contact Methods

Part B 17.2

same process is responsible for inaccurate positioning of the AFM probe when a small region within a larger AFM image is selected for imaging. The scanner applies a voltage offset that results in creep and an error in locating the probe at the desired coordinates. Immediate scanning results in lateral image distortion that lasts for a few minutes. Similarly, scanner creep affects the outof-plane motion of the scanner, especially when sharp surface steps are traversed. After an ascending motion at a surface step, the scanner continues to contract despite the application of constant voltage. Since it is in feedback, a counter voltage is applied to maintain the force interaction with the surface. Similarly, when the PZT extends to descend the step the scanner overshoots due to creep and a counter voltage is applied to prevent contact with the sample. As a result, a perfect surface step is represented by Fig. 17.13a,b and only wide surface steps have a relatively flat top surface. The scanning rate affects the severity of creep. Slow scanning rates reduce the effect of creep but amplify the effect of thermal drift. Lateral distortion due to creep may be mediated by (a) including a buffer area at the beginning of each scan, (b) begin imaging at the line of completion of the last image rather than retuning the PZT to its zero position, (c) allowing the PZT to rest at the image starting point, (d) using the data from the retrace rather than a rapid return of the scanner to the next scanning line (although this approach introduces hysteresis). PZT nonlinearities are inseparable as they occur simultaneously, Fig. 17.13b. The figure shows the AFM image profile of a surface step. The different (and finite) slopes of the step sidewalls are due to hysteresis, i. e. the voltages required for the same amount of PZT tube extension and contraction were not the same. Thus, the overshoot at the beginning of the step and the crevice at its end are due to creep. These features are further accentuated by fast scanning or large controller gain. a)

Cross-coupling is also present, as seen in Fig. 17.12 that shows a larger profile of the same sample. The finite slope of the step sidewalls is subject to tip-sample convolution. There are other nonlinearities that depend on the PZT material and its age. The voltage-scanner displacement relationship is intrinsically non-linear being S-shaped although the calibration of the PZT is usually a linear function. Furthermore, the orientation of individual dipoles in the PZT material becomes random over the period of years when no bias voltage is applied. This effect is termed scanner aging because the piezoelectric actuator can extend for only part of its original range. Scanner aging can be avoided by regular use of the scanner, frequent calibration, and thermal shielding when the AFM is used near hot surfaces. The scanner can be restored to its original performance by re-poling. This process realigns the PZT dipoles by heating above the Curie temperature (350 ◦ C) to allow for dipole mobility and subsequent application of a DC bias for long periods of time. Slow cooling followed by removal of the DC bias force the dipoles to remain aligned. Linearization of the PZT Actuator Actuator nonlinearities may be partially compensated by software linearization. For instance, the effect of cross-coupling can be determined by scanning a surface of known radius of curvature that is subtracted from images of unknown surfaces. A calibration grating is imaged and the image is compared to the true geometry of the grating. A correction matrix is generated to remove scanner nonlinearities. This method is applicable when the lateral and vertical image features are comparable to those of the calibration grating and the direction and scanning rate are the same. Software corrections cannot account for PZT aging and, thus, calibration

b)

Vz

(nm) 200 150 100 50 0

Vx

4

8

Height profile (μm)

Fig. 17.13 (a) Effect of PZT creep on imaging a sharp surface step [17.5], and (b) AFM trace of a surface step of a calibration grating showing the combined effect of creep, cross-coupling, and hysteresis

Atomic Force Microscopy in Solid Mechanics

Limitations in Scanning Rate The maximum scanning rate of commercial AFMs is about 10 Hz. While rough specimen surfaces reduce this rate to 1–2 Hz, the limiting scanning rate is related to the frequency response of the components of an AFM, Fig. 17.3. The A\D electronics have large bandwidth that lies in the MHz regime. The same is true for the photodetector that responds to frequencies up to 600 kHz in tracking the cantilever RMS amplitude in non-contact AFM. The control electronics and PZT scanner amplifier are slower and have a few kHz bandwidth. The limiting factor is the PZT actuator resonance frequency that is on the order of a few hundreds of Hz to a couple of kHz. A rough surface would require numerous extensions and contractions of the PZT tube that are directly multiplied by the scanning rate, thus reaching the scanner’s natural frequency at relatively small scanning rates. In general, the cantilever speed should not be faster than 30 μm/s to allow for proper response of the PZT tube. This speed is related to the response of the feedback loop and the size of the AFM image in pixels. The larger the pixel count, the longer the time to complete an AFM image at a given scanning rate. If the lateral dimensions of surface features to be resolved are on the order of the tip radius, i. e. R = 10 nm, according to Nyquist criterion and for sampling frequency of f = 2 kHz and 1000 Hz scanner bandwidth, the cantilever speed can be at most 2 f R = 40 μm/s [17.62] (no data sampling in retrace). Obviously, small image sizes allow for faster imaging, although the total pixel count remains the same. If the sample surface is flat, fast imaging rates can be achieved by scanning in constant height mode that is described in the next section.

17.3 Imaging Modes by an Atomic Force Microscope The two most common imaging methods by an AFM are the contact and the non-contact modes. The different amplitudes of forces involved in each method make them appropriate for various sample and surface conditions as well as classes of materials.

17.3.1 Contact AFM Imaging by contact AFM is conducted in DC mode, i. e. the force exerted on the AFM cantilever is measured at every point on the surface. The forces in contact AFM are short-range repulsive forces on the order of 10−6 –10−8 N. Despite their small values, they are suf-

ficient to generate tens of nanometers of cantilever deflection. The magnitude of tip forces and the proximity to the sample surface are such that large stresses are applied to the tip causing wear. Similarly, the stresses applied to the sample during imaging may induce deformation that alters the surface profile. Thus, contact AFM is appropriate for imaging hard materials with hardness similar to that of the tip so that tip wear is minimized. An AFM image is generated in contact AFM by two different methods: a) Constant cantilever deflection (constant applied force): The cantilever rasters the surface sampling

423

Part B 17.3

must be conducted frequently. Software linearization may improve the scanner accuracy to 10% [17.5]. When precise metrology is important, the PZT actuator must be outfitted with a closed-loop hardware linearization as the smallest change in imaging parameters, e.g. an offset in scanning coordinates, may result in distorted image features [17.54]. Hardware linearization provides an independent measurement of the PZT scanner position at every image point and corrects for most nonlinearities. A displacement sensor is integrated onto the PZT to monitor its x, y, z motion. The true scanner position is compared to that calculated by the bias voltage, and a correction voltage is applied to accurately position the cantilever. Common methods to implement a closed loop feedback are optical, capacitive, and strain gauges [17.53]. Optical correction is realized by depositing a line pattern or a reflective material on the scanner tube and by reading this pattern interferometrically or by a laser beam reflected onto a photodetector [17.55, 56]. Capacitive linearization is conducted by measuring capacitance changes between an electrode attached to the scanner and a second electrode fixed on a stationary point [17.57–60]. This method has better signal-to-noise ratio than optical linearization. Finally, strain gauge hardware correction, a common approach in PZT precision actuation systems, is based on resistance measurements. Hardware compensation methods can improve the scanner accuracy to about 99%. A study of the repeatability of a linearized scanner has shown less than ±0.4 pixel random uncertainty [17.61] that is attributed to random sources of error.

17.3 Imaging Modes by an Atomic Force Microscope

424

Part B

Contact Methods

predetermined locations whose position is calculated from the physical dimensions of the image and the scanning parameters (number of pixels in each scanning direction.) The surface roughness results in changes of the tip-sample force and, thus, cantilever deflection. The latter is maintained constant and the PZT scanner is displaced to respond to surface height changes, Fig. 17.14b. The motion of the PZT actuator is used to construct the surface topography. This method of image construction is appropriate for rough surfaces that could cause rapid tip wear. In most commercial AFM instruments this is the default mode of operation. b) Constant cantilever height: The PZT actuator is held at fixed height while it rasters the sample surface. The cantilever deflection varies at every surface point as it responds to the surface topography. Successive changes in cantilever deflection are used to construct the surface topography Fig. 17.14a. This method is a) Height of PZT PZT Cantilever

Cantilever deflection

Part B 17.3

Sample

b) Height of PZT

PZT Cantilever Cantilever deflection

Sample

Fig. 17.14a,b Operating modes in contact AFM. (a) Constant cantilever height mode, (b) constant force mode [17.5]

preferred for flat surfaces with a few nanometers of roughness that is much smaller than the range of allowable cantilever deflections. In contact AFM the tip remains in contact with the surface at all times, so it could actually be considered as being immersed in a water monolayer which covers hydrophilic surfaces in atmospheric conditions. This monolayer generates an adhesive force on the cantilever probe, which is not considerably disrupting in contact AFM imaging. If the wetting properties of the surface are uniform, then their adhesive force acts as an offset to the interaction between the probe and the surface. However, spatially variable surface hydrophilicity results in imaging artifacts. The spatial and force resolution of contact AFM imaging is limited by the contact forces at the interface [17.63] and the tip dimensions. The first requirement can be overcome in ultra-high vacuum (UHV) [17.64, 65] or in water [17.26], which minimize the attractive van de Waals and other long-range forces and increase the contribution of short-range forces to the total force applied on the cantilever tip. Longrange interactions can be reduced by minimization of the Hamaker constant [17.66]. Elimination of capillary forces (10−8 –10−7 N [17.67]) can be accomplished in a liquid environment to allow for small repulsive forces (10−7 N) contact AFM allows for atomic resolution of periodic surfaces, such as mica [17.68] or graphite [17.26], but not true atomic resolution [17.63, 64]. The resolution is better than the tip radius because it is the result of frictional forces between the tip and the surface steps [17.69], which is possible only for atomically flat surfaces. However, at such high frictional forces the surface is damaged during imaging [17.70]. Evidently, the van der Waals contribution can be further minimized by reducing the tip radius as indicated by (17.3).

Atomic Force Microscopy in Solid Mechanics

17.3 Imaging Modes by an Atomic Force Microscope

A typical tip radius is approximately 10 nm. A focused ion beam or isotropic oxidation with further etching are among the methods to sharpen AFM silicon probes. Other approaches call for growth of a carbon tip at the end of a silicon tip producing probes with 1–2 nm radii. Single wall carbon nanotubes have been attached [17.71] or grown [17.72] at the end of conventional cantilever tips. Although their small radii are desirable, their slenderness promotes buckling or adhesion at sidewalls of surface steps and trenches. Due to the magnitude of the forces exerted in contact AFM, the aforementioned high-resolution probes are more appropriate for non-contact AFM imaging where the tip forces are 2–3 orders of magnitude smaller.

as

17.3.2 Non-Contact and Intermittent Contact AFM

where m ˆ is the effective mass of the cantilever and m t is the mass of the tip. The mass of the tip is directly included in this calculation as the tip is assumed to be at the free end of the cantilever. However, the effective mass of the latter is different from its true mass, m c , as it is equivalent to a spring with uniformly distributed mass. Starting with the resonance frequency of a simple harmonic system

kc (17.26) ω0 = m ˆ the resonance frequency of a cantilever with uniform cross-section excited at its fixed support is [17.73]

EI 2 , (17.27) ω0 = (1.875) m∗ L 4

Fdriver = FPZT cos(ωt) = kc Aexcitation cos(ωt) , (17.23)

where ω is the frequency of the forced oscillation. For small cantilever deflections linear spring behavior is assumed: F = −kc z .

(17.24)

The energy stored in the cantilever is periodically converted from kinetic to elastic strain energy (m ˆ + mt)

d2 z = −kc z , dt 2

(17.25)

where m ∗ is the effective mass of the cantilever per unit length. Taking into account its stiffness given by (17.4), a comparison between (17.26) and (17.27) gives m ˆ = 0.24m ∗ L = 0.24m c .

(17.28)

Thus, the effective mass of the cantilever-tip harmonic oscillator is m = 0.24m c + m t and (17.27) can be written as  E t ω0 = 1.014 2 . ρ L

(17.29)

(17.30)

The energy dissipation due to damping must be considered dz (17.31) , Fdamping = −η dt

Part B 17.3

Non-contact AFM imaging takes advantage of attractive van der Waals forces. The surface topography is a force gradient mapping in the attractive force regime of the plot in Fig. 17.2. An alternative method of operation, the intermittent contact mode, employs the weak repulsive force exerted on the cantilever tip when it is brought in the proximity of the sample (1–2 nm). In non-contact AFM the force applied on the cantilever ranges between 1–100 pN. For most common AFM cantilevers with spring constants 10−2 –102 N/m these forces would result in maximum cantilever free end deflections of the order of 0.1–10 nm. Such deflections are subject to very low signal-to-noise ratios, or are indistinguishable from Brownian motion. Instead of recording the cantilever free end deflection continuously with time (DC mode), non-contact AFM is realized by an AC method of measurement. The cantilever resonance frequency is sensitive to changes in the gradient of the tip-surface force and, thus, the RMS amplitude of the modulated cantilever is monitored. In this mode, the cantilever is excited to a free oscillation amplitude of 50–200 nm and then it is brought to a distance of a few nanometers from the sample surface to perform imaging. The dynamics of the freely oscillating cantilever-tip system in free space can be modeled as a driven, damped harmonic oscillator, i. e. a springmass system with spring constant equal to that of the cantilever and mass equal to the effective mass of the cantilever-tip system. Consider a cantilever subjected to a forced oscillation far from the sample surface where it does not experience any surface forces. If a periodically varying force Fdriver is applied via a small PZT actuator

425

426

Part B

Contact Methods

where η is the viscosity of the medium. The force balance results in [17.74] m

d2 z = Fspring + Fdamping + Fdriver dt 2 dz = −kc z − η + FPZT cos(ωt) dt

(17.32)

or m

d2 z dz + kc z + η = FPZT cos(ωt) . 2 dt dt

(17.33)

The solution of (17.33) provides the damped resonance frequency ωd as   1 η 2 2 (17.34) ωd = ω0 − 2 m where Q is the quality factor that describes the coupling between input and output energy is given by Q=

≈

mω0 Q

Part B 17.3

⇒ Q 2 + 12   1 ω0 2 ⇒ ωd = ω20 − 2 Q   1 1 2 ωd = ω0 1 − . 2 Q

(17.36)

Finally, (17.33) is written as m z(t) ¨ +

0

1+Q 2 [(ω/ω0 )−(ω0 /ω)]2

  −1 × sin ωt − tan

ω0 ω  Q . 2 ω0 − ω2

(17.38)

The maximum amplitude at resonance frequency is proportional to Q, as derived from the first term of (17.38), and, thus, high Q would result in higher sensitivity in force measurements (but slower instrument response). At resonance the cantilever amplitude is  1 η 2 FPZT Q 2 2 ⇒ Ares =  ω = ω0 −  . 2 m 1 2 kc 1 − 4 Q

(17.35)

Q depends mostly on the damping factor of the imaging medium. Q = 50 000 in vacuum, Q = 50–300 in air, and Q ≈ 1 in water. For operation in air (ωd 1) or in vacuum (ωd ≈ ω0 ), the combination of (17.34) and (17.35) gives mω0

FPZT m z(t) = A(t) =  2  2 2 ω0 − ω2 + ωQ0 ω    ω0 /ω FPZT A= ηω √

(17.39)

ωd m . η

η= 

The steady state solution of (17.37) is

mω0 z(t) ˙ + mω20 z(t) = FPZT cos(ωt) . Q (17.37)

The solution of (17.37) has a transient term with decay time 2Q/ω0 . High Q results in long transient times and, thus, slow response of the AFM system, while a low Q gives faster response times when the cantilever amplitude is the feedback parameter. In this case, imaging in air allows for faster scanning rates compared to vacuum.

Parameter Selection in NC-AFM Imaging NC-AFM imaging is conducted by modulating the cantilever in free space near ω0 . At ω ω0 the acceleration forces are small compared to the elastic deformation forces and the cantilever motion follows that of the driver. At ω ω0 the term kc z in (17.33) is small compared to d2 z/ dt 2 and the cantilever response is controlled by inertia producing small oscillation amplitudes and phase shift of 180◦ since the acceleration of the harmonic oscillator is 180◦ out-of-phase with respect to its displacement. The first step in NC-AFM imaging is a frequency sweep of the cantilever to determine its natural frequency. Qualitatively, the amplitude-frequency behavior is shown in Fig. 17.15. According to (17.38), the RMS value of A(ω) and the width of this curve depend on Q. Increasing Q (i. e. smaller viscosity, operation in relative vacuum) results in higher oscillation amplitude and sharper resonance peak. From (17.38) it is derived that larger Qs result in reduced phase shift from the driver frequency. Conversely, a viscous medium results in larger phase shift. The quality factor can be estimated from Fig. 17.15 as √ ω0 (17.40) , Q= 3 Δω where Δω is the width of the resonance curve at A(ω) = A0 /2. Alternatively, the Q factor can be calculated by the cantilever ring-down method [17.75].

Atomic Force Microscopy in Solid Mechanics

In this approach, the driver signal is interrupted and the cantilever free oscillation amplitude is measured as a function of time. The characteristic decay time 2Q/ω0 is that of the transient solution of (17.37). The cantilever RMS amplitude is mostly detectable near ω0 . Changes in RMS amplitude while the probe is modulated at fixed frequency, ωsp , can be used as the feedback variable to maintain the tip at constant distance from the sample surface, an operation that is referred to as amplitude modulation (AM-AFM) [17.76]. The modulation frequency is constant during imaging and NC-AFM images are actually contours of constant force gradient. An alternative approach is frequency modulation (FM-AFM) detection where the cantilever is modulated at constant amplitude by varying the drive amplitude in response to changes in damping and tip-sample forces [17.77]. In this mode of operation, the instrument maintains constant frequency shift ϕ = π/2 between the driver and the cantilever. The driver frequency, ω, becomes a function of the mechanical characteristics of the cantilever and its environment, ω0 and Q, as well as ϕ = π/2. It is the shift in the natural frequency due to changes in the local force gradient that is used as feedback to generate the FM-AFM images. Reference [17.78] is suggested for a thorough description of FM-AFM. Among the two methods, AM-AFM is most commonly used in media with medium or small Q including liquid environments, while FM-AFM is mostly appropriate in ultra high vacuum [17.79]. The insignificant air damping in UHV A (ω) A(ω0)

1

0.5 Δω

0

ω0

ω (Hz)

Fig. 17.15 RMS amplitude versus frequency response of an

AFM cantilever

427

results in very high Q (≈ 104 ) and, as a consequence, very long feedback response times in amplitude modulation. FM-AFM requires two feedback loops and when it is applied for imaging in air or liquids the electronics become slow. In AM-AFM, changes in RMS amplitude are monitored at a set point modulation frequency, ωsp , where the slope of A(ω) is maximum so that small frequency shifts result in large changes in amplitude. For this reason, this method is also termed as slope detection. It is common practice to select a set point modulation frequency, ωsp , at which A0 = 0.8Ares . The maximum slope of the A(ω) curve is approximately located at [17.76]   1 ∂F 1 (17.41) . 1± √ ωsp = ω0 1 − kc ∂z 8Q The application of surface forces on the cantilever alter the A(ω) curve for free oscillation, not due to changes in absolute force amplitude but changes in their gradient. When the cantilever is brought near the surface, its effective spring constant is ∂F (17.42) . ∂z As a consequence, the resonance frequency of the cantilever near the surface, ωs , is  keff 1 ∂F 2 ωs = (17.43) ⇒ ωs = ω0 1 − . m kc ∂z keff = kc −

Considering the Lennard–Jones potential, and that the zero z-axis coordinate is at the cantilever rest position, one concludes that (a) repulsive forces ( dF/ dz < 0) stabilize the cantilever and increase its resonance frequency, (b) attractive forces ( dF/ dz > 0) destabilize the cantilever and decrease the resonance frequency [17.8, 79]. Thus, a uniform force in the zdirection ( dF/ dz = 0) has no effect on the frequency of a modulated AFM cantilever. Upon engagement of the AFM tip to a surface a shift in the resonance curve occurs. If the cantilever is modulated at ωsp , when brought near the sample surface it will follow a new resonance curve with smaller oscillation amplitude, as seen in Fig. 17.16a. According to (17.42) and (17.43), the set point frequency dictates the type forces that must be employed in AFM imaging. If ωsp < ω0 then further approach to the sample surface must result in shift of the A(ω) curve to higher frequencies, which is possible when the applied force is repulsive. This shift is equal to Δω = ωs − ω0 where ωs can be computed from (17.43). If ωsp > ω0 then further

Part B 17.3

RMS

17.3 Imaging Modes by an Atomic Force Microscope

428

Part B

Contact Methods

approach to the sample surface would require a shift of the A(ω) curve to smaller frequencies and, thus, the tip-sample force interactions must be attractive. In either case, the change in A(ω) must be monotonic as the tip approaches or is retracted from the sample surface so that the cantilever does not become resonant. Selection of ωsp > ω0 is preferable for soft samples but it is prone to loss of feedback, while ωsp < ω0 provides stable imaging. Alternatively, the change in the cantilever phase lag, calculated as the difference between the signal of the photodetector and the oscillation driver, may be used to monitor the tip-sample spacing, Fig. 17.16b. The same considerations as those in Fig. 17.16a apply in Fig. 17.16b. The second NC-AFM imaging parameter is the peak force applied on the cantilever tip near the sample surface. A strong sample surface-cantilever tip interaction would result in surface deformation, tip wear, and poor resolution of fine surface features. A small force would result in low resolution and the risk of feedback loss during imaging. Since both in the attractive or the rea) Cantilever oscillation amplitude Attractive

Repulsive

pulsive force regimes, the force varies monotonically with distance and so does dF/ dz, the steady-state cantilever amplitude is given by (17.38). Instead of directly defining the tip-sample force that is not a measurable quantity in NC-AFM, the cantilever RMS, set point amplitude, Asp , is defined by the AFM operator. Alternatively, the ratio Rsp = Asp /A0 with the RMS free oscillation amplitude A(ωsp ) = A0 is used. Rsp = 0.5 is a starting value to engage the cantilever probe onto the sample surface with subsequent corrections to obtain an improved image. As shown in Fig. 17.17a–c corrections in Rsp may be conducted while imaging. The absolute value of Asp is also important. The restoring elastic force is proportional to Asp and large cantilever free oscillation amplitudes are used to overcome strong capillary adhesion forces. The drawback of large Asp is the large force inflicted to the sample and the associated rapid cantilever tip and sample wear. Upon engagement of the AFM cantilever onto a surface, a feedback loop monitors the ratio R = A/A0 that changes from R = 1 far from the sample surface to Rsp at the sample surface. During engagement onto the sample, the van der Waals forces are added to the excitation force applied on the cantilever [17.47, 80] mω0 z(t) ˙ + mω20 z(t) Q = FPZT cos(ωt) + F(z c , z) ,

m z(t) ¨ + ΔA

ΔA

where z c is the cantilever equilibrium position. The van der Waals interaction can be described by

Part B 17.3

F(z c , z) = − ωsp

ω0

ωsp

Driver frequency

b) Cantilever phase lag (φ) Attractive

Repulsive

Δϕ 0 Δϕ ωsp

ω0

ωsp

Driver frequency

Fig. 17.16 (a) Amplitude and (b) phase shift frequency sweep curves upon engagement of the cantilever probe to a sample surface in different force regimes

(17.44)

HR , 6(z c + z)2

z c + z ≥ a0 ,

(17.45)

where a0 is the intermolecular spacing, often taken equal to 0.165 nm, z is the tip-sample distance, and H = π 2 Cρtip ρsurface is the Hammaker constant that depends on tip and sample materials, the surrounding medium, and the molecular densities of the tip and the surface, ρtip , ρsurface , respectively [17.29,80]. Control of the tip-sample surrounding medium reduces the attractive van der Waals interaction and it may even result in repulsive van der Waals forces depending on the dielectric constants since H ≈ (ε1 − ε3 )(ε2 − ε3 ) [17.81]. When attractive van der Waals forces are harnessed to construct an AFM image the interaction force in (17.45) is sufficient. If the overall interaction force is repulsive, the forces applied to the cantilever are a combination of van der Waals and repulsive forces. This mode of AFM imaging is intermittent contact mode, or tapping mode. The interaction term F(z c , z) in (17.44),

Atomic Force Microscopy in Solid Mechanics

a)

17.3 Imaging Modes by an Atomic Force Microscope

b)

0

ΔRsp /R0 = +15 %

1 μm

429

c)

0

1 μm

0

ΔRsp /R0 = – 75 %

Rsp = R0

1 μm

Fig. 17.17 (a–c) Selection of Rsp for NC-AFM imaging of a polymer nanofiber. Increase or decrease in Rsp compared to an optimum R0 suppresses the fine surface features. In (c) Rsp = R0 below the dividing line

is then given by √

3 HR 4E R (a0 − z − z c ) 2 , + 6a02 3(1 − ν2 ) z c + z ≤ a0 . (17.46)

F(z c , z) = −

where E 0 is the binding energy and r0 is the equilibrium distance. When the cantilever reaches z = z sp at which R = Rsp its oscillation amplitude is FPZT /m Asp =  , (ω2s − ω2 )2 + (ωs ω/Q s )2 where

(17.48)

 1 dF (z sp ) kc dz ωs (z sp ) Qs = Q0 . ω0

ωs (z sp ) = ω0 1 −

and

A(ω, z) ωs (z)/ω FPZT  . = 2 ηω0 1 + Q(z) [(ω/ωs (z)) − (ωs (z)/ω)]2 (17.51)

Asp − A(ω, z) is calculated at every grid point in Fig. 17.4, and the piezoelectric actuator contracts or extends to minimize this difference. The relative vertical adjustment of the PZT actuator between adjacent points is the local change in surface height. Flat surfaces may be imaged by simply recording the relative changes in A(ω, z). A calibration of the cantilever spring constant is required to compute surface heights. Several parameters affect the measurement sensitivity including Brownian motion that is demonstrated as thermal noise. The signal-to-noise ratio in the measurement of A(ω, z) is [17.76]  ∂F A0 ∂F ω0 Q S ∂z ∂z =  = , (17.52) 1 ∂F ∂F N 27k 1 − c (kB T )B Δ k ∂z ∂z

(17.49)

As the AFM scanner rasters the sample surface, it encounters variations in surface height that change the

c

where B is the bandwidth of the thermal noise. NC-AFM cantilevers have high resonance frequency (100–300 kHz) and high stiffness (5–40 N/m) compared to those designed for contact AFM. A combination of (17.26) and (17.52) shows that high stiffness

Part B 17.3

The first term in (17.46) can also be replaced by 2πγ R that represents the adhesive force according to the Derjaguin–Muller–Toporov (DMT) model [17.82]. If the interaction is limited to the front atom of the AFM probe and an atom on the sample surface, the continuum description by (17.46) may be replaced by the Lennard– Jones force [17.83]  13  7  12E 0 r0 r0 − (17.47) , F(z c , z) = r0 z z

tip-surface distance, and thus ωs (z), so that  1 dF (z) and ωs (z) = ω0 1 − kc dz ωs (z) Q(z) = Q 0 (17.50) ω0 with the cantilever oscillation amplitude becoming

430

Part B

Contact Methods

affects the S/N ratio negatively. This is often outweighed by other advantages, such as the strong elastic restoring force required to overcome adhesive forces at hydrophilic surfaces. Additionally, the precision and accuracy of NCAFM depends on the responsiveness of the PZT actuator to changes of surface morphology. The AFM feedback system uses proportional and integral gains to respond to the photodetector signals and maintain constant cantilever deflection [17.24]. The choice of proportional-integral-derivative (PID) controller parameters for optimum imaging is a function of surface roughness, imaging speed [17.84], and the mechanical properties of the sample.

17.3.3 Phase Imaging In addition to surface topography, NC-AFM provides a phase image, which is constructed from changes in the phase lag between the cantilever oscillation and the driver signal. Figure 17.16b shows the effect of excitation frequency on the phase shift. At resonance the phase shift is π/2, excitation frequencies well below ω0 result in zero phase shift, while excitation frequencies well above resonance produce a phase shift equal to π. The phase shift according to (17.38) is ϕ = tan

−1



ω0 ω/Q ω20 − ω2

.

(17.53)

Part B 17.3

As the AFM cantilever approaches the sample, surface repulsive forces, ωsp < ω0 , increase its resonance frequency and the curve of phase lag as a function of excitation frequency shifts to higher frequencies and the phase lag at ωsp changes. When the cantilever operates under attractive forces, ωsp > ω0 , a shift of the phase lag curve to lower frequencies occurs and the phase lag increases, Fig. 17.16b. In both the repulsive and the attractive force operation regimes, the amplitude of cantilever oscillation decreases near the surface and this is used as the feedback parameter to restore the tip-surface interaction. Phase images produced by AM-AFM show the cantilever phase lag, which, however, is not used as feedback parameter. The contrast in a phase image is a result of surface geometry, adhesion between the tip and the surface, variation in the viscoelastic properties of the surface, and, in some cases, the surface elasticity. In AM-AFM changes in ϕ should be attributed to energy dissipated by tip-sample forces, E dis , per oscillation period

as [17.85, 86] sin ϕ =

Q E dis ω Asp (ω) + . ω0 A0 (ω0 ) πkc A0 (ω0 )Asp (ω)

(17.54)

Equation (17.54) is comprised of two terms: the elastic (Asp /A0 ) and the dissipative (Q and E dis ). When the cantilever amplitude is used as the feedback parameter, the forced oscillation amplitude is fixed at Asp and, as a result, the first (elastic) term remains constant throughout imaging. It is the second term that provides the contrast in a phase image [17.87]. Higher damping improves the image contrast as more energy is transferred from the tip to the sample. However, significant energy exchange results in deformation of the sample surface and suppressed surface features. Equation (17.54) has been verified for a variety of samples with different stiffness [17.86]. It should be noted that changes in tan ϕ can be caused by topographic variations whose effect must be deconvoluted from that of the dissipative terms. A relevant discussion is presented in [17.88]. The excitation frequency, amplitude, and the damping ratio, Rsp = Asp /A0 , determine the features resolved in a phase image. The effect of Rsp on topographic and phase images can be significant. When Rsp approaches 1 surface details become unclear because the interaction force between the tip and the surface is very small. On the other hand, Rsp 0.5 provides a strong interaction between the surface and the tip which results in noisy images and suppressed image details, especially for soft materials. A small Rsp limits the imaging resolution because of the suppressed cantilever amplitude at ωsp which allows only small relative changes of Asp that are difficult to resolve by a photodetector. Rsp = 0.5 represents a good starting value that can be modified to produce an improved image. Figure 17.17a–c demonstrates the effect of Rsp on resolved image details. R0 is an optimum value for the damping ratio that is used to engage the cantilever on the sample surface, Fig. 17.17b. If Rsp increases to ΔRsp /R0 = +15%, the tip interacts weakly with the surface and fine surface details disappear, Fig. 17.17a. An reduction in Rsp , Fig. 17.17c, has two effects: (a) the material surface is subjected to a larger force that suppresses fine surface features, (b) the segment of the amplitude curve in Fig. 17.16a that is used to compute the feedback parameters has very small gradient and therefore, small sensitivity to variations in surface height. As a result, a significant reduction in Rsp at ΔRsp /R0 = −75% has the same effect as ΔRsp /R0 = +15%, Fig. 17.17a–c. A small increase

Atomic Force Microscopy in Solid Mechanics

431

Fig. 17.18 Atomic steps in annealed sapphire resolved in intermittent contact mode (Image courtesy of Mr. Scott Maclaren, CMM-UIUC)

(μm) 5

(nm)

4

17.3 Imaging Modes by an Atomic Force Microscope

2 1

3

0 2 –1 1

0

–2

0

1

2

3

4

5

(μm) Height (nm) 2 1 0 –1 –2 0

1

2

3

4

17.3.4 Atomic Resolution by an AFM At room temperature, the resolution of NC-AFM is mostly limited by thermal noise that accounts for cantilever deflections equal or larger than 0.2 Å. At UHV and at temperatures ≈ 4 K NC-AFM resolution reaches that of an STM. Vacuum conditions guarantee that the surface remains free of oxides throughout imaging while low temperatures minimize the cantilever random motion, which in turn enhances the signal-to-noise ratio and improves on the minimum force gradient that can be sensed by the instrument [17.77]. The effect of noise due to Brownian motion is amplified by small cantilever spring constants, becoming apparent when flat surfaces are imaged. In general, the resolution of an AFM in the out-of-plane direction is sufficient to measure atomic layer steps as long as they are significantly wide. Fig-

6

ure 17.18 shows atomic steps in annealed sapphire with step size of 3 Å. While contact AFM cannot provide true spatial atomic resolution, the latter is possible by NC-AFM because the measurement of the gradient of force is more sensitive than its absolute value, even at the pN level. Such small forces require sampling in very small material volumes and, thus, at small tip-sample separations. Imaging with atomic resolution in UHV is conducted by FM-AFM [17.77,78,90,91]. True atomic resolution can be verified by detecting individual point defects on the surface of freshly cleaved specimens [17.92, 93], contrary to perfectly periodic atomic structures that were initially thought of as evidence of atomic resolution imaging by an AFM. In order to improve the spatial resolution, the cantilever is modulated at reduced amplitude to decrease the force on the sample while the tip approaches the surface at distances comparable to those required for STM [17.94]. Stiff NC-AFM cantilevers support this requirement as they suppress the snap-in instability to very small tip-sample separations. The forces that are required to achieve true atomic resolution are not fully understood. It has been suggested that atomic resolution in NC-AFM originates in the attractive force between the sample and the front tip atom [17.91]. Other studies have shown that repulsive forces are the reason for atomic resolution [17.95]

Part B 17.3

in Rsp has a more pronounced effect than a reduction because it gradually leads to loss of cantilever feedback. Further examples of the application of phase imaging in polymers are found in [17.89]. Finally, phase images are generated in other modulated cantilever modes, such as magnetic force microscopy (MFM), force modulation microscopy (FMM), and scanning near field optical microscopy (NSOM).

5

432

Part B

Contact Methods

and as a result tapping mode is appropriate [17.80, 96], Fig. 17.2. Obtaining atomic resolution requires that tip-surface reactivity be minimized because at UHV conditions and contaminant free surfaces the tip and

the sample may temporarily form covalent bonds at distances equivalent to interatomic separation [17.97]. For instance, atomic imaging of pure Si surfaces requires silicon nitride cantilever tips.

17.4 Quantitative Measurements in Solid Mechanics with an AFM As inherent PZT actuator nonlinearities and thermal drift, the main obstacles in quantitative AFM imaging, are being resolved by hardware linearization and imaging at low temperatures, new opportunities in experimental mechanics are generated. Scanner linearization allows for repeatable images at the subpixel level [17.61] and in situ full-field strain measurements extracted from AFM images [17.98, 99]. Understanding and controlling force interactions during imaging, as described in the previous section on atomic resolution, permitted the measurement of pN level forces and comprehensive molecular force spectroscopy [17.23, 24]. In this section, the quantitative measurement of force interactions between a surface and an AFM tip, and the measurement of in-plane fields of displacement and strain with nanometer spatial resolution are described.

in the inset of Fig. 17.19a. The gradient of the applied attractive force with respect to the tip-sample spacing increases at small tip-sample separations to overcome the cantilever spring constant resulting in a snap-in instability in the F–d curve in Fig. 17.19a. The amplitude of this jump is smaller for stiff cantilevers, which however, reduces the force resolution. a)

Force PZT

PZT displacement Sample

17.4.1 Force-Displacement Curves

PZT PZT

Part B 17.4

A distinct feature of an AFM compared to other high-resolution electron microscopy methods is the measurement of the force applied by the probe to the surface. Typically, the AFM software provides a curve of the force applied on the cantilever as a function of the vertical position of the piezoelectric actuator. Actually, the data recorded by the instrument are the RMS photodetector voltage versus the voltage applied on the PZT actuator. The force applied on the cantilever is calculated from a calibration of the cantilever deflection as a function of photodetector signal and subsequent application of (17.4) or (17.7). The PZT motion is calculated following the instrument calibration with a calibration grating. The resulting plots are cantilever force – PZT displacement (F–d) curves. In the absence of capillary forces, the interaction between the cantilever tip and a surface includes attractive van der Waals and repulsive electronic forces that follow a combined potential such that in Fig. 17.2. When the cantilever probe is held far from the surface, the interaction force is zero. At separation distances of a few nanometers, the force on the cantilever becomes attractive and results in detectable deflection as seen

Attractive forces

b)

Repulsive forces

Force

PZT

PZT displacement Sample

PZT

Water layer

PZT

Capillary forces

Repulsive forces

Fig. 17.19a,b F–d curves recorded from (a) a hydrophobic surface, and (b) from a hydrophilic surface with enhanced capillary interaction [17.5]

Atomic Force Microscopy in Solid Mechanics

This snap-in instability may be used to obtain an estimate for the spring constant of soft cantilevers given the value of the Hamaker constant H. The spring constant of a cantilever with a conical tip of half cone angle θ and radius R [17.30] approaching a flat surface is

 ∂F

∂ HR tan2 θ

= ⇒ − kc =

∂z z=d ∂z 6z z=d

tive interaction between the tip and the water meniscus. Given a tip geometry, this adhesive force can be estimated by existing adhesion models [17.82, 100, 101]. Additional interactions between the surface and the tip because of surface contaminants, electrostatic, or capillary forces that act concurrently with van der Waals further complicate Fig. 17.19b [17.23, 24]. The measurement of F–d curves in Fig. 17.19 employs a DC scheme. Similarly an AC method could be applied to record dF dz –d curves. F–d curves may be used to obtain an estimate of the elastic/plastic mechanical properties of a surface. The surface deformation is calculated from the PZT actuator displacements and the cantilever deflection. The scanner motion, z, with reference to an arbitrary position away from the surface is given by [17.36] z = d + (dc + ds ) ,

(17.56)

where d is the original tip-sample distance, ds is the sample deformation, and dc is the deflection of the cantilever free end, (Fig. 17.20). a) Cantilever deflection (μm) 0.35

Descend Retract

0.25

Air

0.15 0.05 – 0.05 – 0.15

0

0.3

0.35

0.4

b) Cantilever deflection (μm)

0.45 0.5 PZT motion (μm)

0.5

PZT

Cantilever after contact with surface

Descend Retract

0.4

Water

0.3 δc PZT D

0.2 Z

0.1 0

δs Sample

Fig. 17.20 Experimental parameters in the determina-

tion of surface mechanical properties using F − d curves [17.36]

433

– 0.1

0

0.2

0.4

0.6

0.8 1 1.2 PZT motion (μm)

Fig. 17.21a,b F–d curves in (a) air and in (b) water. The snap-in and snap-off instabilities are absent in water

Part B 17.4

HR tan2 θ . (17.55) 6d 2 As the tip further approaches the sample, attractive forces are gradually balanced by repulsive forces that stem from electron repulsion and Pauli’s exclusion principle [17.29]. At sub-nanometer distances the repulsive force becomes dominant. In the absence of surface penetration, additional approach of the PZT actuator to the surface results in strong repulsive forces and cantilever bending. For small cantilever deflections these forces are a linear function of the PZT actuator displacements. A force-displacement trace is acquired when the actuator is retracted. The approach and retraction curves overlap including the snap-in instability, which in the case of retraction is called snap-off instability and has larger amplitude compared to the snap-in instability. F–d curves similar to that in Fig. 17.19a are typically possible only in UHV conditions. In ambient conditions, capillary adhesion due to the water monolayer that covers hydrophilic surfaces dominates the F–d curves that resemble that in Fig. 17.19b. Specifically, the snap-in instability is equivalent to that recorded on a dry surface but it is due to the attraction between the water layer and the cantilever tip. The retraction component of the curve, however, demonstrates significantly larger hysteresis due to the strong attrackc =

17.4 Quantitative Measurements in Solid Mechanics with an AFM

434

Part B

Contact Methods

The cantilever deflection measured by the photodetector and the free space motion, d, are subtracted from z to compute ds . The retracting component of a plot dc – ds , or F–ds can be used together with models, such as that by Sneddon [17.102], to compute the reduced modulus of elasticity for the surface material. This approach is subject to several limitations and rarely produces consistent and repeatable results. In the majority of AFM instruments the cantilever is mounted at an angle with respect to sample surface and the contact area is not well defined. Special cantilevers with inclined tips may be employed to correct this problem. Furthermore, this method is accurate for small cantilever deflections where the force-deflection relationship is linear and the photodetector calibration is accurate. The snap-off instability can be avoided in UHV or by conducting experiments in water. Figure 17.21a,b shows two F–d curves recorded on a glass surface in air and in water. The snap-in and snap-off instabilities are clear in Fig. 17.21a where the latter is pronounced due to the hydrophilic nature of glass. On the contrary, none of the two instabilities are discernible in Fig. 17.21b because of the medium surrounding the surface and the cantilever.

17.4.2 Full Field Strain Measurements by an AFM Images obtained with a hardware linearized AFM have been used for quantitative strain measurements on thin films fabricated for microelectromechanical systems (MEMS) [17.99]. With proper digital image analysis

tools, in-plane deformations and strains can be extracted with 1–2 nanometer spatial resolution from AFM images of specimens subjected to external loading. In this method, a sample is loaded in situ and AFM images of the sample surface are acquired before and during loading. When the sample deforms, the distance between surface features changes as a function of applied loading and sample properties. If the surface roughness features are orders of magnitude smaller than the specimen thickness, then they act as non-intrusive strain markers. A difficulty in measuring strains directly from AFM images arises from their digital nature: Images of a sample subjected to a stress increment do not precisely include the same material points. Figure 17.22 shows a cross-section of an AFM image before and after the sample was loaded along this cross-section. After deformation, a material point in the original profile does not always move to an AFM grid point in the deformed configuration that is determined according to the rastering procedure in Fig. 17.4. Instead, only very few material points that move to imaging grid locations will be identical in the two line profiles. Since local deformations are smaller than the pixel-to-pixel distance, a method with subpixel resolution is required to calculate such deformations from a set of digital AFM images. Calculation of displacements and strains can be conducted by Digital Image Correlation (see related chapter in this volume.) This method, originally developed for optical imaging [17.103], employs a pair of digital images to extract local in-plane displacements and their gradients (strains) simultaneously via a least square

Part B 17.4

(μm) 0.1

After deformation Before deformation

0.08 0.06 0.04 0.02 0

0

1

2

3

4 (μm)

Fig. 17.22 Line profile before and after deformation is applied. Original image pixels move to non-grid points, which

necessitates sub-pixel resolution in extracting local displacements [17.99]. The reference in the displacement of the surface profile after deformation is at the left of the image

Atomic Force Microscopy in Solid Mechanics

images. The ambient temperature should be maintained within a tenth of a degree centigrade during imaging, while accelerations must be smaller than 50 μg. Due to the latter considerations, displacements and strains in the fast imaging direction, where each line of data is collected within one second or faster, are more reliable than those computed in the slow imaging direction, where data are sampled in the course of minutes. To increase the fidelity of AFM images used in DIC, two image sets with x and y as the fast scanning directions should be collected. Thus, the axial u and the transverse v displacements are accurately calculated with x and y as the fast imaging direction, respectively. The advantages of the AFM/DIC strain measurement method are (a) the spatial displacement and strain resolution is better than other experimental methods that employ optical data, (b) nanometer-scale resolution is possible for soft and organic materials, otherwise not suitable for high-resolution electron microscopy imaging, (c) is a non-intrusive method because it benefits from the natural surface roughness to monitor surface deformations; sub-nanometer RMS surface roughness is sufficient [17.104], (d) it provides direct spatial correlation between the material microstructure and local displacement and strain, and (e) it is a multiscale experimental method that allows for full-field measurements in areas ranging from 1 × 1 μm2 to 100 × 100 μm2 with 1024 × 1024 pixel count. An example of the last advantage is shown in Fig. 17.23 where the representative volume element of polycrystalline silicon for MEMS was determined [17.107]. The freestanding test films were comprised of columnar grains with 600 nm average grain size. The application of uniform far field stress resulted into uniform strain at the scale of 15 μm but the local displacement field at the scale of individual grains (1000 × 500 nm2 see inset in Fig. 17.23) was highly heterogeneous. Thus, a 15 × 15 μm2 polysilicon sample, on average containing 25 × 25 columnar grains, behaves isotropically and it is a representative volume element. By reducing the field of view, it was actually determined that a 10 × 10 μm2 specimen domain, on average containing 15 × 15 columnar grains, was the minimum representative volume element for this polysilicon. On the other hand, displacement fields in 4 × 4 μm2 or 2 × 2 μm2 areas were highly inhomogeneous and the effective behavior of these domains could deviate significantly from that described by isotropy. It was established that the homogeneous material elastic constants (Young’s modulus and Poisson’s ratio) of polysilicon are applicable to MEMS components com-

435

Part B 17.4

optimization. The accuracy in resolving local displacements is as good as 1/8-th of a pixel and it is achieved by, non-linear, bi-cubic spline interpolation between pixels. This implies that strains as small as 0.01% can be resolved with current AFM instrumentation that provides 1024 × 1024 pixel images [17.99, 104]. The calculation of displacements by DIC is more accurate than their gradients, and thus, the former are usually reported. The application of DIC for nanometer-scale mechanical deformation measurements from AFM images has been reported in several studies for the mechanical behavior and properties of MEMS [17.99,104–107]. In all studies, the required speckle pattern is provided by the natural surface roughness that functions as distributed surface markers. In polycrystalline materials this pattern is readily contributed by grain boundaries, Fig. 17.22, and the intragrain surface roughness that is of smaller amplitude. The roughness/grain boundary frequencies dictate the spatial resolution of the AFM/DIC method while the amplitude variation of surface markers affects the signal-to-noise ratio. A description of the mathematical formulation and analysis of the limitations of the DIC method as a function of surface features and noise contained in AFM images is provided in [17.108]. The application of AFM/DIC requires hardware linearization of the PZT scanner so that surface deformations are not convoluted with scanner nonlinearities. The height data are used to create the surface pattern but not to determine the out-of-plane displacements; thus, z-axis linearization is not imperative and should be avoided for 1 × 1 μm2 or smaller images in order to improve the signal-to-noise ratio. Although, scanner nonlinearities are mostly removed by hardware, it is important that basic procedures are followed when AFM images are to be used for quantitative strain measurements. Contrary to optical images, AFM images may not be averaged out to improve the signal-to-noise ratio because of inevitable rigid body motions in the AFM field of view with respect to the absolute specimen position. This rigid body motion is easy to recognize if it amounts to several pixels but it is not detectable if it is at the subpixel level. Furthermore, operation of the PZT actuator with different voltage offsets subjects an image to hysteresis and cross-coupling, which may not be fully corrected by hardware linearization. Thus, a zero offset voltage must be applied to the piezoelectric actuator during sample imaging. Temperature fluctuations, ambient noise, and instrument vibrations limit the accuracy of nanoscale displacements computed from AFM

17.4 Quantitative Measurements in Solid Mechanics with an AFM

436

Part B

Contact Methods

prised of 15 × 15 or more grains (10 × 10 μm2 ) but the same elastic constants do not provide an accurate description of the mechanical response of smaller MEMS components [17.107]. An application of the AFM/DIC method in inverse problems in mechanics is demonstrated in Fig. 17.24 [17.105]. A polycrystalline silicon thin film with a central circular hole was subjected to uniaxial tension and AFM images were recorded in the area next to the 6 μm hole indicated by the shaded square in the leftmost schematic. DIC was applied to portion of the AFM image in the dashed square, and the contour plot shows the u-displacement field as computed by DIC. The displacement field data were then incorporated in a least square scheme to determine the homogeneous material elastic constants, Young’s modulus and Poisson’s ratio, of polycrystalline silicon (average grain size 300 nm) by solving the inverse hole problem. The calculated property values were compared to those measured in uniform tension experiments of microscale tension specimens with the application of the AFM/DIC

method [17.107, 109]. In the latter experiments, the u and v displacements were computed from 5 × 15 μm2 AFM images that were collected with x and y as the fast scanning directions, respectively [17.105]. The agreement in Young’s modulus and Poisson’s ratio between the two methods was better than 5%, which validates the AFM/DIC strain measurement method. For very hard materials, such as amorphous diamond-like carbon, this mechanical property measurement method is the only one available that allows obtaining all elastic constants from direct strain records [17.104, 110]. AFM images are suitable to investigate local phenomena near a crack tip, such as the fracture behavior of polycrystalline silicon films fabricated for MEMS [17.106, 111]. In a relevant study, mathematically sharp edge pre-cracks were generated by indentation of the substrate next to polysilicon specimens that were bonded onto the substrate. The pre-cracked polysilicon specimens were subsequently removed from their substrate and tested as 2 μm thick freestanding fracture specimens under mode I and mixed mode loadU displacement (nm) 18.3 17.1 15.9 14.6 13.4 12.2 11 9.8 8.5 7.3 6.1 4.9 3.7 2.4 1.2

200 nm

Part B 17.4

U displacement (nm)

(nm) 3000

F

2000 1000 0

0

2000

4000

6000

8000

10 000

12 000

(nm)

159 148 137 127 116 106 95 85 74 63 53 42 32 21 11

Fig. 17.23 Axial displacement contour of a 3 × 15 μm2 segment of a 2 μm thick freestanding polycrystalline silicon specimen subjected to uniaxial tension. The inset shows the local displacement field superposed on the grain structure in a 1000 × 500 nm2 area. There is direct correlation between the local displacements and the grain structure. The left-to-right order of gray levels in the contour plots is the same as the bottom-to-top order in the contour legends [17.107]

Atomic Force Microscopy in Solid Mechanics

a)

b)

0

x (nm) 12 000

Area of image correlation

Load direction

F

17.4 Quantitative Measurements in Solid Mechanics with an AFM

0

x (pixels) 100 200 300 400 500 600 700 800 900

10 000

5

c) U (displacement) y (pixels) 800

21.3

1.4

700

18.9

1.3

16.6

1.1

14.2

0.9

11.8

0.8

400

9.5

0.6

300

7.1

0.5

4.7

0.3

2.4

0.2

600 8000 500

F

F 6000

10

4000

200 2000

100

Fixed side 15 0

5

10

15 (μm)

0

0

2000

4000

6000

8000

10 000 12 000 x (nm)

437

0 (nm)

0

0 (pixels)

Fig. 17.24 (a) Schematic of a 2 μm thick polysilicon film with a central perforation subjected to uniaxial tension. The film contains a central circular perforation with 6 μm diameter. (b) AFM image of the area next to the perforation including a portion of the

central circular hole. DIC analysis was conducted in the 14 × 12 μm2 area next to the hole, which is marked with a dashed box. (c) Transverse displacement field in the marked area. The local displacements are plotted relatively to the displacement of the lower right corner of the contour plot. The left-to-right order of gray levels in the contour plot is the same as the top-to-bottom order in the contour legend [17.105]

a)

⎯

b)

KI, PolySi = 0.81 MPa√ m

⎯

⎯

c)

KI, PolySi = 0.83 MPa√ m

KI, PolySi = 0.89 MPa√ m

Crack tip

F

Crack tip

0.5 μm

0.5 μm

U displacement (nm) –23.7

d)

–16.9

–10.2

–3.4

3.4

10.2

e)

16.9

23.7

f)

F

2 μm

2 μm

2 μm

Fig. 17.25a–c AFM micrographs of sub-critical crack growth in a freestanding polysilicon film and the corresponding displacement fields. (a) The crack tip is below the field of view at the specimen centerline, (b) crack tip location after the first step of crack growth, and (c) crack tip location after second crack √ growth with the crack arrested at a grain boundary. After (c) the crack grew catastrophically at K I,PolySi = 1.06 MPa m. The displacement contours (d–f) correspond to AFM images (a–c) respectively but span wider fields of view (6 × 4 μm2 ). The left-to-right order of gray levels in the

contour plots is the same as the left-to-right order in the contour legend [17.106, 111]

Part B 17.4

0.5 μm

438

Part B

Contact Methods

ing by a custom-made apparatus that was positioned under an AFM [17.106,111]. It was shown that the fracture process in polysilicon was mainly controlled by the grain where the crack tip resided. This local, grain level, control of crack initiation allows for incremental, subcritical, crack growth in polysilicon, (Fig. 17.25). This mechanism was directly evidenced via spatially resolved crack length measurements by an AFM. Figure 17.25a–c shows the details of a crack traversing a large grain. The crack initially resided below the field of view of Fig. 17.25a. An increment in the √ effective stress intensity factor K I,PolySi = 0.81 MPa m, at which Fig. 17.25a was recorded, resulted in crack growth and arrest inside the large grain √ at the center of Fig. 17.25b, at K I,PolySi = 0.83 MPa m. The presence of the large grain in the crack path (with not as unfavorable orientation of its weak cleavage plane as that where the crack originally resided) was instrumental in crack arrest. An increase in the effective √ stress intensity factor to K I,PolySi = 0.89 MPa m re-

sulted in further crack growth and arrest at a grain boundary as seen in Fig. 17.25c. These values of K I,PolySi were close to those for Si {111} cleavage planes. After the two increments of subcritical crack extension, catastrophic √ crack propagation occurred at K I,PolySi = 1.06 MPa m, which was close to the average K Ic,PolySi determined from a large number of specimens. Further support to this crack growth mechanism is provided by the local displacement fields recorded via the in situ AFM/DIC method in the vicinity of the cracks. Figure 17.25d–f shows the near crack tip deformation fields as derived from the AFM images in Fig. 17.25a–c. The crack tip is located at the root of the u-displacement jump in the centerline of the contour plots and it is in agreement with the location of the crack tip in images 17.25a–c. This experiment illustrates the importance of thin film microstructure (grain size and geometry) that influences the details of the fracture process.

17.5 Closing Remarks

Part B 17.5

This chapter focused on the operating principles of AFM and an evaluation of AFM data in view of scanner nonlinearities and tip artifacts. AFMs are not, however, limited to surface imaging. In addition to contact and NC-AFM, a number of other imaging modes [17.48] have been developed as derivatives of the two basic methods. These methods provide high resolution in spatial mapping of charge, thermal properties, temperature, electric potential, magnetic flux, frictional and adhesive properties, etc. In many cases these methods are only qualitative or their results are convoluted with spatial surface features. In parallel, existing AFM imaging techniques have been expanded to non-traditional applications such as high resolution imaging in liquid environments. To date, AFM is the only available high-resolution method for soft biological materials, organisms [17.24], and polymeric materials [17.112]. Figures 17.26, 27 show two examples of human tumor cells [17.113, 114] and bacteria imaging to obtain insight into their physical morphology and functions. Modeling of the dynamics of AFM cantilevers have allowed for optimization of the strongly coupled imaging parameters [17.78, 90]. Major advances have been accomplished in the use of SPMs in quantitative force [17.23, 30] and deformation measurements [17.99, 104], while other aspects of in-

strumentation development are still under way, such as nanoindentation [17.25] and single electron spin detection [17.115]. In the aforementioned applications, the PZT rastering speed remains the limiting factor. Answers to this problem may come from recent advances in multiple cantilever processing. The positioning accuracy of the AFM piezoelectric actuator facilitates nanolithography methods such as scanning probe lithography [17.116]. The latter may be used to pattern surfaces by parallel writing by using thermomechanical cantilevers that range in numbers from 1024 [17.22] to 55 000 [17.117] for large area patterning and reading. The ultimate objective of these technologies is data storage and nanolithography with sub 100 nm resolution where large arrays of cantilevers will perform in parallel without feedback control on individual cantilever deflection. The concept of cantilever-based data storage, often referred to as “Millipede”, might also provide solutions for fast large area/high resolution AFM imaging with application to flaw detection and analysis of inhomogeneous surfaces. However, positioning accuracy remains subject to hardware linearization as described in Sect. 17.2. The millipede approach has not provided yet a final solution to the limitation of SPMs, which is their slow imaging speed, mostly

Atomic Force Microscopy in Solid Mechanics

17.6 Bibliography

439

200

Invasive tumor extension

75 –50 –175

Target cell

–300

0

1.5

3

4.5 (μm)

Fig. 17.27 Intermittent contact mode image of E.coli show-

ing the extended flagellum (Image courtesy of Mr. Scott Maclaren, CMM-UIUC)

Tumor cell Fig. 17.26 NC-AFM phase image of fixed T98 brain tu-

mor cells. Part of one cell is seen in the lower left corner with a cellular extension (invadopodia) projecting toward a cell in the upper right corner. The field of view is 80 × 80 μm [17.113, 114]

affected by the bandwidth of the piezoelectric actuator. An alternative approach that provides a tenfold enhancement in imaging speed involves positioning

feedback of a cantilever by depositing a piezoelectric film on its back surface [17.32, 33]. The PZT layer replaces the fine positioning control of the probe by the main PZT actuator and it performs most efficiently with relatively flat surfaces where otherwise constant height AFM imaging is applied. It is expected that advances in the aforementioned open problems on SPM instrumentation will significantly impact the ways mechanical strain measurements are conducted in the future.

• • • • • • •

P.F. Barbara, D.M. Adams, D.B. O’Connor: Characterization of organic thin film materials with near-field scanning optical Microscopy (NSOM), Ann. Rev. Mater. Sci. 29, 433–469 (1999) D. Bonnell: Scanning Probe Microscopy and Spectroscopy, 2nd edn. (Wiley, New York 2001) B. Cappella, G. Dietler: Force-distance curves by atomic force microscopy, Surf. Sci. Rep. 34, 1–104 (1999) B. Bhushan (Ed.): Springer Handbook of Nanotechnology Part C, Scanning Probe Microscopy 2nd edn. (Springer, Berlin, Heidelberg, 2006) pp. 591–784 R. Garcia, R. Perez: Dynamic AFM methods, Surf. Sci. Rep. 47(6–8), 199–304 (2002) U. Hartmann: Magnetic force microscopy, Ann. Rev. Mater. Sci. 29, 53–87 (1999) J. Israelachvili: Intermolecular and Surface Forces, 2nd edn. (Academic, New York 1992)

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A. Majumdar: Scanning thermal microscopy, Ann. Rev. Mat. Sci. 29, 505–585 (1999) S.N. Magonov, D.H. Reneker: Characterization of polymer surfaces with atomic force microscopy, Ann. Rev. Mat. Sci. 27, 175–222 (1997) E. Meyer: Atomic force microscopy, Prog. Surf. Sci. 41(1), 3–49 (1992) E. Meyer, H.J. Hug, R. Bennewitz: Scanning Probe Microscopy: The Lab on a Tip, 1st edn. (Springer, Berlin, Heidelberg 2003) S. Morita, R. Wiesendanger, E. Meyer: Non-Contact Atomic Force Microscopy (Springer, Berlin, Heidelberg 2002) V.J. Morris, A.P. Gunning, A.R. Kirby: Atomic Force Microscopy for Biologists (Imperial College Press, London 2004) B.D. Ratner, V.V. Tsukruk: Scanning probe microscopy of polymers, American Chemical Society

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•

ACS symposium series Ser., Vol. 694 (Oxford University Press, Washington 1998) T. Thjundat, A. Majumdar: Microcatilevers for physical, chemical, and biological sensing, In: Sensors and Sensing in Biology and Engineering, ed. by F. Barth, J.A.C. Humphrey (Springer, Berlin, Heidelberg 2003)

•

P. Vettiger, M. Despont, U. Drechsler, U. Dürig, W. Häberle, M.I. Lutwyche, H.E. Rothuizen, R. Stutz, R. Widmer, G.K. Binnig: The “Millipede” – more than thousand tips for future AFM storage, Direct. Inform. Technol. 44(3), 323–340 (2000)

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17.15

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17.27

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17.29 17.30

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Part C

Nonconta Part C Noncontact Methods

18 Basics of Optics Gary Cloud, East Lansing, USA

24 Holography Ryszard J. Pryputniewicz, Worcester, USA

19 Digital Image Processing for Optical Metrology Wolfgang Osten, Stuttgart, Germany

25 Photoelasticity Krishnamurthi Ramesh, Chennai, India

20 Digital Image Correlation for Shape and Deformation Measurements Michael A. Sutton, Columbia, USA

26 Thermoelastic Stress Analysis Richard J. Greene, Sheffield, UK Eann A. Patterson, East Lansing, USA Robert E. Rowlands, Madison, USA

21 Geometric Moiré Bongtae Han, College Park, USA Daniel Post, Blacksburg, USA

27 Photoacoustic Characterization of Materials Sridhar Krishnaswamy, Evanston, USA

22 Moiré Interferometry Daniel Post, Blacksburg, USA Bongtae Han, College Park, USA

28 X-Ray Stress Analysis Jonathan D. Almer, Argonne, USA Robert A. Winholtz, Columbia, USA

23 Speckle Methods Yimin Gan, Kassel, Germany Wolfgang Steinchen (deceased)

447

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Many powerful methods of experimental mechanics are based on optical phenomena. Approaches in this class include, for example, photoelasticity, speckle interferometry, laser Doppler, and moiré interferometry. Typically, these techniques are noncontact, relatively noninvasive, and they yield data over the entire field of interest. Informed application of these techniques requires understanding of basic optics. The purpose of this chapter is to provide the necessary background knowledge in wave optics. The exposition begins with the nature and description of light and its interactions with materials, then progresses to the first cornerstone of optical methods, interference, and its manifestation in classical interferometric applications. The second cornerstone, diffraction at an aperture, is then treated along with its application in optical spatial filtering. Geometrical optics is not discussed. Where possible, reliance is on physical reasoning in place of mathematical manipulation. The main goals are utility, clarity, brevity, and accuracy.

18.1 Nature and Description of Light ............. 18.1.1 What Is Light? ............................ 18.1.2 How Is Light Described? .............. 18.1.3 The Quantum Model ................... 18.1.4 Electromagnetic Wave Theory ...... 18.1.5 Maxwell’s Equations................... 18.1.6 The Wave Equation ..................... 18.1.7 The Harmonic Plane Wave ...........

448 448 448 448 448 449 449 449

18.2 Interference of Light Waves ................... 18.2.1 The Problem and the Solution...... 18.2.2 Collinear Interference of Two Waves............................. 18.2.3 Assumptions ..............................

449 450 450 451

18.3 Path Length and the Generic Interferometer ............. 451 18.3.1 Index of Refraction..................... 451 18.3.2 Optical Path Length .................... 452

18.3.3 18.3.4 18.3.5 18.3.6 18.3.7

Path Length Difference ............... A Generic Interferometer ............. A Few Important Points............... Whole-Field Observation............. Assumptions ..............................

452 452 453 453 453

18.4 Oblique Interference and Fringe Patterns 453 18.4.1 Oblique Interference of Two Beams ............................ 453 18.4.2 Fringes, Fringe Orders, and Fringe Patterns .................... 454 18.5 Classical Interferometry......................... 18.5.1 Lloyd’s Mirror ............................ 18.5.2 Newton’s Fringes ....................... 18.5.3 Young’s Fringes.......................... 18.5.4 Michelson Interferometry ............

455 455 456 458 459

18.6 Colored Interferometry Fringes .............. 461 18.6.1 A Thought Experiment ................ 462 18.7 Optical Doppler Interferometry .............. 18.7.1 The Doppler Effect ...................... 18.7.2 Theory of the Doppler Frequency Shift..... 18.7.3 Measurement of Doppler Frequency Shift .......... 18.7.4 The Moving Fringe Approach........ 18.7.5 Bias Frequency .......................... 18.7.6 Doppler Shift for Reflected Light... 18.7.7 Application Examples .................

464 464 465 466 467 467 467 468

18.8 The Diffraction Problem and Examples ... 18.8.1 Examples of Diffraction of Light Waves ........................... 18.8.2 The Diffraction Problem .............. 18.8.3 History of the Solution ................

468 468 470 470

18.9 Complex Amplitude............................... 18.9.1 Wave Number ............................ 18.9.2 Scalar Complex Amplitude ........... 18.9.3 Intensity or Irradiance ................

470 471 471 472

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Basics of Opt 18. Basics of Optics

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18.10 Fraunhofer Solution of the Diffraction Problem..................... 472 18.10.1 Solution of the Problem .............. 472 18.10.2 Summary .................................. 474 18.11 Diffraction at a Clear Aperture ............... 18.11.1 Problem and Solution ................. 18.11.2 Demonstrations ......................... 18.11.3 Numerical Examples and Observations .......................

474 474 475 476

18.12 Fourier Optical Processing ..................... 18.12.1 The Transform Lens..................... 18.12.2 Optical Fourier Processing or Spatial Filtering...................... 18.12.3 Illustrative Thought Experiment ... 18.12.4 Some Applications ......................

476 476 477 478 478

18.13 Further Reading ................................... 479 References .................................................. 479

18.1 Nature and Description of Light 18.1.1 What Is Light? As we begin to study experimental methods that utilize light and optical components, this question is an obvious place to start. The fundamental nature of light seems undefinable. We can say that light is a form of energy that is marked by two characteristics.

• •

It moves. When it ceases to move, it is no longer light. It carries a wealth of information.

Beyond that, we are reduced to telling what light is by observing how it behaves and how it interacts with matter (e.g., tanning your skin, exposing a photo film) and with itself (e.g., interference). Further, we define some rather artificial but useful boundaries to distinguish light from similar but different forms of energy (electricity, heat). These distinctions are made primarily on the basis of interactions and behavior rather than on the basis of a fundamental definition. Fortunately, we do not need to understand exactly what light energy is in order to use it and to describe and predict its creation, propagation, and interactions with materials.

18.1.2 How Is Light Described? The fact that we might not totally understand the fundamental nature of light leads to a problem in describing its creation and behavior. We have two quite different experience-based systems, namely, quantum mechanics and electromagnetic wave theory. The application (photography, interferometry) dictates which system to use. The systems are not as differentiated as they seem; they actually are parts of a unified comprehen-

sive theory called quantum electrodynamics. Yet, this bifurcation annoys tidy thinkers.

18.1.3 The Quantum Model The quantum model tells us that light consists of bundles of energy called photons. The generation and behavior of photons can be predicted through statistical mechanics. The photons have characteristics of both waves and particles. This approach is needed to explain various phenomena such as photoelectricity, lasers, and photography. We must rely on it in the detection and creation of light, both important in optical methods of measurement. However, we do not need to be experts in quantum mechanics in order to use a mercury lamp, a laser, or a television camera.

18.1.4 Electromagnetic Wave Theory This model considers light to be energy in the form of electromagnetic waves. Proof of the wave nature of light was provided through the elegant experiment by Thomas Young in 1802, of which more will be said presently. This theory is not totally adequate; it does not explain, for example, photoemission and photoresistivity. Most of the phenomena that are used by experimental mechanicians, including refraction, interference, and diffraction. can be predicted and explained by wave theory. The behavior of the waves and their interactions with matter are conveniently described by Maxwell’s equations. In order to understand the relationships between observables (e.g., light intensity) and mechanics response (e.g., displacement), we need to develop some facility with the equations describing waves.

Basics of Optics

James Clerk Maxwell (1831–1879) collected the work of several other scholars, notably Faraday, added some inspired ideas of his own, and developed the systematic set of equations that are named after him. The wave train of radiation is described in terms of two vectors that are perpendicular to each other and perpendicular to the axis of propagation of the wave. These wave vectors are E, the electric vector, and H, the magnetic vector. If E and H are known as a function of time and position in the electromagnetic field, then the wave and its interactions with materials are completely described. The relevant field quantities that may be seen as responses of materials are the electric displacement, the current density, and the magnetic induction. The material quantities in the field include the specific conductivity, the dielectric constant (related to index of refraction), the magnetic permeability, and the electric charge density. Maxwell’s equations relate the wave vectors, the field quantities, and the material properties. Fortunately, for most experimental mechanics applications, we need not be concerned with general solutions. A special case is sufficient.

18.1.6 The Wave Equation In a nonconducting medium that is free of charge, Maxwell’s equations reduce to the wave equation for the vector E as a function of time and position. The wave equation in one of several possible forms is ∇2 E = K μ

∂2 E , ∂t 2

(18.1)

where ∂2 ∂2 ∂2 ∇ = 2 iˆ + 2 ˆj + 2 kˆ ∂x ∂y ∂z 2

and K is the dielectric coefficient of the medium, μ is the magnetic permeability, K μ = 1/v2 , and v is the speed of propagation of the wave.

Equation (18.1) shows how, for the restrictions accepted, the electric vector, the material properties, the wave speed, and spatial coordinates are related. For most of our work, we need not consider general solutions to the wave equation. The simplest solution suffices.

18.1.7 The Harmonic Plane Wave A simple solution to the wave equation, the harmonic plane wave traveling along the z-axis, allows us to do most of the calculations required in basic experimental mechanics. It is   2π (18.2) (z − vt) , E = A cos λ where A is the vector giving the amplitude and plane of the wave, λ is the wavelength, and v is the wave velocity. Visualization of the wave is accomplished in two ways. If a snapshot could be taken at some fixed time, the wave would look like a cosine plot along the z-axis. Alternatively, one might sample the amplitude of the wave at some fixed z over a period of time. The amplitude plotted as a function of time would also be a cosine graph. These experiments are nearly impossible to execute because the wavelengths of light are very small – on the order of a half a micrometer, and the wave frequencies are large – in the vicinity of 1014 Hz. Recall that, in the sensory domain, wavelength and frequency appear as color. So, the basic wave function contains in one simple expression all the important data about the wave, including its strength, polarization, wavelength, direction, and speed. In order to perform calculations for more complicated processes such as holography, (18.2) is generalized by

• • •

allowing it to travel along an arbitrary axis instead of just the z-direction, changing it to an exponential form, realizing that we need use only the complex amplitude.

These subjects are treated in Sect. 18.9 of this chapter.

18.2 Interference of Light Waves Interference of two light waves is one of the two cornerstones of optical methods of measurement, the other being diffraction. If this phenomenon is thoroughly

understood, and if its implications are grasped, then almost all of the methods of optical measurement, from photoelasticity through moiré interferometry and on to

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18.1.5 Maxwell’s Equations

18.2 Interference of Light Waves

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Part C 18.2

speckle interferometry, will be seen to be merely applications and variations of the interference idea.

18.2.1 The Problem and the Solution The objective is to use light energy as a measuring stick. The problem is how to measure the absolute phase of a wave of light, or how to measure the phase of one wave relative to another. Our eyes and all other light detectors are not quick enough to see oscillations at optical frequencies, so they are insensitive to phase. We can detect only a long-term (relatively speaking) average intensity. The solution to this problem is somehow to convert phase difference, which cannot be sensed directly, to an intensity change that can be seen or detected. Mixing two waves together in the process called interference accomplishes this task. The question then becomes, What is observed when two identical waves of light energy are mixed together? For now, attention is confined to the case where the two waves are traveling along the same axis. Broadbeam interference and oblique crossings of waves are discussed subsequently. For much of this study of basic optics, we rely on physical reasoning and observation, but some simple math is required in order to understand the interference concept thoroughly. We are interested in both the fundamental result and the process by which the result is obtained and interpreted, because the process is ubiquitous in interferometry calculations.

18.2.2 Collinear Interference of Two Waves Begin with the electric vector for a harmonic plane wave traveling along the z-axis 

 2π E1 = A cos (z − vt) . λ E

λ

(18.3)

This wave is mixed with another wave that is identical except that it lags behind the first wave by some distance r in spatial units. This lag between the waves will be found to be relative retardation, path length difference, or phase difference; and it is the quantity that is to be measured. The  electric vectorfor the second wave is 2π (18.4) (z − vt − r) . E2 = A cos λ Figure 18.1 shows these two waves and the relevant variables. Now, we simply add these two wave functions together, then use the trig identity for the sum of two cosine functions to obtain the electric vector of the resulting wave    πr  2π  r . (18.5) cos z − vt − Es = 2A cos λ λ 2 This result is already in a form that is common in interferometry measurements, and these equations are always looked at in the same way, namely, Electromagnetic vector = [Amplitude] × [Wave function] . The second cosine expression in (18.5) is seen to be a wave function that is identical to the first wave except for the lag term; it is just another optical wave, as plotted in Fig. 18.2. So, the entire first portion must be the amplitude of the wave and it contains both the amplitude of the original wave and a cosine function whose value depends on the lag term r and the wavelength. This result is important and thought provoking. It is our first indication that the lag term r might be measured by examining the amplitude of the wave that is created by mixing two other waves. Recall that the amplitude of a light wave cannot be measured directly. The irradiance or intensity may, however, be observed by using our eyes, a photographic film, a photocell, or a charge-coupled device (CCD) camera. The irradiance is given by the following relationship for this simple case Irradiance = [Amplitude]2 ,

υ

A Es Amplitude Z Z r

Fig. 18.1 Interference of two waves having differing

phases

Fig. 18.2 The wave obtained by the interference of two waves having a phase difference

Basics of Optics

Is = 4A cos 2

2

 πr  λ

Is

.

(18.6)

The irradiance varies between zero and some Imax = 4A2 that depends on the strengths of the original waves. Figure 18.3 is a plot of the irradiance as a function of the lag term r. If the two original waves were mixed with r = 0, λ, 2λ, . . ., then the irradiance is maximum. Further, if r = λ/2, 3λ/2, 5λ/2, . . ., then the irradiance is zero. More important, we now see that information about the phase lag between the two original waves can be obtained by merely measuring the intensity of the combined waves. The invisible has been made visible by converting phase difference to intensity. This process makes interferometric measurement possible. A significant problem remains. For a given measured value of the intensity, the phase lag r is not single-valued. The possible values differ by multiples of the wavelength. Additional steps, such as fringe counting, are needed to establish which cycle of the irradiance plot is the correct one.

Imax

0

λ/2

λ

3λ/2

r

Fig. 18.3 Irradiance as a function of phase lag between interfering waves

18.2.3 Assumptions In this analysis, the interfering light waves were assumed to be of equal amplitude and polarization. If the amplitudes differ, then the minimum irradiance produced by interference is not zero. The waves also are required to be able to interfere, which, for all practical purposes, means that they come from the same source and that the source has appropriate properties. Also, the terms irradiance and intensity mean the same thing in this treatment.

18.3 Path Length and the Generic Interferometer We have learned how a phase difference between two waves can be converted to an intensity variation through the phenomenon of interference, thereby making measurement of the phase difference possible. Now, we focus attention on what might cause the phase lag and, in general terms, the essential components of an interferometric arrangement for measuring it. To get at the phase difference, some means of describing the length of the path that a wave travels is required. The optical path length depends on both the physical length of the path and the wave speed, which is most easily quantified by the refractive index.

18.3.1 Index of Refraction The wave equation (18.1) showed that the speed of travel of a wave depends on material properties. For practical purposes, we normalize the speed of light in a material to the speed of light in a vacuum (c ≈ 3 × 105 km/s) by defining an absolute index of re-

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which yields

18.3 Path Length and the Generic Interferometer

fraction as: Absolute index of refraction Speed of light in vacuum . = n1 = Speed of light in material 1 Sometimes, it is convenient to define a relative index of refraction between two materials as: Relative index of refraction Speed of light in material 2 = n 12 = . Speed of light in material 1 This relative index might be used to describe the speed of light for material 1 when it is immersed in material 2. In this case, two subscripts are used, with the first referring to the object and the second to the immersion medium. Be careful to get these relationships right-sideup; it is natural to invert them. Many writers do not carefully distinguish between relative and absolute indexes, particularly when describing experiments where

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the immersion medium is air. To reduce confusion, this article uses only the absolute index. Note also that, as far as we know, the speed of light is maximum in a vacuum; so the index of refraction for materials is greater than 1.

18.3.2 Optical Path Length The length of the path traveled by a light wave, for measurement purposes, depends on the physical length of the path and the speed at which the wave travels over the path. Conceptually, it is the time it takes for the light to cover the path. However, since we usually normalize the speed relative to vacuum, the path length has units of distance. Optical path length =[Physical distance traveled]×[Index of refraction] .

18.3.3 Path Length Difference In experimental mechanics measurement, we are interested in the change in a path length or else the difference between two path lengths; this is called the path length difference (PLD). The PLD is what causes the phase lag discussed in Sect. 18.2. PLD can be determined directly by using the definition given above, but let us instead conduct a thought experiment. Imagine a race between two waves, as depicted in Fig. 18.4. The entire event is immersed in a medium that has a refractive index n 0 . The first wave leaves the starting line, and arrives at point A at time t1 . The second wave leaves the start line at the same instant the first wave left, but this wave is required to pass through a slab of refractive material having index of refraction n 1 and thickness

d. At time t1 , this second wave will have attained only point B, because it was slowed down for a spell while traversing the slab. The distance by which the second wave lags behind the first is the PLD for this problem. It is also called the absolute retardation R1 . The calculation of the PLD is straightforward since it involves only velocity–time–distance considerations. It is the difference between d and the distance that the first wave travels in the time required for the second wave to traverse the refractive slab. The result is   n1 − n0 d. (18.7) PLD = n0 A nice aspect of this approach is that it automatically corrects for the immersion medium, a factor that might not be intuitively obvious by the more direct calculation. Clearly, a change in either of the refractive indexes or a change in the distance d will create a PLD. This notion brings some unity to the various optical methods of experimental mechanics. Finally, it is easy to extend this result to the case where each of the waves traverses a separate slab of refractive material.

18.3.4 A Generic Interferometer All but one of the methods of optical measurement in experimental mechanics are based on interferometry, which is the measurement PLD by interference. The exception is geometric moiré. A unifying conceptual model of the process of interferometry is depicted in Fig. 18.5. This model shows all the necessary elements that are common to interferometry measurements, meaning that the devices shown appear in one form or another in all interferometry setups. Since the waves in the two paths must be able to interfere, they must come from a common source. A divider or beam splitter separates the two waves, and the waves travel along different paths, one

Immersion medium: n0

υ0

Path 1 A PLD υ1

υ0

Combiner

Source

Detector

n0

B

n1 d

Refractive slab: n1

Fig. 18.4 Model for calculating path length difference

(PLD) caused by a refractive slab

Splitter

Path 2

Specimen

Fig. 18.5 The generic interferometer

Basics of Optics

18.3.5 A Few Important Points A particular aspect of interferometric measurement deserves great emphasis. The discussion so far has centered on the idea that we can measure the difference between two path lengths, and this, indeed, is one way that interferometry is used. However, there is another way that is even more useful. Suppose that we hold one of the paths constant and call it a reference path. Then, we induce a change in the second path, taking intensity measurements before and after the change. Subtraction of the post-change intensity from the pre-change intensity gives us the path length change in the second path. Such a procedure is very common in experimental mechanics measurements. In some instances, changes in both paths must be accounted for in the course of an experiment. Bear in mind that interferometry is a differential form of measurement, and that is one reason for its versatility and usefulness. The principles outlined here apply to other forms of waves. Interferometry can be performed with microwaves and at radio frequencies, for example.

18.3.6 Whole-Field Observation An oft-cited advantage of optical measurement techniques is that they yield a whole-field picture of the displacement or strain field. So far, we have considered only the case of two waves interfering, which implies that we obtain the PLD for only one small area of a specimen. The measurement is point-by-point. How can this idea be generalized to the entire field? The answer is simple: utilize a great many pairs of waves to interrogate all points in a specimen simultaneously using a multitude of interferometers. This is parallel processing to an advanced degree. At the conceptual level, we add to the generic interferometer a beam expanding device near the source to create a beam of, perhaps, parallel waves; or else we can use two beam expanders downstream from the beam splitter. The combiner joins the broad beams to create an intensity map of the field; this map shows the local intensities that are created by the local PLDs. The detector is a device, such as a camera, that can record and display this intensity map.

18.3.7 Assumptions The index of refraction is actually a complex number. We assumed that the materials involved are nonconductors, so we can ignore the imaginary part of the complex refractive index. This is not a serious restriction for most applications, but it has some implications when taking observations by reflection from metallic surfaces, as when interpreting Newton’s rings or when performing optical measurements on semiconductors.

18.4 Oblique Interference and Fringe Patterns So far, we have considered only collinear interference of two waves as we grappled with the basic ideas of interferometry. We broaden our horizons, so to speak, by examining what happens when two waves, or collections of waves, cross and interact with one another. Then, some terminology issues are settled.

18.4.1 Oblique Interference of Two Beams This phenomenon is extraordinarily useful, and it must be understood. Many forms of interferometry, including moiré interferometry for instance, are based on interference of two beams that meet at small crossing angles. The process gives us a way of creating the very fine grat-

ings required in moiré experiments. In the subsequent readout stage, the moiré pattern is created by two-beam oblique interference of diffracted beams. A hologram is actually a grating that is produced by two-beam interference. In fact, the process described below creates a simple hologram. The analysis given here also parallels almost exactly that used to predict the formation of fringes in geometric moiré. Define, for now, a beam as a collection of waves that are related to one another in coherence (same source and polarization), phase, and direction of travel. A planeparallel beam is one that has all its waves traveling along parallel axes and synchronized in phase to form a plane wavefront.

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or both of which might contain various optical elements or objects that are specimens. At the ends of the paths, the waves will be out of phase because of the differing PLDs. Some sort of beam combiner is needed to bring the waves together so that they interfere. The interference, as we have learned, creates an intensity that is a function of the PLD. A device to measure intensity completes the setup.

18.4 Oblique Interference and Fringe Patterns

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Part C 18.4

Imagine that two such plane-parallel beams that are able to interfere cross each other as suggested in Fig. 18.6. To facilitate analysis, assume that the crossing angle is symmetrical with respect to a horizontal plane. Further, consider the problem as two dimensional, meaning that we are looking at a cross section in the plane of the paper. The mathematics of this situation will be explored thoroughly when holography is discussed. Here, purely physical reasoning explains what happens, and, much insight is gained from the exercise. Apply to the case at hand what was learned about collinear interference. Wherever a maximum positive electric vector of one beam meets a maximum positive of the other wave, they will combine constructively. Two maximum negatives will also reinforce one another. Wherever a maximum positive of one wave meets a maximum negative of the other, the waves will interfere destructively. In the figure, maxima of waves in the two beams are represented by solid lines, and minima are represented as dashed lines. Thus, where maxima of the sine waves cross, a bright spot will be created, and the same is true where minima cross. These bright spots are represented as gray dots. A dark spot is created wherever a minimum and a maximum are mixed so as to interfere destructively, and these spots are shown as black dots. What will we see if we map this field of crossing waves with an intensity detector or by placing a screen such as a ground glass in the field? To answer this question, evaluate the horizontal spacing dh between spots and the vertical spacing dv λ/2

Screen

Dark spots

A Ψ

D

B C A Ψ

λ/2 λ/2

D

B

Ψ/2 C dh

Fig. 18.6 Oblique interference of two beams

Bright spots dv

between the rows of bright (gray) spots. Refer to the enlarged sketch of the diamond ABCD and remember that the distance from a maximum to a minimum is one-half the wavelength. Simple trigonometry shows that λ , dh = 2 cos Ψ2 λ . (18.8) dv = 2 sin Ψ2 To extract physical meaning from this result, consider the case where the crossing angle Ψ is small, say of the order of 2◦ . The horizontal distance between bright patches is about half the wavelength, implying that it is too small to be resolved. The vertical spacing, on the other hand, is approximately 30 times the wavelength (about 0.018 mm), and this spacing can be resolved by photographic films and high-resolution instruments. With even smaller crossing angles, the vertical spacing can be discerned by eye. This finding means that, at least for small crossing angles, the bright patches will appear to blend into horizontal flat layers (like slats) of alternating bright and dark. A screen or photographic film placed so as to record the cross section of this volume of slats will yield a pattern of alternating bright and dark lines (interference fringes), as suggested in the sketch. More precise analysis shows that the variation of amplitude on any cross section is actually sinusoidal. Notice that if the phase relationship between the two beams is changed, as by inducing a path length difference (PLD), then the positions of the slats in space will change. If the angle between the beams is changed, then the slat spacing changes. If the phase relations inside one beam or the other change (warped wavefront), then the slats become curved. These behaviors are important in quantitative interferometries. It is instructive to figure out the spacings of the bright patches for a range of crossing angles, particularly the extreme values. If the crossing angle approaches zero, then the horizontal spacing becomes one-half the wavelength, and the vertical spacing becomes infinite. This case corresponds exactly to the collinear one discussed earlier when it is extended to a broad beam. A similar result is obtained for crossing angles approaching 180◦ ; the bright and dark slats approach the vertical and are very close together.

18.4.2 Fringes, Fringe Orders, and Fringe Patterns Described above is one version of whole-field interferometry. Instead of monitoring the variation of

Basics of Optics

The concept of fringe order has, perhaps, more meaning when the interferometric process is extended to a large field, as discussed in Sect. 18.3 and examined in some detail above. In this case, there will be many points in the field where the PLD will give rise to a specific fringe order. An example would be all points that have a PLD of three wavelengths, so we see a third-order bright (or dark, depending on our starting intensity) spot at each of these points in the image. In the general case, the whole-field interference pattern will be a random pattern of light, gray, and dark points, as is seen in a laser speckle pattern. The PLDs have a random distribution so the fringes are random. However, if the PLDs are created by some sort of spatially continuous process such as deformation of a solid, then all those spots that have a common PLD join up to create a patch of uniform intensity in the image. Such a patch is called an interference fringe. Interference fringes are loci of points having constant PLD, but we see them as loci of points of equal intensity. A typical interference fringe picture, such as the one shown in Fig. 18.8, contains several such loci, and it is called a fringe pattern. Assigning appropriate orders to the fringes in such a pattern is often tricky because, for example, a third-order fringe looks much the same as a fifth-order fringe, and the pattern might not even contain a zeroth order. We are helped by the realization that, for spatially continuous processes, adjacent fringes will differ by no more than one order.

18.5 Classical Interferometry Described here are four examples of classical experiments in interferometry. These are important historically, they provide instructional paradigms, they assist us in understanding interferometry, they provide the basis for several of the techniques that are used in experimental mechanics, and they are useful in their own right.

glass is placed so as to intercept part of the beam and deflect it so that this part crosses the undeflected part. A screen such as a piece of white cardboard is placed in

Beam expander

18.5.1 Lloyd’s Mirror This demonstration, named after Humphrey Lloyd (1800–1881), is easy to set up. It illustrates clearly the phenomenon of oblique interference and the relationships between PLD and fringe order. Figure 18.7 shows a sketch of the setup. The laser beam is expanded into a narrowly diverging beam by using a lens such as a low-power microscope objective. A mirror or a good flat piece of

Laser Mirror

Screen with interference fringes where direct and reflected beams overlap

Fig. 18.7 Setup for interferometry by Lloyd’s mirror

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intensity at one point, we observe the variation of intensity over an extended volume or, at least, a cross section of the volume. This extension of concept forces us to deal with some terminology issues that might seem tedious but that will help us in the future, especially when dealing with laser speckle methods. We have learned that interference between waves converts the PLD between two waves to a visible intensity. If the relationship between intensity and phase difference is known, then observation of intensity yields the PLD. In actual practice, we rarely deal with absolute intensity measurements, but rather work in the more favorable differential mode wherein we correlate changes of intensity to changes of phase difference. A possible exception, depending somewhat on semantics, is when we rely on observation of maximum and minimum intensities in a fringe pattern, as was done above for two-beam interference. We found that, as the PLD changes, the intensity varies between maximum and minimum according to a cosine-squared law. One complete cycle of intensity variation means the PLD has changed by one wavelength of light, so we call it an interference fringe cycle. As we increase the PLD and monitor the intensity, we observe successive fringe cycles. They can be numbered as they pass, and the numbers are called fringe orders.

18.5 Classical Interferometry

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Part C 18.5 Fig. 18.8 Monochrome rendition of fringes produced dur-

ing Lloyd’s mirror experiment using an argon-ion laser

the crossing region. Interference fringes will be seen on the screen. A monochrome reproduction of such a fringe pattern appears in Fig. 18.8. The only difficult aspect of this experiment is to keep the mirror at a shallow-enough slope (grazing incidence) so that the angle of intersection of the two beams is very small. If the angle is too large, then the fringes will be too close together to be visible. Use a fairly large mirror set so as to intercept approximately half the expanded beam. Then orient the mirror so that the beams cross at least 3 m from the mirror. The fringes will then be spaced widely enough to see with the eye, although a magnifier is helpful. Do not be confused by spurious fringes caused by the beam expander and by diffraction around the edges of the mirror. The parallel fringes in the crossing region are the ones of interest. Lloyd’s mirror is an example of the category of interferometry that is based on wavefront division, because the interfering beams come from different portions of the cross section of the beam emitted by the source. Point sources other than the laser–lens combination can be used (Lloyd certainly did not have a laser), but the laser does make such experiments easier. Several similar experiments illustrate oblique interference. Billet’s split lens technique cuts a convex lens across a diameter, then separates the two halves slightly. This arrangement seems to be a precursor of shearing interferometries such as speckle shearography.

Sir Isaac Newton (1643–1727) did not invent the fringes that are named after him, neither was he the first to notice or describe them. He was, however, the first to offer a reasonable explanation of this phenomenon and to quantify the relevant parameters. This success is very interesting, because Newton did not subscribe to the wave theory of light; yet, he devised a workable model for the formation of interference fringes based on “fits of transmission and obstruction” of the particles that he thought constituted light. That is, Newton managed to incorporate wavelike periodicity into his particulate theory. Demonstration of Newton’s Fringes One easy way to see Newton’s fringes is to place a clear dish of water on a black cloth and let it settle down. Switch off the lights except for a diffuse lamp a few meters away, such as your kitchen ceiling light fixture. Stand so that the you can see the reflection of the light in the surface of the water. Dip a pencil or a toothpick into some oil (cooking oil works), then touch the pencil to the surface of the water. You should see Newton’s fringes form and drift over the water as the oil film spreads. Nice patterns can be created by touching the oil to the water in various places or by giving the water a little stir. Figure 18.9 shows a monochromatic reduction of an example of the result. A full-color rendition of the original pattern can be found in an article by the author Cloud [18.1]. Theory of Newton’s Fringes Figure 18.10 shows a bare-bones setup to explain Newton’s fringes and their relationship to what we have learned about interference. In this instance, only one il-

18.5.2 Newton’s Fringes Newton’s fringes (also known as Newton’s rings) are ubiquitous and they are easy to spot in everyday life. Examples include the lovely colored pattern seen where a film of oil floats on water. Perhaps you have been annoyed by the fringes that appear when you mount a valuable color slide between glass plates or when you mount a glossy photo behind glass. Most of us have enjoyed the colors seen in soap bubbles.

Fig. 18.9 Mono-

chrome rendition of Newton’s fringes in a film of oil on water

Basics of Optics

A collimating lens is used to create a parallel light beam that falls on the surfaces at normal incidence. The reflected waves bounce back close to the normal. A partial mirror is used to separate the incident and reflected waves. A field lens converges the reflected waves so that all of them reach the eye or camera aperture in order that the entire field can be viewed at once. Considerable light is wasted by this system; the main advantage is that interpretation of the fringes is very simple. The incidence and viewing angles are both zero, so the relationship between gap and fringe order reduces to w=

Nλ , 2n

where w is the local gap between the surfaces, N is the fringe order, n is the index of refraction in the gap, and λ is the wavelength of light. Newton’s fringes fall into the broad class called interferometry by amplitude division, because each incident wave is split into two parts that are recombined after following separate paths. Applications As suggested above, Newton’s fringes create a map of the gap between two surfaces. They can used to Light source

Collimating lens Partial mirror Field lens Imaging device

Transparent slab

w

Fig. 18.10 InterTest object

(18.9)

ference model for Newton’s fringes

Transparent specimen Reflective flat plate

Fig. 18.11 Apparatus for using Newton’s fringes to create

a contour map

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luminating wave is shown, and the surfaces are assumed smooth and partially reflective, as would be the case if the plates were smooth glass or plastic. The incident wave impinges on the first surface of the glass plate, where part of it is reflected. The rest goes to the second glass surface, where another portion is reflected. The remainder strikes the third surface, where the same thing happens unless this surface is polished metal so that the entire remaining wave is reflected. This third surface can also be diffusive or scattering in nature as long as the roughness is very small scale. Now, if the gap between the second and third surfaces is small enough, the wave portions reflected from these surfaces will be nearly collinear; at least they will be close enough together to enter the iris of the eye and also interfere with one another. The path length difference (PLD) between the waves is found by trigonometry and converted to fringe order using the ideas presented in previous sections. The fringe order will depend on the thickness of the gap, the index of refraction in the gap, the viewing and observing angles, and the wavelength. If the first glass layer is thick enough, the portion reflected from the first surface will be relatively far from the other waves and, so, will not contribute to the interference process. This is not always the case. The same is true for other fragments of waves that will be created by various partial reflections. The implication is that Newton’s fringes can be used to measure the separation between surfaces over a large field, thereby obtaining a fringe map of the difference of contours between the surfaces. To accomplish this quantitative measurement, broad beams must be used, the angles of incidence and observation must be under control, and the nature of the surfaces must be appropriate. These goals can be achieved in several different ways. One classic setup is shown in Fig. 18.11.

18.5 Classical Interferometry

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test the contours of lenses and mirrors, although more refined interferometers are usually used for this purpose. Carefully lapped glass slabs, called optical flats, are used by machinists and others to test the flatness of surfaces by observation of these fringes. Another application, among many, is to measure the crack opening displacement in transparent fracture specimens. One further reason that Newton’s fringes are important is that the relation between fringe order and separation of the surfaces is exactly the same as that discovered for certain examples of more sophisticated techniques such as holographic interferometry and speckle interferometry. Two qualifications must be mentioned. First, the term Newton’s fringes is typically used, as is the case here, in a sense that is larger than implied by history. Second, thorough analysis of this phenomenon is quite complex. For example, the phase changes that occur upon reflection from, say, metallic surfaces, must be considered in numbering the fringe orders. Newton’s fringes seem to have been somewhat forgotten by the experimental mechanics community, which is unfortunate since they can provide a simple solution to certain measurement problems.

18.5.3 Young’s Fringes In the history of physics, Young’s experiment was critically significant and it is also a good fundamental example of diffraction at an aperture. It is of basic importance in several measurement techniques, including speckle interferometry. Aperture screen

Young’s Experiment In 1801 (25 years after Newton’s death) Thomas Young (1773–1829) conducted an elegantly simple experiment that confirmed for the first time the wave theory of light that had been put forward by Christiaan Huygens (1629–1695) a century or so earlier and well in advance of Newton’s work in optics. Young’s results were rejected, even derided, for many years because they contradicted the then-dominate particulate theory that had been espoused by Newton (1643–1727), who was, by then, approaching sainted status. Young was an epic prodigy. Among his awesome contributions were demonstration of the elastic properties of materials and the first decoding of the Rosetta stone. Figure 18.12 shows a schematic of Young’s setup with the wavefronts added. An opaque aperture screen containing two tiny holes or slits very close together is illuminated. One might expect that mere shadows of the slits would appear on the viewing screen that is placed downstream from the apertures. Instead, a broad illuminated halo is seen, and this halo contains interference fringes such as are shown in Fig. 18.13. How can this happen? Theory of Young’s Fringes Two different approaches predict the formation, orientations, and spacings of the fringes seen in Young’s experiment. The sophisticated method is to treat the problem as one involving diffraction at an aperture, which implies that the pattern observed is the Fourier transform of the aperture function. Diffraction theory was not available to Young, and it has not yet been discussed in this Chapter. A simpler analysis is based on Huygens’ principle, which allows an illuminated aper-

Viewing screen

y p/2 p/2

d

Fig. 18.12 Arrangement for Young’s experiment

Fig. 18.13 Digital photograph of Young’s fringes obtained through a simple experiment using a helium–neon laser

Basics of Optics

y=

Nλd , p

(18.10)

where y is the distance to a bright fringe, d is the distance from aperture plane to observing screen, p is the distance between the slits, λ is the wavelength of light, and N = 0, 1, 2, . . . is the fringe order. Diffraction theory shows this solution to be correct except for a missing obliquity factor, meaning that this simplified approach does not explain the observed fact that the fringe brightness diminishes with increased distance from the optical axis. Young’s fringes are of the wavefront division category, because different waves from the cross section of the beam are brought together to interfere. In addition to teaching the wave nature of light, Young’s experiment demonstrates basic two-beam interference; in this case, the wavefronts are actually spherical or cylindrical depending on whether holes or slits are used. Demonstration Young’s experiment is quite easy to reproduce with minimal equipment. Use a fine pin to punch a pair of small holes as close together as you can manage in a slip of aluminum foil. A magnifier helps, and, while you are at it, punch several pairs of holes in the foil and mark their locations with ink so you can find them easily. The reason for making several pairs of holes is that most of them will not work very well because the holes are not round, do not match, or are too far apart. If you would rather use slits than holes, you can scratch parallel lines

through the emulsion in a fogged and developed photographic plate or film. Put the foil in front of a light source in a darkened room and look for the fringes on a screen that is placed a meter or so downstream. If using a point light source, you will need to mask off the unused holes and somehow contain the excess light that is scattered around. It is far easier to use a laser – an inexpensive laser pointer serves very well in giving highly visible fringes. The only problem with the laser is that it can be difficult to zero in on the hole pairs. Figure 18.13 shows a monochrome example of a result from this simple experiment. A more meaningful composite color rendition is to be found in a related article by Cloud [18.2]. The central portion of the pattern is overexposed to show the modulation of Young’s fringes by the diffraction pattern from circular apertures that results in a rapid decrease of intensity with distance from the optical axis and interruption of the fringes. Application A significant application of Young’s fringes is in speckle photography for measuring displacement. The speckle pattern in the image of a specimen illuminated with a laser is captured on film. The specimen is then displaced and a second exposure is taken. The doubly exposed film contains a multitude of aperture pairs. The separation of the apertures in each pair is equal to the local surface displacement vector in image space. When the doubly exposed speckle photograph is interrogated with a laser beam, Young’s fringes form. These fringes allow determination of both the magnitude and direction of the local specimen displacement.

18.5.4 Michelson Interferometry Like Young’s experiment, Michelson interferometry is important in the history of physics and engineering, it teaches much about the behavior of light, and it is the conceptual basis of several measurement techniques including certain forms of speckle and holographic interferometry. Michelson and His Experiments Albert Abraham Michelson (1852–1931) was an ingenious and energetic physicist who was born in Prussia and graduated from the US Naval Academy in 1873. He was awarded the Nobel Prize in physics in 1907. Michelson published in 1881 the description of an interferometric arrangement that used a partial mirror to create the interfering beams, thus overcoming some disadvantages of the other interference schemes that were

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ture to be replaced by an array of point sources of light. In applying the principle to the problem at hand, the two slits are assumed to be linear arrays of point sources of coherent light (rather a leap of faith). Cylindrical wavefronts radiate outward from these sources. The figure offers a hint as to what happens. The two sets of waves eventually overlap and interfere in a way that is similar to oblique interference of two beams as discussed in Sect. 18.4. This interference creates a three-dimensional (3-D) system of fringes, a cross section of which appears on the viewing screen. To quantify the result, calculate by the Pythagorean theorem the PLD between the waves that arrive at any point on the screen from each of the two slits. The solution is simplified by assuming that the distance to the screen is much larger than the separation of the slits (a paraxial approximation). Since slits are used, the problem is merely two dimensional. The result is

18.5 Classical Interferometry

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being used at the time. The instrument was used by Michelson in at least three projects that are of lasting importance, namely (1) the ether-drift experiments that were conducted with E. W. Morley; (2) the first systematic study of the structure of spectral emission lines; and (3) the definition of the standard meter in terms of wavelengths of light. The ether-drift research is especially interesting in that it sought to answer questions about the existence of a luminiferous ether that was postulated by Maxwell as necessary for the propagation of electromagnetic waves. Michelson and Morley used interferometry to determine how the speed of light was affected by the velocity of the experimental reference frame relative to the ether. No such effect was discovered. This null finding led to Albert Einstein’s proposal that the speed of light is a fundamental constant and contributed to the development of the theory of relativity. The Michelson Interferometer Michelson’s original interferometers did not use collimated light from the source. This scheme is still used in some measurement contexts, including certain versions of laser Doppler interferometry. The usefulness of the device was greatly enhanced by the introduction of lenses to create broad-field collimated radiation, Light source

Partial mirror

Collimating lens

A

Field lens

O

Imaging device

C

notably by F. Twyman and A. Green. The Twyman– Green configuration, patented in 1916, is still used for inspection of lenses and mirrors. Several other interferometric measurement systems that are modifications or extensions of Michelson’s original have been devised. Discussed here is the basic whole-field version of instruments of this class. We refer to it as simply the Michelson interferometer in accord with common practice. This device is an example of amplitude division interferometry, as are Newton’s fringes. The configuration of a large-field version of a Michelson interferometer is shown in Fig. 18.14. A lens is used to collimate radiation that is projected upon a beam splitter. A portion of this beam is directed to the flat (for now) fixed mirror M1, where it is reflected back through the splitter to the observation screen. Some of the incident light passes directly through the beam splitter to fall upon another flat mirror or test object M2. It is reflected or scattered back to the splitter, and part of this light is redirected to the viewing screen. Of course, each interaction with the beam splitter causes another amplitude division, meaning there are some extra waves bouncing around in there that can cause problems with imaging and measurement. These spurious waves are often directed out of the system using optical wedges and such. Only the important divisions and reflections are shown in the sketch. Also, the beams are shown slightly separated for convenience of illustration. For actual experimentation, the viewing screen is usually replaced by some sort of whole-field imaging system, shown here as a field lens and the eye. The imaging system will typically be focused on one of the mirrors. Observables and Interpretation If the system is originally adjusted to be perfectly square, two separate but parallel wavefronts arrive at the screen. The path length difference (PLD) for the labeled wave path will be

(OA + AO + OC)−(OB + BO + OC)=2(OA − OB) . 2θ Reflective plate M1

θ

Screen for direct viewing if desired B

Reflective plate M2

Fig. 18.14 Schematic of large-field Michelson (Twyman–Green) interferometer

That is, the PLD for any interfering pair of waves is twice the local difference between the distances from the beam splitter to each of the mirrors. Clearly, this effect might be useful for measuring motion of one of the mirrors relative to the other one, or to compare the profiles of the mirrors. If the PLD is a multiple of the wavelength of the radiation, then we expect the screen to be light. If either of the mirrors M1 or M2 is translated axially, then the irradiance at the screen will be alternately bright and dark

Basics of Optics

Application Concepts Much of the usefulness of this interferometer derives from the fact that it is a differential or comparison-type measuring device. The two paths are separate in space, but the path lengths are still subtracted from one another. If only one of the mirrors is absolutely flat, for example, the resulting fringe pattern will be a contour map of the other mirror. If neither are flat, then the fringe map will give the difference between the two profiles. If both move, then the difference between the motions will be measured. Clearly, this device is both flexible and powerful.

While being able to compare two physically separate paths offers advantages, they come at a serious price. Any spurious or unwanted change of one PLD with respect to the other appears as a change in the fringe pattern. The desired fringe data are contaminated by effects that are caused by vibrations or air currents. One must minimize unwanted PLD changes by careful setup and isolation of the device, or, alternatively, by actually evaluating the noise-induced data and subtracting it out later. Often, the sensitivity of this interferometer to vibrations is put to good use in assessing the stability of certain environments. An example is in testing the mounting pads for sensitive apparatus such as electron microscopes. While the experimental mechanician is not likely to need a classic Michelson or Twyman–Green interferometer, the type is very important. Many of the techniques that are used in experimental mechanics, including holographic interferometry, speckle interferometry, and laser Doppler methods utilize the separate-path Michelson configuration. Common-Path and Separate-Path Interferometers In general, a useful way to classify interferometers is whether they are common-path or separate-path devices. Common-path interference setups, including photoelasticity, basic shearography, and Newton’s fringes, for example, are very tolerant of vibrations because noise-induced PLD changes affect both paths equally, and, so, are subtracted out. Separate-path devices, including those for holography, ordinary speckle interferometry, and Lloyd’s mirror, are highly susceptible to disturbances because the PLDs can be affected unequally. Photoelasticity is performed easily on an ordinary table or in the classroom with a transparency projector. Holography is usually done in the laboratory with the components mounted so they are isolated from the environment.

18.6 Colored Interferometry Fringes Here, questions as to why interference fringe patterns are sometimes monochrome and sometimes brilliantly colored are addressed, and some uses of these patterns are explored. Suppose you are demonstrating some interferometric process, such as photoelasticity, using an overhead projector, and you are projecting the fringe pattern on

a screen. Experience suggests that the audience will be delighted by the rippling flow of brightly colored interference fringes, and some might be so impressed that they decide to follow a career in experimental mechanics. Later in the demonstration you might use a laser to show another type of interferometry, and the fringes will be monochromatic; or you might show a black-

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according to expectations from our study of collinear interference. The same will happen if a test cell is placed in the arm OB and the refractive index of the cell content is changed. If either object M1 or M2 is tilted by some angle θ, the reflected beam will undergo a deviation of 2θ relative to the undisturbed beam. The beams will interfere according to the rules discussed above for oblique interference and create a system of parallel fringes. Simple collinear interference calculations give the same result if one realizes that tilting the object creates a linearly varying PLD over the field that is proportional to 2 sin θ. The spacing of the fringes indicates the tilt of the object. If the object is then translated axially, the tilt fringes will move across the screen. If a point radiation detector is placed at some location in the screen, an oscillating signal proportional to the changing irradiance will be produced. These ideas are applied in determining displacements and velocities of objects. The fringes discussed above are so-called fringes of equal thickness (constant PLD), otherwise called Fizeau fringes. A different fringe system, called Haidinger fringes, will be seen if the imaging system is focused at infinity. These are fringes of equal inclination and, for the square setup, will take the form of concentric circles.

18.6 Colored Interferometry Fringes

Noncontact Methods

Black

Black

2640

3960

Purple Red Purple Blue Red Purple Purple Black Purple Red Purple Blue Purple Red 1760 1980 2200

1320

To facilitate understanding of the formation of colored fringes, consider a thought experiment. Set up a system that is similar to the generic interferometer that was discussed in Sect. 18.2. Imagine that we can control the path length difference (PLD). Use for illumination a single wavelength of light such as from a laser. Suppose that we begin by using a laser that emits blue light at a wavelength of 440 nm. We connect the detector in the interferometer to a plotter that displays intensity as a function of the PLD. We know from our study of this problem that we will obtain a plot of intensity versus PLD, such as is shown by the light brown trace in Fig. 18.15. Whenever the PLD is a multiple of the 440 nm wavelength, the intensity will be zero. Now, suppose we exchange the blue laser for one that emits red light at a wavelength of 660 nm. The experiment is repeated. The trace of intensity versus PLD will match the dark brown trace in the figure, with zero intensity whenever the PLD is a multiple of 660 nm. Suppose, finally, that we put both lasers into the setup and do the measurements again with simultaneous red and blue illumination. The plotter will display the sum of both the traces obtained when the lasers were used separately. This resultant intensity trace is not reproduced in the figure, but it is easy to visualize qualitatively. More illuminating is to imagine that we replace the intensity detector with a sensor that responds to color, such as a color camera or our eye. This sensor will yield, at each PLD, the color that is created by adding the blue and red components that were recorded individually at that PLD. For example, at a PLD of around 300 nm, approximately equal intensities of blue and red are passed through, and they add to create purple or magenta, which is the color that is perceived. For a PLD of around 700 nm, most of the red is eliminated by interference, so we are left with a lot of blue, meaning that we perceive a blue with a slight purple cast. A full-color representation of the intensity traces and the observed colors appeared in [18.3]. The top of the figure names the resultant colors that are perceived as the PLD is increased and when the

440 660 880

18.6.1 A Thought Experiment

Color seen

Black

Part C 18.6

and-white fringe picture that you have used for precise quantitative analysis. The audience will likely ask why some patterns are colored and some are monochrome, and they might wonder why you waste time with the monochromatic patterns when you can create amusing colored ones.

Intensity

Part C

0

462

Path length difference (nm)

Fig. 18.15 Fringe colors resulting from the use of two

wavelengths in interferometry

two wavelengths are used simultaneously, as suggested above. Observe first that the sequence of colors is not all that simple, even though we are using only two wavelengths that were purposely chosen to keep the color pattern as tidy as possible (660 nm = 1.5 × 440 nm). Let us explain the different colors observed by visualizing the superimposition of the two intensity plots while remembering that we are dealing with light and not pigments.

• • • • • • • • • •

At zero PLD, there is no light, so we start from black. As the PLD increases from zero to about 220 nm, approximately equal amounts of red and blue are mixed, so we see a slowly changing purple that becomes more intense as the amplitudes increase. Beyond 220 nm PLD, the blue begins to drop off while red stays strong, so the purple fades quickly into red. At 440 nm, the blue is canceled by destructive interference, so only red remains. Between 440 and 660 nm, the blue increases while red drops off, so we have a short band of rapidly changing purple that is getting more blue. At 660 nm, the red is canceled, so we perceive pure blue. Between 660 nm and 1320 nm the sequence just described is reversed. At 1320 nm = 2 × 660 nm = 3 × 440 nm, both red and blue undergo total destructive interference, so we finally get back to black. Between 1320 nm and 2640 nm PLD, the whole sequence of colors is repeated. And so on.

Basics of Optics

Black

Black Black

Purple Red Blue Purple Red Purple Purple Blue

Black

Purple

Red Red

Fig. 18.16 Detail from light-field photoelasticity pattern obtained using simultaneous illumination from two laser sources: argon at 488 nm (turquoise) and helium–neon at 633 nm (red). The granular appearance in the digital photo is caused by laser speckle

have nice mathematical proportions. Better to go to the laboratory with color filters or spectral sources, a photoelasticity model, and some means of judging colors. With continuous-spectrum illumination the complement of whatever colors are canceled by destructive interference is seen. For PLDs beyond the first cycle, the problem gets very tricky. For example, the third cancellation of blue matches the second cancellation of red, as we found in our experiment, and you have left white light that has both red and blue removed, with the remaining colors mixed in uneven proportions. Black will not be seen again. The sequence of colors is no longer cyclic as it was in the thought experiment, and the colors lose saturation, tending to pastel shades and eventually to white. Various investigators have published charts of resultant color versus PLD for ideal white-light illumination (sunlight or equivalent). One such chart appeared in a classic reference by Mesmer [18.4], and has been reproduced in [18.5]. The catch is that it is very difficult to reproduce sunlight. The colors observed are dependent on the color spectrum in the actual illuminating beam, the transmittance spectra for the optical components and the specimen, and also the precision with which one is able to record or judge color, itself a difficult task. These difficulties are the second reason why colored fringes have not often been used for quantitative measurement. The third difficulty with color methods results from the way refractive index varies with wavelength, a phenomenon called dispersion. That said, colored fringes do have important uses. For one thing, the color sequence is very useful in establishing the direction of the gradient of fringe orders in whole-field work, thereby helping solve the vexing problem of fringe counting. Interpolation between whole-order fringes can also be facilitated by using colors. In recent years, the advent of color video cameras and digital processing has led to a resurgence of interest in the use of colored fringes in interferometry, especially in what is often called RGB photoelasticity. Such systems can be calibrated to account for the color balance in the light source and sensors, to eliminate the effects of color absorption in the specimen, and also to eliminate errors caused by dispersion. These methods are now practical for obtaining accurate results, especially when fringe orders are small, from colored interferometric patterns that once were used primarily to impress layfolk and add cosmetic appeal to engineering reports.

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The photograph appearing in Fig. 18.16 suggests that these predictions are valid. It is a monochrome reduction of the color picture that appeared in [18.3]. The color sequence in the experiment is not quite the same as predicted in the thought experiment described before because the two wavelengths used were not as simply proportioned. An obvious question is, for any given PLD, what is the fringe order? No truly meaningful answer to this question presents itself. At PLD = 3960, we have migrated through 3 cycles of pure blue, 6 cycles of pure red, 4 cycles of black (counting the zero), and no fewer than 12 cycles of purple. Unless you name a wavelength or color, the concept of fringe counting has no meaning. Furthermore, notice that the cycles of certain colors (purple in this case) are not necessarily evenly spaced. This is one reason why colored fringe patterns, while very nice to look at and impress managers with, have not often been used for quantitative analysis. Imagine doing the thought experiment with some form of interferometry that creates a large field where the PLD varies continuously over the field. The experiment can involve the oblique intersection of two beams, Newton’s fringes, Young’s fringes, photoelasticity, or any one of several other possibilities. The PLD at each point in the field will give rise to a specific color according to the rules described above. Further, if the PLD varies smoothly over the field, then the colors will also vary smoothly, hence, a colored fringe pattern results. A vexing question remains. What happens if more than two wavelengths are used, or even a continuous spectrum? Repeating the thought experiment with three separate wavelengths is entertaining, instructive, and time consuming. For four or more wavelengths, the problem gets out of hand, partly because you run out of wavelengths in the narrow visible spectrum that

18.6 Colored Interferometry Fringes

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In a way, the story of colored fringes has come full circle. It is likely that the earliest fringe patterns observed by Thomas Young and others were created with near-white light (sunlight, candles, arc lamps). Then, spectral sources and narrow-band filters became available; and, given the problems in judging color and so on,

most serious interferometric measurements were done with monochromatic light. Color video cameras and digital processing make it possible to analyze accurately the spectral content of light; and, so, we return to the use of broadband illumination and colored fringes for accurate analysis.

18.7 Optical Doppler Interferometry The interferometric processes studied so far in this chapter have primarily utilized light to measure optical path length differences or changes of optical path length. The path length changes of interest are caused either by changes in the physical path or else by variations of wave speed resulting from changes of index of refraction. The Doppler technique differs in that velocities of objects are measured through interferometric observation of changes of optical frequency. Path length data are not relevant. The output, whether viewed as a frequency-modulated intensity or as moving interference fringes, indicates the shift in the frequency of radiation that is emitted by a moving source, received by a moving observer, or reflected from a moving test specimen. Laser Doppler interferometry, often called laser Doppler velocimetry (LDV), is widely used in investigating the dynamic behaviors of solid materials, fluids, and structures; and sophisticated instruments that utilize this principle are readily available. In common with most optical techniques, it is remote and noncontact, and only optical access to the specimen is needed. Its sensitivity and dynamic range are remarkable. The first task is to relate the Doppler frequency shift to object motion. Then, we will discover how the frequency shift can be observed and interpreted. Finally, some applications are mentioned.

18.7.1 The Doppler Effect The acoustic Doppler shift is familiar to anyone who has noticed the change in pitch of the whistle of a train that is passing on a nearby track. The phenomenon is used to good effect by handbell ringers when they swing a rung bell to create a slow vibrato in an otherwise static pitch. It is easy to distinguish between the sound of a steeple bell that swings against its clapper and the sound of a stationary carillon bell that is struck. What we hear is the increase or decrease of the frequencies of the emit-

ted waves depending on whether the vibrating source is moving toward or away from us. Reliance on the acoustic model to explain the optical manifestation of the Doppler effect misleads if carried to extreme. Some clarifications of history and the differences between acoustic and optical Doppler shifts deserve brief mention. The Austrian scientist Christian Johann Doppler (1803–1853) explained how the velocity of the source affects the perceived frequency of sound waves. He also theorized that, if the pitch of sound from a moving source varies according to source speed, then the color of light from a star should vary according to the star’s velocity relative to earth. The French physicist Armand-Hippolyte-Louis Fizeau (1819–1896) later clarified the theory of the Doppler effect in starlight and demonstrated how it could be used to measure relative velocities of stars. In the context of astronomy and applied optics, the phenomenon is often called the Doppler–Fizeau effect. A color astronomical photograph reproduced in [18.6] illustrates the correctness of the expectations of Doppler and Fizeau. This photograph is very unusual because it shows the wavelength shift in the visible spectrum. For most waves, including sound, the observed wave velocity differs depending on the velocity of the observer. The notable exceptions are electromagnetic waves, including light, for which the speed of the wave is constant irrespective of observer speed. This means that the Doppler frequency shift for sound waves differs from the shift for light waves for any case where the observer is moving. In particular, for light waves, the frequency recorded by a moving observer who intercepts waves from a fixed source differs from the frequency received by a fixed observer from a moving source, even if the relative velocities are the same. The reason for this apparent paradox is found in the theory of relativity and derives from the facts that the speed of light is invariant and that the clocks for moving and stationary observers are dif-

Basics of Optics

ct1 D Fixed source A–D emitted

C

B

x λ

18.7.2 Theory of the Doppler Frequency Shift

ct1

Before learning how to utilize the Doppler effect, the relations between frequency shift and velocity must be understood. Two space diagrams that illustrate the mechanics appear in Fig. 18.17. In the upper diagram, the source is at rest, and it emits a harmonic wave of wavelength λ that travels at speed c. At time t = 0, wave point A is emitted by the source. After an interval t1 , wave point A has traveled the distance ct1 , and wave point D is just then being emitted at the source. For this pictorial example, only three wave cycles, identified by wave points A–D, are emitted at equal intervals during the time t1 to fill the space between A and D. In general, if n cycles are emitted during some interval t1 , then the space diagram suggests that nλ = ct1 .

(18.11)

Use the relationship ν = c/λ between wave speed c, frequency ν, and wavelength λ to obtain ct1 (18.12) = νt1 . n= λ The lower space diagram in Fig. 18.17 illustrates the situation if the source is moving to the right with constant speed v, but with all other conditions remaining unchanged. Wave point A still leaves the source at time t = 0. Since the wave speed is invariant, it travels the same distance as before in time t1 . The second wave point B is emitted downstream at time t = t1 /3; wave point C leaves the source at time t = 2t1 /3; and wave point D will only just be emitted at the source location at time t = t1 . Clearly, all three wave cycles must fit within the now-shorter space between the end point A and the final position D of the source. The source seems to have caught up to the waves it has previously emitted. So, the wavelength must be changed to some new value λ , and a fixed observer senses a new frequency ν . In general, if n wave cycles are emitted by the moving source during the interval t1 , the new wavelength is calculated as nλ = ct1 − vt1 ,

A

(18.13)

υ Moving source at t = 0, A emitted B emitted here at t = t1/3

D

C

B

A x

Source at t = t1

λ'

C emitted here at t = 2t1/3

Fig. 18.17 Space diagrams to explain the Doppler effect

which yields for the value n     (c − v)t1 ν v  n= = (c − v)t1 = ν 1 − . λ c c (18.14)

The number of wave cycles emitted during a given interval is the same whether or not the source is moving, so the results from (18.12) and (18.14) are equated to obtain the new frequency ν . (18.15) ν = 1 − vc Note that a specific location for the observer need not be declared; we simply take a snapshot of the space that the source is moving into. This result is the new frequency ν that would be seen when the source is moving toward the observer with velocity v. In this case ν > ν. Repetition of the calculation for the case where the source is moving away from the observer, using a snapshot of the space that the source is moving out of, shows that the new frequency is less than the original frequency, with the result ν . (18.16) ν = 1 + vc The difference between the old and new frequencies Δν = ν − ν is the Doppler frequency shift. More is to be said about the sign of this quantity. Clearly, if the speed of light and the original frequency of the radiation are known, and if the new

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ferent. The calculation for the moving observer is not particularly difficult, but it will not be pursued here. The difference between moving-source and movingobserver predictions is not measurable for speeds that are well below the speed of light, so it is not significant in most terrestrial applications.

18.7 Optical Doppler Interferometry

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frequency can be ascertained, then the speed of the source can be calculated. In the acoustic domain these quantities are quite apparent to us. However, optical frequencies are on the order of 1014 Hz; so, if the source velocities are not great, then the Doppler shift is relatively small. One faces the difficult problem of measuring the small change in a large quantity. As is usual in such an experimental situation, the measurement is performed in differential or comparison mode. The difference between the original frequency and the new frequency is measured directly through optical interferometry.

18.7.3 Measurement of Doppler Frequency Shift Determination of the Doppler frequency shift by measurement of the original and shifted frequencies and then subtracting the one from the other is not appropriate unless the velocities and resulting frequencies are very large, as might be the case for certain astronomical observations. For the speeds involved in typical engineering applications, the problem is poorly conditioned in that the small change in a large quantity is sought, so the potential error is large. As is usual and wise in this situation, the measurement is instead performed in differential or comparison mode. Interferometry offers a natural approach to the direct determination of the Doppler shift. Several interferometric schemes satisfy the need. For this discussion, consider the Michelson arrangement shown in the Fig. 18.18. Rather than a broad collimated beam, only a single wave train or small

pencil of light, as from a laser, is used to interrogate the moving target, which carries a reflective or scattering surface that bounces some of the incident radiation back to the photodetector. The target motion causes a shift in the frequency of that radiation. The mirror in the second interferometer arm is shown as fixed, although it can also be moving if the difference between two target velocities is sought. The two lightwave trains, which combine interferometrically at the detector, have different frequencies ν and ν , as was shown above. The electric vector for the sum of these waves at the detector is E S = A1 cos(2πνt + φ1 ) + A2 cos(2πν t + φ2 ) . (18.17)

The irradiance will be I = |E S |2 = A21 cos2 (2πνt + φ1 ) + A22 cos2 (2πν t + φ2 ) + 2A1 A2 cos(2πνt + φ1 ) cos(2πν t + φ2 ) . (18.18)

We use the identity for the product of two cosine functions to convert the irradiance to the following form, where φ = φ1 − φ2 is the phase difference, I = A21 cos2 (2πνt + φ1 ) + A22 cos2 (2πν t + φ2 ) + A1 A2 cos[2π(ν + ν)t + φ] + A1 A2 cos[2π(ν − ν)t + φ] . (18.19)

The first three terms in the irradiance (18.19) describe waves that oscillate at frequencies equal to, slightly less than, or greater than the basic radiation frequency for Moving target object velocities in the range considered here. If light waves are being used, then these frequencies are too large to be tracked, so only a time-average value, esBeam sentially a constant, would be displayed by the detector splitter for these three waves. The final expression in the irradiance is the one of interest. It describes an output that is proportional to the cosine of the difference between the original freCoherent light source quency and the new Doppler-shifted frequency. In other words, it is an irradiance oscillation at what is usually Fixed thought of as the beat frequency. This difference fremirror quency will be low enough that it can be tracked by ordinary photodetectors. Detector As an elementary example, suppose that the target Fig. 18.18 Michelson interferometer to measure the Doppler has a constant velocity and its motion is analyzed with frequency shift the Michelson interferometer. The output from the de-

Basics of Optics

18.7 Optical Doppler Interferometry

18.7.5 Bias Frequency 0

t

b) Amplitude

0

t

Fig. 18.19a,b Target motion and resulting irradiance record for two cases of laser Doppler interferometry. (a) Constant speed. (b) Varying speed

tector will be a harmonic wave of constant frequency as suggested in Fig. 18.19a. The dashed line is the target speed, and the solid trace is the detector output. That the frequency of the output signal is the indicator of target velocity should not be forgotten, because our habit is to think of the amplitude of the output as the important data. To underline this idea, suppose that the target is vibrating instead of moving at constant speed. The detector output will then be a frequency-modulated wave of the sort shown in Fig. 18.19b. As before, the dashed line is the target velocity as a function of time. The solid trace is the frequency-modulated signal from the detector on the same time scale. At any instant, the frequency is proportional to the instantaneous velocity of the target. For the example shown, a bias frequency has been incorporated for reasons mentioned below, and it must be taken into account in interpreting the signal.

18.7.4 The Moving Fringe Approach An alternative approach to understanding the Michelson interferometer and its use in measuring velocities by the Doppler method is to calculate the rate at which interference fringes cross the detector. Think of one of the mirrors as being tilted slightly so that a system of parallel interference fringes is created near the detector by oblique interference. If, now, one of the mirrors is given a velocity, a time-varying path length change is introduced, which causes the interference fringes to move across the detector to create an irradiance variation. Phase-shift calculations give a result that is the

A problem with the basic Doppler technique described so far is that the sign of the target velocity is not easily distinguished. This difficulty and artifacts derived from the processing of low-frequency signals are eliminated by introducing a bias frequency signal, which, in this case, functions as a frequency modulation (FM) carrier wave, as suggested in the example discussed above. The bias frequency is created by introducing into one arm of the interferometer a frequency-shifting device. A constant velocity of the reference mirror accomplishes this aim, but it cannot go on for long. Oscillation of the reference mirror can be used, but it introduces a frequency modulation itself. One practical method is to introduce into the optical path a diffraction grating that is rotated at constant speed. The angular deviation of the diffraction grating coupled with its rotation creates a constant Doppler shift in the reference arm. A better approach is to use an electro-optic device such as a Bragg cell to create the bias frequency deviation.

18.7.6 Doppler Shift for Reflected Light Most engineering applications of Doppler motion measurement involve interrogation of the target object with a light beam that is reflected or scattered back to the detector, as suggested by the Michelson model discussed above. We expect that the apparent source velocity as indicated by the Doppler shift and our equations will be different from the actual target velocity, because the light traverses the object arm of the interferometer twice and a reflection is involved. Thinking about this problem is aided by Fig. 18.20. In the upper part of the figure, the reflective target is at a distance d from the source. The laws of reflection prescribe that the image of the source will be at the same distance behind the target. In the lower part of the figure, the target has moved forward by amount Δx so that it now lies a distance d − Δx from the source. As before, the image of the target is now at this same distance from the target. The image of the target has moved forward a distance 2Δx, or twice the target motion. Differentiation with respect to time demonstrates that the velocity of the image is twice the velocity of the target. For this reflection case, then, the Doppler shift indicated by the

Part C 18.7

same as that presented above. This mathematical development is not pursued here.

a) Amplitude

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Image of source in original position

d

d

Source 2Δx

Image of source in new position

Original object position

Δx

Source d –Δx

d –Δx

New object position

Fig. 18.20 Relationship between target motion and image motion for light reflected from a moving target

interferometer is twice what it would be for a simple moving source.

18.7.7 Application Examples Doppler interferometry is widely used in engineering to measure the motions, vibrations, and mode shapes of structures and machine components as well as fluid flow

fields by what is usually called laser Doppler vibrometry or laser Doppler velocimetry (both LDV). Powerful scanning or imaging laser Doppler interferometry systems, called Doppler picture velocimetry (DPV), yield a full-field picture of the specimen velocity field. Laser Doppler methods are so sensitive and accurate that they have become techniques of choice for the calibration of other motion measurement devices such as accelerometers. Doppler interferometry is also fundamental to the function of the laser gyroscope that senses rotations and that has no moving parts. Mirrors or fiber optics are used to direct light waves in opposite directions around a ring or a triangular path to converge at a detector. If the ring is stationary, no frequency shifts are created, so no beat frequency is observed. If the device rotates, the frequencies of the two beams will undergo equal and opposite Doppler shifts. The beams are mixed at the detector to produce an oscillating signal whose frequency is proportional to the rotational speed of the gyroscope. While not interferometric in function, applications of the Doppler effect in such fields as medical diagnosis, for example, in measuring blood flow rates inside the body, should not go unnoticed. These techniques have saved the lives of many.

18.8 The Diffraction Problem and Examples The first seven sections in this article on the basics of optical methods of measurement dealt with the nature of light and the first of the two cornerstones of optical methods of measurement, namely the interference of light waves. There is much more to be said about types and applications of interferometry, and, these topics are discussed extensively in the appropriate chapters of this Handbook. It is time to augment our understanding of the behavior of light and to expand our kit of tools through examination of the diffraction of light waves, the second of the two cornerstones of wave optics. This first section on diffraction provides some observations about our physical world that motivate study of the problem, mentions some potential applications, gives some background of the long history of the solution, and defines the problem in its simplest form. Study of the problem is facilitated by development of some useful mathematical tools, particularly the concept of scalar complex amplitude that is pre-

sented on Sect. 18.9. Section 18.10 then quantifies the problem and presents the most useful form of the solution. Important example applications are discussed in Sects. 18.11 and 18.12.

18.8.1 Examples of Diffraction of Light Waves Outlined here are simple experiments and observations that illustrate various aspects of diffraction of light at an aperture and also give some idea of its ubiquitous presence in our activities. First, project light through a sharp-edged hole (aperture) in a piece of cardboard or shim stock (aperture plate). An ordinary lamp, a flashlight, or, even better, the sun can be used as the source; but a practical alternative is to use an inexpensive laser pointer. A piece of cellophane mending tape stuck to the end of the pointer will serve to expand the beam to cover the hole if needed. What is observed on a viewing screen that is

Basics of Optics

Fig. 18.21 Diffraction pattern created by passing laser light through a small circular aperture

based on Huygens’s construction requires a substantial leap of faith. A better explanation is developed from diffraction theory. Modify the setup again by passing the laser beam through a very fine sieve or mesh. For best results, the mesh density should be on the order of at least 100 threads or lines per inch. Fine fabric such as hosiery (not the stretch kind) or other sheer nylon, preferably black, can be used, in which case the viewing screen (maybe the wall) should be a few meters distant. The screen exhibits an array of bright dots. The geometry of the dot pattern depends on the way in which the mesh or fabric is woven. The dot spacing increases as the thread spacing is decreased, which is contrary to uninformed expectation. Figure 18.22 shows a result from such an experiment. Change the experiment further by allowing the beam that falls on the aperture to carry some sort of information. The difference is that the beam carries the data and the aperture is just a hole, rather than having a uniform beam illuminating intelligence in the aperture. For example, you might use the light that is scattered from a distant object. If the aperture is pinhole-small (about 0.4 mm), an inverted image of the object will be observed on the screen. Light intensity will be very low, so the screen must be in a relatively dark place. This phenomenon was evidently known in China by the 5th century BCE and was described in the notebook of Leonardo da Vinci. It has long been used to safely observe solar eclipses. A device based on this discovery, called the camera obscura (dark room), was used by renaissance artists. Nowadays it is called the pinhole camera and is used by children and serious photographers alike. How can a tiny hole function as an imaging lens?

Fig. 18.22 Portion of diffraction pattern from photographic replica of a crossed bar-space grating (grid) having spatial frequency 1000 lines/inch (argon-ion laser, no enhancement, orders visible to eye range from −15 to +15 in both directions)

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Part C 18.8

placed downstream from the hole? Lifelong experience suggests that you should see a simple shadow of the aperture plate; that is, the viewing screen will be dark except where it is lit by the energy coming through the hole. Close observation shows something more complex. The edges of the shadow will be fuzzy. If using a very small light source, the sun, or a laser for illumination, you might even be able to see interference fringes at the edge of the shadow. Investigate further by continuing the experiment with consecutively smaller apertures. One will notice that the fringes near the edge of the shadow become stronger. If the aperture is made really small, approaching a pinhole, the bright area on the screen will be larger than the aperture, and you might notice some welldefined concentric rings in the pattern. As you shrink the aperture ever smaller, the bright patch on the screen becomes ever larger, suggesting an inverse relationship that seems totally contrary to experience and expectation. Figure 18.21 shows an example of the concentric rings observed downstream when laser light is passed through a small round aperture. Now, modify the aperture so that it contains some simple but ordered intelligence. One way of doing this is to repeat Young’s experiment, as was described in Sect. 18.5.3. The aperture is just a pair of tiny holes close together. One sees on the viewing screen a large illuminated patch that is modulated by a system of parallel interference fringes. As was mentioned, the model

18.8 The Diffraction Problem and Examples

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Part C

Noncontact Methods

Part C 18.9

Detector P Source Q

Plate with aperture

Fig. 18.23 The diffraction problem

This example requires a little more equipment. Set up a lens to image a backlit window screen onto a ground glass or just a piece of white cardboard. Place an iris diaphragm near the lens to control its aperture. Better, set up a view camera or a single-lens reflex if you have one. Use a magnifier to examine the image of the mesh as you decrease the aperture of the imaging lens. At some small aperture setting (large f -number), you will find that the mesh image loses contrast, and it eventually disappears. The implication is that decreasing the aperture diminishes the ability of the lens to transmit information that has high spatial frequencies (fine detail); that is, a lens is a tunable low-bandpass filter. Think about the conventional wisdom that instructs us to use extremely small apertures for the sharpest pictures. Are we properly instructed?

18.8.2 The Diffraction Problem Experiments and observations such as those mentioned above led to the formulation of a classic problem in mathematical physics. Consider the sketch shown in Fig. 18.23, which illustrates the problem in qualitative terms. Light emitted by a source at location Q falls on an opaque plate containing some kind of aperture (hole). The aperture may or may not contain intelligence such as a transparency or an optical element such as a lens.

The task is to describe the light field that will be received at point P downstream from the aperture. This deceptively simple problem is of fundamental importance in optics, including electron microscopy, and in the propagation of radio waves. It provides understanding of the formation of images by optical components, and it leads to ways of specifying and measuring the performance of optical systems. Diffraction theory allows us to utilize certain optical elements as Fourier-transforming devices, thereby giving us the ability to perform whole-field optical processing of spatial signals (pictures) to modify content and improve signal-to-noise ratio. The results are critical to experimental mechanicians who seek to thoroughly understand geometric moiré, moiré interferometry, speckle interferometry, holo-interferometry, and so on.

18.8.3 History of the Solution The diffraction problem is much more complex than it might seem, and it has not been solved in general form. The oldest solution rested on the assumption by Huygens in 1678 that the illuminated aperture can be replaced by an array of point sources. This equivalent problem was solved correctly by Fresnel and by Fraunhofer in the early 1800s, although not all observed phenomena were explained. The Huygens– Fresnel–Fraunhofer approach is still useful because it provides an easy way to visualize diffraction phenomena. Rigorous solution of the diffraction problem was accomplished in 1882 by Kirchhoff, who treated it as a boundary-value problem and incorporated several severe simplifying assumptions. Kottler and Sommerfield used mathematical theory developed by Rayleigh to refine, extend, and correct Kirchhoff’s elegant analysis. These solutions incorporated several of the ideas that were developed in the earlier era by Fresnel and Fraunhofer.

18.9 Complex Amplitude The emphasis in this Handbook chapter is on the physical phenomena that underpin optical methods of measurement, so mathematical manipulation is held to a minimum. As we begin study of the diffraction problem, however, and then go on to consider more complex optical techniques such as

holographic interferometry, we require a way to deal with light waves that is more compact and less unwieldy than the trigonometric approach used so far. This need is satisfied by a descriptor called the complex amplitude, which is developed in this section.

Basics of Optics

Extract from the first section of this article the expression that represents the electric vector for a single wave traveling in the z-direction. Change the sign in the argument, which makes no difference to the meaning   2π (18.20) (vt − z) , E = A cos λ where A is the vector giving the amplitude and polarization direction of the wave, λ is the wave length, and v is the wave velocity. This equation is easily generalized for a wave propagating along some axis specified by unit vector α having directions l, m, and n in Cartesian space. The caret distinguishes the unit vectors α = l iˆ + m ˆj + n kˆ .

(18.21)

The wave traveling in the α direction can be written as   2π (18.22) E = A cos (vt − (lx + m y + nz)) . λ A general position vector locating some arbitrary point in space is r = x iˆ + y ˆj + z kˆ ,

(18.23)

so the equation for a wave traveling in the specified direction can be written with a dot product term that contains wavelength and direction, E = A cos(ωt − k · r) ,

(18.24)

where k is the vector wavenumber = kα = 2πα λ , ω is the angular frequency of radiation = 2πv = 2πν, and ν is λ the optical frequency (Hz). Caution, do not confuse the scalar wave number k with the unit vector kˆ .

18.9.2 Scalar Complex Amplitude The representation of the wave in (18.24) is physically meaningful and sufficient for many optical calculations. However, it tends to be cumbersome when describing the propagation and interactions of waves in the more complicated optical systems, such as those that are used for holographic interferometry. A complex number representation is better, although more difficult to interpret. We use the identity eiθ=cos θ+i sin θ to convert the electric vector to the equivalent exponential form. Only the cosine part is needed E = Re Aei(ωt−k·r) ,

(18.25)

√ where Re means real part of and i is −1 (often represented by j). Henceforth, the Re will be understood to be applicable wherever it is appropriate, so it is dropped from the equations. Now, it is a simple matter to include a phase-angle term φ, which is how we get the entire path length (PL) or the path length difference (PLD) into the picture. The phase angle could have been introduced into the cosine wave function above, of course. We multiply the physical PLD by the wavenumber to convert it to the equivalent phase angle 2π (18.26) (PLD) = k(PLD) . λ The expression for the electric vector with the PLD contained becomes φ=

E = Aei(ωt−k·r+φ) .

(18.27)

This representation of the general wave can also be written as a product, E = Aeiωt eiφ e−ik·r .

(18.28)

The first exponential in this equation represents the oscillation at optical frequencies. We have no way of tracking signals at these high frequencies, so it makes sense to leave that term out. The amplitude data, the space variables, and the phase are the only quantities of interest. Also, for interference to take place, the polarization direction in a setup must be uniform, so we usually drop the vector designations on the electric vector E and the amplitude vector A. What is left is a much simplified representation of the wave that contains all the important information. It is called the scalar complex amplitude U U = A eiφ e−ik·r = ei(φ−k·r) .

(18.29)

In many instances, the amplitude and the phase angle are functions of location in the coordinate system, so we would write A(x, y, z) and so on. As an example, a bundle of waves traveling in the y-direction and having amplitude and phase varying with position could be expressed as U = A(x, y, z) ei(φ(x,y,z)−ky) .

(18.30)

Note that there is nothing implicit here that indicates the breadth of the wave bundle. The same equation serves for just one wave or for a broad beam of many waves. A detail that often proves useful in optics calculations is that twice the real part of a complex number is the sum of the complex number and its complex conjugate. The factor of 2 is often absorbed in the other

471

Part C 18.9

18.9.1 Wave Number

18.9 Complex Amplitude

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Part C 18.10

constants and does not usually appear explicitly. An asterisk is used here to indicate complex conjugate. U + U ∗ = 2 Re(U) .

(18.31)

18.9.3 Intensity or Irradiance We learned in the sections on interference that the essence of interferometry is in determining invisible phase quantities by converting them to palpable intensity or irradiance. The irradiance is defined, somewhat arbitrarily, as twice the average over many optical oscillations of the square of the magnitude of the electric vector. That is, where T is several optical oscillation periods and r represents the coordinates of a point in space where the intensity is to be determined 2 I (r) = T

which is to say, the irradiance at a point turns out to be the square of the amplitude at that point. To establish the relationship between intensity and complex amplitude, write the electric vector in terms of complex amplitude and its complex conjugate as mentioned in (18.31)

1 iωt (18.35) U e + U ∗ e−iωt . E(r, t) = 2 Put this identity into the definition of intensity (18.32) or (18.33), leave an extra 12 out as is usual, and obtain I (r) = U 2 ei2ωt + U ∗ 2 e−i2ωt + UU ∗ .

(18.36)

Recognize that,  e±i2ωt = 0

(18.37)

and the intensity reduces to

T E 2 (r, t) dt .

(18.32)

0

Taking the trigonometric form for a simple harmonic wave for example, and letting  represent the average over several oscillations, I (r) = 2A2 (r)cos2 [ωt − k · r + φ(r)] .

(18.33)

The time average over many periods of the cosinesquared function is just 12 , so I (r) = A2 (r) ,

(18.34)

I (r) = UU ∗ = |U|2 .

(18.38)

This result shows that the intensity of a wave at a point is the square of the modulus or amplitude of the complex amplitude at that point, which agrees with the result of (18.34). Interference and diffraction calculations require tracking the complex amplitude through the system and then determining the intensity distribution in the field by multiplying the resulting complex amplitude by its complex conjugate. Often, the result is transformed back into trigonometric form to make it easier to interpret.

18.10 Fraunhofer Solution of the Diffraction Problem The objective of this section is to present the Helmholtz–Kirchhoff–Rayleigh–Sommerfield–Fresnel– Fraunhofer solution to the diffraction problem. A pedagogical problem is to find an acceptable balance between key concepts, mathematical detail, and physical understanding. The mathematical development is important, because physical reasoning is tightly woven into evolution of the theory, and assumptions and simplifications that affect the utility of the solution are incorporated. However, Handbook space limitations do not permit full development of the theory, so we settle for a definition of the problem, a presentation of the result, a summary of the restrictions, and examples of useful applications. Practitioners sufficiently involved, needful, and sophisticated should make themselves aware of the nuances of diffrac-

tion theory through study of appropriate resources included in the Further Reading section of this chapter and [18.5].

18.10.1 Solution of the Problem The diffraction problem was outlined qualitatively in Sect. 18.8. Figure 18.24 facilitates a quantitative statement of the objective. Light having a certain complex amplitude is emitted from source point Q and falls on an aperture in the x–y plane that might contain some intelligence such as a transparency. The task is to predict the complex amplitude of the light arriving at observing point P. This isometric sketch establishes the geometry and defines the position vectors r1 and r2 that locate the

Basics of Optics

Source point Q (x1, y1, z1)

x ξ

Observing point P(x2, y2, z2)

Screen with aperture

z Q (x1, y1, z1)

r2

r1

φ1

r1 ds

r2

y η

P(x2, y2, z2) φ2

z'

Element of aperture area ds

Screen with aperture

Fig. 18.24 Geometry of source, aperture, and receiver for

solving the diffraction problem

Fig. 18.25 Definition of parameters involved in the diffrac-

tion solution

source point and receiving point relative to a small surface element ds within the aperture. The two-dimensional illustration of Fig. 18.25 clarifies the parameters and defines the two direction angles, ϕ1 and ϕ2 , taken from a positive z  -axis through the aperture element, for the position vectors. ξ and η represent coordinates in the plane of the aperture and serve as dummy variables for the needed integrations, while x, y, z remain as global coordinates. The aperture area element within the aperture is ds = dx dy = dξ dη for all practical purposes. Assume that the aperture might contain a signal screen or transparency whose transmittance function can be described as T (ξ, η). The solution of the diffraction problem starts with Stokes’s theorem and evolves for functions that satisfy the wave equation into a relationship between integrals extending over the surface of a vessel that contains the aperture. Kirchhoff showed that, under certain conditions, the integration needs to extend only over the expanse of the aperture, and the rest of the vessel can be ignored. The resulting expression, called the Kirchhoff diffraction integral, is very difficult to evaluate even for simple cases because the position vectors and angles vary widely as the integration surface element roams over the aperture. The genius of Fresnel and Fraunhofer is evident in assumptions they made that greatly simplify the geometry and result in an integral equation that is well known and easily evaluated for many cases that are of practical interest. The Fraunhofer approximation requires that the maximum dimension of the aperture be much smaller than its distances from the source point Q and the observing point P. An approximate quantitative statement of this limitation is 1 λ 1 + 2 , (18.39) z1 z2 w

where z 1 is the horizontal distance from source to aperture, z 2 is the horizontal distance from aperture to receiver, λ is the wavelength of light, and w is the maximum aperture dimension. Implied is that the source and receiving points must be very far removed from a very small aperture. It is a so-called paraxial approximation. This restriction is severe in physical terms, and its implications are not widely acknowledged, but it is satisfied reasonably well in many useful applications. More will be said about these limitations presently. With the described restrictions in place, after reducing the problem to two dimensions for convenience, and after lumping various geometric constants and rearranging a bit, the diffraction integral reduces to the Fraunhofer approximation for the complex amplitude at the receiving point 

UP = C e

ik 2

x 12 x 22 z1 + z2





−ik

T (ξ) e



x1 x2 z1 + z2

 ξ

dξ .

aperture

(18.40)

The integral is recognized as a Fourier transform of the aperture transmittance function and it is often written as 

UP = C e

ik 2

x 12 x 22 z1 + z2



F{T (ξ)}

f = λ1



x1 x2 z1 + z2



.

(18.41)

The leading exponential contains inclination factors that are often ignored. The long subscript appended to the transform symbol specifies the spatial frequency or dimension metric in transform space. Physically, the x/z terms are angular deviations from the optical axis (tangent φ), the restriction to small angles having already

473

Part C 18.10

xξ

18.10 Fraunhofer Solution of the Diffraction Problem

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Part C 18.11

been imposed. The extension to three dimensions gives a two-dimensional Fourier transform.

18.10.2 Summary Diffraction at an aperture is a Fourier transforming process that decomposes optical information (e.g., a picture) into its constituent space-frequency components (e.g., lines per millimeter) that appear at some distance downstream from the aperture. Spatial frequencies in the input plane are translated into illumination at corresponding distances off-axis in the transform pattern. The relative strengths (intensities) of the illumination patches in the transform pattern correspond to the relative weights of the in-

dividual spatial frequency components in the original signal. That an aperture is a physical Fourier transforming device, or spectrum analyzer, is thought provoking, powerful and far reaching. All of our mathematical lore about Fourier transforms can be called into service. Most of us have been introduced to transforms of time-varying signals. Substitute space (distance) for time, and our learning transfers to the optical domain. Applications include optical data processing, moiré interferometry, and holography, to name only a few. In order to fix and test these findings, the Fraunhofer approximation of the diffraction integral will be applied to some simple cases that are easily reproduced in the laboratory.

18.11 Diffraction at a Clear Aperture A useful example application of the diffraction integral is to predict what should be seen when light is passed through either a long narrow slit or a small circular hole in a plate and to determine whether the predictions are supported by laboratory observation. These two cases are mathematically equivalent and require only a two-dimensional analysis. The circular hole is axisymmetric, so one needs to examine only the medial plane. This problem is not as trivial as it might seem, and the results are important in many applications, including optical imaging, speckle methods, and spatial filtering. The findings, when compared with experience, illustrate some implications of the Fraunhofer approximations.

This transmittance function is substituted directly into the Fraunhofer integral (18.40). Since the illuminating waves are parallel, the source is at infinity and the x1 /z 1 expressions are zero. The transform integral for the scalar complex amplitude at observing point P becomes 

UP = C e

ik 2

x 22 z2

 +w/2 

−ik



e

x2 z2



ξ

dξ .

(18.43)

−w/2

The integral is recognized as the Fourier transform of the aperture function and is evaluated as follows, ignoring for the moment the multipliers outside the integral

18.11.1 Problem and Solution For convenience, choose a plane wave front (collimated light) that is illuminating the slit or circular hole at normal incidence as shown in Fig. 18.26. The width of the slit or the diameter of the hole is w. The problem is to predict the light intensity at some general point P that is a considerable distance away from the aperture. Figure 18.27 shows the transmittance function for the clear aperture, which, from now on, will be called a hole. The mathematical description of this aperture function, which is a rectangular pulse in space, is    λ 1 for |ξ| ≤ w2 . (18.42) = T (ξ) = rect w 0 for |ξ| ≥ w2

x ξ P w

z

Fig. 18.26 Geo-

metry for diffraction by a clear aperture

T

Fig. 18.27 Trans– w 2

+

w 2

ξ

mittance function for a clear aperture

Basics of Optics

f= –8 w

1 2 3 w w w

x2 λz2

8 w

Fig. 18.28 The sinc function for an aperture of width w

sign ⎛

−ik

⎝e



x2 z2

 ξ

⎞+w/2

 ⎠ −ik xz 22 −w/2   x2 sin kw 2z 2 = . k 2zx22

F{T (ξ)} =

(18.44)

expressions outside the integral sign in (18.43). These multipliers can be ignored for development of an understanding of the intensity distribution in a transform plane where z 2 is much larger than x2 . A graph of the square of the sinc function is shown in Fig. 18.29. Note that the intensity distribution is plotted at an expanded scale in the left-hand portion in order to show clearly the way in which the intensity oscillates and diminishes off-axis. The graph shows that, for a long narrow slit, the diffraction pattern will be a system of alternating light and dark bands (fringes) that are parallel to the slit and that rapidly decrease in contrast with increasing distance from the optical axis. For the circular hole aperture, the diffraction pattern will be a central bright patch, called the Airy disc, surrounded by concentric light and dark rings that also rapidly decrease in visibility with increasing distance from the center.

18.11.2 Demonstrations (18.45)

Recall that kx2 /2z 2 is just π f , where f is the spatial frequency parameter that is the distance dimension in transform space, x2 . (18.46) f = λz 2 With this definition in place, the transform can be written in terms of spatial frequency as

This clear-aperture diffraction pattern can be generated easily by passing light from a laser or even a laser pointer through a small pinhole in a sheet of foil. The photograph already presented in Fig. 18.21 is an example. Qualitatively at least, these results demonstrate the validity of diffraction theory and the approximations that have been made. That these ideas apply in the astronomical domain is shown in Fig. 18.30.

sin πw f (18.47) = w sinc(w f ) . πf A plot of the sinc function appears in Fig. 18.28. The intensity or irradiance is essentially the square of the sinc function multiplied by the square of the F{T (ξ)} =

I = w2 sinc2 (wf )

Expanded scale

–8 w

Full scale

1 2 3 w w w

8 w

x2 f= λz2

Fig. 18.29 Irradiance distribution in the far field for an

aperture of width w

Fig. 18.30 Hubble Space Telescope NICMOS image of NGC 2264 IRS mother star and baby stars in the Cone Nebula. The rings and spikes emanating from the image form diffraction patterns that demonstrate the near-perfect optical performance of the camera. Portion of image no. STScI-PRC1997-16 (Image by R. Thompson, M. Rieke, G. Schneider of University of Arizona and NASA)

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Part C 18.11

w sinc (wf ) w

18.11 Diffraction at a Clear Aperture

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Part C 18.12

18.11.3 Numerical Examples and Observations The results obtained allow us to understand and predict, among other things, the resolution limit of lenses and the size of laser speckles. As part of the process, one calculates the diameter of the central spot, called the Airy disc, of the diffraction pattern. Examination of the intensity graph (Fig. 18.29) suggests that this diameter will be where     1 x2 (18.48) =2 , 2 λz 2 w which gives for the diameter of the central bright patch, λz 2 (18.49) . w Take, for example, an aperture diameter w of 0.2 mm, use red light at λ = 0.6 μm, and satisfy the Fraunhofer restriction by placing the viewing screen quite far from the aperture at z 2 = 1000 mm. The diameter of the central disc turns out to be 6 mm. If the aperture diameter is reduced to 0.02 mm, then the central disc diameter expands to 60 mm. The inverse reciprocity between aperture size and expanse of the diffraction pattern is evident. The smaller the aperture, the larger the illuminated area in the diffraction pattern observing plane. This idea deserves to be looked at more closely. Consider the case where the aperture is vanishingly small. In this case, the aperture function would be the impulse function or Dirac delta. The Fourier transform of the delta function is a constant, meaning that, in this unattainable ideal case, the entire transform plane is illuminated by the central patch of the diffraction pattern. The other extreme seems troublesome. If the aperture is large, the theory suggests that the bright patch d = 2x2 = 2

in the diffraction pattern should be small. That is, the Fourier transform of a constant extending broadly in the aperture plane would approach a delta function in transform space. Stated another way, as the aperture expands, the theory shows that the sinc function becomes narrower. However, if you project light onto an aperture of, say, 20 mm diameter and if you place a viewing screen at 1000 mm behind the hole, as we did before, you will see on the screen a bright patch that is roughly the size of the aperture. In other words, one sees what we usually recognize as the mere shadow of the aperture screen with maybe some visible fuzziness around the edges of a bright spot of 20 mm diameter. Yet, diffraction theory predicts for this case an illuminated central spot having a diameter of only 0.06 mm. What is wrong here? The problem is that the large-aperture example chosen does not satisfy the Fraunhofer requirement that the distance to the viewing plane must be much larger than the aperture diameter squared divided by the wavelength (18.39). Specifically, for this case, the viewing distance would need to be much greater than 700 m in order to see the proper Fraunhofer pattern from a 20 mm aperture. This aspect of the Fraunhofer approximation, which led us to the useful Fourier transform result, is easy to forget; and the forgetting can cause much trouble. This is a problem that is inherent in most treatments of subjects such as moiré interferometry and holography that involve diffraction by gratings of broad extent. The question then arises as to how one might create within the limits of a laboratory of finite size a diffraction pattern from a broad spatial signal such as a photograph. The answer is to use a lens that causes the nearly collimated diffracted beam to converge more quickly. This idea is studied next, as is its application in optical spatial filtering.

18.12 Fourier Optical Processing This section shows that a lens can be used to create the transform within the confines of the laboratory. Then, the use of the optical Fourier transform in modifying the frequency content of a picture is described.

18.12.1 The Transform Lens The Fraunhofer approximation that was imposed during the development of the diffraction integral implies

that the true optical Fourier transform is visible only far away from a small aperture. No problem appears when the transform of a tiny aperture is sought, as is the case with Young’s experiment. If, however, the aperture is broad, the transform appears, perhaps, several kilometers distant from the input plane. In the nearer field, the Fresnel equation applies, and the product is contaminated with extra exponential terms. Transforms of wide-aperture signals are often needed, for example, in holography and moiré analysis. The problem is to

Basics of Optics

aperture



=

T (ξ) e−i2π ( λF )ξ dξ . x

(18.50)

aperture

The result is the Fourier transform of the input signal, as before. Now, however, the transform appears in the Aperture with spatial signal T (ξ )

ξ

"Transform lens"

Fourier transform U(x) = F{T (ξ )} x

focal plane of the lens and so is under control. Also, the scaling factor is modified by the focal length of the lens. The spatial frequency metric has become f = x/λF instead of f = x2 /λz 2 as appeared in the Fraunhofer integral. That the input signal can be placed downstream from the lens is quite easy to demonstrate. A useful aspect of that development, which is not undertaken here, is that it is one instance where the Fresnel equation can be integrated. The effect of the lens cancels the extra term that appears in the Fresnel integral. The frequency metric then contains the distance from the signal to the viewing plane. Collimated incident light was chosen here to simplify the mathematics, but it is not required. The source can be located at a finite distance from the lens. In that case, the distance from the lens to the back focal plane is not equal to the focal length of the lens. It is found by locating the point of convergence of the light beam. Again, the frequency metric will be modified. Now that we know how to produce and locate the optical transforms of extended optical signals, two useful examples fix the ideas and begin to suggest some applications. Suppose that the input transparency contains parallel lines with the transmittance varying sinusoidally. The transform plane will exhibit only three bright dots. The center dot shows the strength of the DC component. The other two spots will be symmetrically arranged above and below the center, and the distance from the center to these dots is proportional to the spatial frequency of the sine wave in the signal grating. Now, replace the sine grating by a sharp barand-space grill. Sharp corners imply the presence of high-frequency components. The optical Fourier spectrum will be a row of dots symmetrically arranged from the center. The location of a dot gives the spatial frequency of the corresponding spectral component, and the brightness of the dot gives the relative amplitude of the component. These phenomena are illustrated in Fig. 18.22.

18.12.2 Optical Fourier Processing or Spatial Filtering

F

Fig. 18.31 Creation of an optical Fourier transform with the spatial signal in front of a lens

One practical application of the optical Fourier transform is realized by the addition of two more ideas. The first is that the spatial frequency content of the original input optical signal can be easily modified in the Fourier transform plane. The second is that another lens placed downstream may be used to perform a second transform, which is an inverse transform, to regenerate

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produce these transforms for broad apertures within the confines of the laboratory. The solution is to bring the far-field rendition of the optical Fourier transform closer to the aperture. This task is accomplished by use of a lens in one of many possible arrangements, a simple example of which is illustrated in Fig. 18.31. In this example, an expanded and collimated laser beam is made to pass through what has come to be called the transform lens, which converges the beam to a point that establishes the focal length of the lens. Place a viewing screen or detector array at this distance from the lens in what is called the back focal plane. Now, place the optical signal, probably in the form of a transparency or phase object into the system in front of the lens. The beam entering the lens now contains the aperture data, and the Fourier transform diffraction pattern will be seen on the screen. To see quickly how this works out mathematically, go back to the Fraunhofer integral (18.40). Ignore the multiplier preceding the integral, and take x1 /z 1 = 0 because the source is at infinity for collimated light as is used in this example. Viewing is in the back focal plane, so x2 is just x, and z 2 is the lens focal length = F. The integral simplifies to,  x T (ξ) e−ik( F )ξ dξ UP =

18.12 Fourier Optical Processing

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Input spatial signal

Optical fourier transform

Image of modified input signal

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Fig. 18.32 Schematic of a system for optical Fourier processing or spatial filtering

the original signal, now modified by having its spatial frequency content changed. Such a procedure is called spatial filtering, coherent optical data processing, or optical Fourier processing. Figure 18.32 shows one of many possible arrangements for implementation. An optical signal is placed in a collimated beam, as before, to generate its transform. Added to the system is an inverse transform lens that creates an image of the input plane on a viewing screen or detector array. Also added is a device, called the spatial filter, to block or modify portions of the Fourier transform. The inverse transform image is made of what is left. No additional theory is needed to understand this concept. In the system sketch, the input signal has sharp variations (corners) in its transmittance function. The spatial filter is a hole in a plate, meaning it blocks the high-frequency components. The inverse image is made with the low-frequency data, which implies that the corners on the input are lost.

18.12.3 Illustrative Thought Experiment An exaggerated thought experiment serves to illustrate the process and suggest its usefulness. Figure 18.33 represents the important data at different locations in the spatial filtering setup. Suppose that you have a photographic transparency of a person who is significant to you, but who, for some reason, could only be photographed through a substantial iron grillwork. You would be most happy to eliminate the grill from the picture. Place the photo in a Fourier filtering system. The transform will show a fuzzy bright ball that contains most of the picture in-

Diffraction pattern

Filter mask

Recovered image

Fig. 18.33 Imaginary example of using Fourier optical fil-

tering to remove unwanted information from a photograph

formation. The grillwork, because it is periodic, will yield four bright patches. The two spots on the vertical axis are from the horizontal bars, and the two on the horizontal axis are the data from the vertical bars. Place in the transform plane a transparent sheet that carries four black patches, and arrange these so they block the bright spots from the grillwork data. The inverse transform reconstructs the image, but the grillwork information has been deleted, so the bars no longer appear in the picture. The exaggeration in this example derives from the implication that the portions of the image that are occluded by the bars can be recovered. As usual, one cannot make something from nothing. However, if the bars are finer than shown and not too closely spaced, the filtering process will greatly improve the photograph. Some smoothing of the derived image will further improve it.

18.12.4 Some Applications Sophisticated applications of these ideas have been used to remove the raster scan lines from video images and space photographs. Enhancement of arial photographs for intelligence gathering is another possibility. Fourier processing is also useful for improving fringe visibility and reducing noise in moiré measurements. The sensitivity of moiré methods can be multiplied by selecting only the data contained in the high-frequency domain of the transform, both during the grating photography and during final data processing. Spatial filtering as described here has become less common than it once was. The Fourier transform, filtering, and image reconstruction are instead performed digitally. The analog process described here is, however, still useful in many instances. It also provides a paradigm that helps many of us understand what happens inside the computer.

Basics of Optics

References

This chapter was derived from articles by G.L. Cloud of the series Back to Basics – Optical Methods in Experimental Mechanics that have been published in the journal Experimental Techniques. These articles are all available at www.sem.org/PUBS-ArtDownloadOMTOC.htm. Note specifically those on diffraction, Parts 9–15 of the series.

• •

M. Born, E. Wolf: Principles of Optics, 5th edn. (Pergamon, New York 1975) J.W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York 1968)

• • • • •

R.E. Haskell: Introduction to Coherent Optics (Oakland University, Rochester 1971) G.L. Cloud: Diffraction Theory, Parts I–III, Exp. Tech. 28(2), 15–17; (4), 15–17; (5), 15–18 (2004) F. Jenkins, H. White: Fundamentals of Optics, 4th edn. (McGraw-Hill, New York 2001) P. Rastogi (Ed.): Photomechanics, Top. Appl. Phys. 77 (Springer, Berlin, Heidelberg 2000) F. Träger (Ed.): Springer Handbook of Lasers and Optics (Springer, Berlin, Heidelberg 2007)

References 18.1 18.2 18.3

G.L. Cloud: A classic interferometry: Newton’s rings, Exp. Tech. 27(1), 17–19 (2003) G.L. Cloud: Another classic interferometry: Young’s experiment, Exp. Tech. 27(2), 19–21 (2003) G.L. Cloud: Colored interferometry fringes, Exp. Tech. 27(3), 21–24 (2003)

18.4 18.5 18.6

G. Mesmer: Spannungsoptik (Springer, Berlin 1939), in German G.L. Cloud: Optical Methods of Engineering Analysis (Cambridge Univ. Press, New York 1993 and 1998) G.L. Cloud: The optical Doppler effect, Exp. Tech. 29(2), 17–19 (2005)

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Digital Image

19. Digital Image Processing for Optical Metrology

The basic principle of modern optical methods in experimental solid mechanics such as holographic interferometry, speckle metrology, fringe projection, and moiré techniques consists either of a specific structuring of the illumination of the object by incoherent projection of fringe patterns onto the surface under test or by coherent superposition (interference) of light fields representing different states of the object. A common property of the applied methods is that they produce fringe patterns as output. In these intensity fluctuations the quantities of interest such as coordinates, displacements, refractive index, and others are coded in the scale of the fringe period. Consequently one main task to be solved in processing can be defined as the conversion of the fringe pattern into a continuous phase map taking into account the quasisinusoidal character of the intensity distribution. The chapter starts with a discussion of some image processing basics. After that the main techniques for the quantitative evaluation of optical metrology data are presented. Here we start with the physical modeling of the image content and complete the chapter with a short introduction to the basics of digital holography, which is becoming increasingly important for optical imaging and metrology. Section 19.2 deals with the postprocessing of fringe patterns and phase distributions. Here the unwrapping and absolute phase problems as well as the transformation of phase

Look but don’t touch is an important rule for modern measurement and testing systems in industrial quality control. What is required are measurement and inspection techniques that are very fast, robust, and relatively low cost compared to the products being investigated. Modern full-field optical methods such as holographic interferometry, speckle metrology, moiré, and fringe

19.1 Basics of Digital Image Processing.......... 483 19.1.1 Components and Processing Steps .. 483 19.1.2 Basic Methods of Digital Image Processing............ 484 19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology 19.2.1 Intensity Models in Optical Metrology ..................... 19.2.2 Modeling of the Image Formation Process in Holographic Interferometry ....... 19.2.3 Computer Simulation of Holographic Fringe Patterns....... 19.2.4 Techniques for the Digital Reconstruction of Phase Distributions from Fringe Patterns .. 19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology ........ 19.3.1 The Technology of Optical Nondestructive Testing (ONDT) ........ 19.3.2 Direct and Indirect Problems ......... 19.3.3 Fault Detection in HNDT Using an Active Recognition Strategy .......

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References .................................................. 557 data into displacement values are addressed. Because image processing plays an important role in optical nondestructive testing, finally in Sect. 19.3 some modern approaches for automatic fault detection are described.

projection provide a promising alternative to conventional methods. The main advantages of these optical methods are their: noncontact nature, nondestructive and fieldwise working principle, fast response, high sensitivity and accuracy (typical displacement resolution of a few nanometers and strain values of 100 microstrain), high resolution of data points (e.g., 1000 × 1000 points

Part C 19

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Part C 19

for a sub-mm field of view), and the advanced performance of the systems, i. e., automatic analysis of the results and data preprocessing in order to meet the requirements of the underlying numerical or analytical model. Consequently, optical principles are increasingly being considered in all steps of the evolution of modern products. Relevant implementations cover such important fields as testing of lenses and mirrors with respect to optical surface forms, design optimization of components by stress analysis under operational load, on site investigation of industrial products with technical surfaces with the purpose of material fault detection and, last but not least, optical shape measurement as an important component in reverse engineering. The basic principle of the methods considered herein consists either in a specific structuring of the illumination of the object by incoherent projection of fringe patterns onto the surface under test or by coherent superposition (interference) of light fields representing different states of the object. A common property of the methods is that they produce a fringe pattern as output. In these intensity fluctuations the quantities of interest – coordinates, displacements, refractive index, and others – are coded in the scale of the fringe period. Using coherent methods this period is determined by the wavelength of the interfering light fields. In the case of incoherent fringe projection the spacing between two neighboring lines of the projected intensity distribution can be tuned very flexible by modern spatial light modulators in order to control the sensitivity and scale of the measurement. Consequently, the task to be solved in fringe analysis can be defined as the conversion of the fringe pattern into a continuous phase map, taking into account the quasisinusoidal character of the intensity distribution and the difficulties of the inverse problem. This aspect is especially addressed in Sects. 19.2 and 19.3. Techniques for the analysis of fringe patterns are as old as interferometric methods themselves, but until into the late 1980s routine analysis of fringe patterns for optical shop testing, experimental stress analysis, and nondestructive testing was mainly performed manually. One example may illustrate this highly subjective and time-consuming process: if only 1 min is estimated for the manual evaluation per measuring point, the total time necessary for the reconstruction of the threedimensional displacement field of a 32 × 32 mesh would take almost six working days because at least three holographic interferograms with 1024 grid points each have to be evaluated. Such a procedure must be qualified as highly ineffective with respect to the requirements

of modern industrial inspection, to say nothing of the limited reliability of the derived data. However, the step from a manual laborious technique only done by skilled technicians to a fully automatic procedure was strongly linked to the availability of modern computer technology. Consequently the development of automatic fringe pattern analysis closely followed the exponential growth in the power of digital computers with image processing capabilities. For several years now various commercial image processing systems with different efficiencies and price levels have been developed, and specialized systems dedicated to the solution of complex fringe analysis problems are now commercially available [19.1–4]. Advanced hardware and software technologies enable the use of desktop systems with several image memories and special video processors near to the optical setup. This online connection between digital image processors and optical test equipment opens completely new approaches for optical metrology and nondestructive testing such as real-time techniques with high industrial relevance. In this sense automatic fringe analysis has developed mainly during the 1980s into a subject in its own right. A number of international conferences and several comprehensive publications are devoted specifically to this subject [19.5–15]. Automatic fringe analysis covers a broad area today. It would be beyond the scope of this chapter to discuss all the relevant methods and trends. Consequently it concentrates mainly on the application of digital image processing techniques for the reconstruction of continuous phase fields from noisy fringe patterns. Optical and analogue electronic methods of fringe analysis are not considered; for this subject the reader is referred to [19.16–18]. The tasks to be solved with respect to the derivation of quantities of primary interest such as coordinates, displacements, vibrational amplitudes, and refractive index are described by using the example of three-dimensional shape and displacement measurement. Concerning qualitative fringe interpretation some modern trends are discussed only with respect to the computer-aided detection of material faults. This chapter starts with some basics of image processing. Afterwards the fundamental intensity relations in fringe-based optical metrology are described and a physical model of the image content that contains the relevant disturbances is derived. Such a modelbased approach is necessary since a priori knowledge of the image formation process is very useful for the development of efficient algorithms. Using the image model, fringe patterns for various metrological meth-

Digital Image Processing for Optical Metrology

tion is only the first step in fringe pattern analysis, the computation of vector displacements using digitally measured phases is described as an example. Image processing is also an important prerequisite for holographic nondestructive evaluation (HNDE). Therefore finally an overview of modern approaches in automatic fault detection based on knowledge-assisted and neural network techniques is given.

19.1 Basics of Digital Image Processing 19.1.1 Components and Processing Steps The elementary task of digital image processing and computer vision is to record images, and to improve and to convert them to new data structures with the objective of obtaining a better basis for the derivation and analysis of task dependent features. Figure 19.1 shows a typical scenario for an image processing system. Images of an object illuminated by a light source are recorded by an electronic camera. After conversion into digital values by a frame grabber, the analog electrical image signals from the camera are stored in a computer. The so obtained digital representation forms the starting point for the subsequent algorithmic processing steps, analyses, and conclusions, otherwise known as image processing and computer vision. Besides the knowledge of the theory of digital image processing, expertise in the fields of optics, image formation, lighting technology, signal processing, electrical engineering, electronics, and computer hardware and software is also necessary for the implementation of image processing tasks. The first steps toward finding a successful image processing solution are not in the selection and implementation of suitable algorithms but rather much earlier on, i. e., in the suitable acquisition of the images. Particular attention must be given to the illumination of a scene. A type of illumination well adapted to the problem produces images that can be analyzed using simple algorithmic methods, whereas badly illuminated scenes may produce images from which even the most refined algorithms are unable to extract the relevant features. It is also equally important to take the dynamic behavior of imaging sensors into consideration. If illumination is too bright, sensor elements may reach saturation and produce overilluminated images that could lead to false conclusions. As far as metrological tasks are concerned, it is essential to understand the imaging properties of imaging systems. For these

reasons, it is most important to consider all aspects of illumination, imaging, and sensing in the planning of the procedure. Some helpful basics are given in [19.19–23]. Image processing is not a one-step procedure. Dependent on the purpose of the evaluation a logical hierarchy of processing steps has to be implemented before the desired data can be extracted from the image. In general an image processing architecture consists of four main steps: acquisition, digitization, processing, and presentation. For the acquisition the appropriate type of illumination (incident light, transmitted light, dark field, bright field, wavelength, coherence, etc.) and sensing (wide angle, telecentric, spectral sensitivity, lateral resolution, dynamic range, etc.) has to be chosen. Digitization converts the image into a form that can be processed by computers. Digitization is a twofold procedure: the image is scanned line by line or column by column and samples of discrete data points, the so-called pixels (picture elements), are taken to be quantized. Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space) [19.20]. Because the sensor is a complex optical system containing the photoelectric detector chip

User Illumination

Object

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Fig. 19.1 The main components of an image processing system

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ods can be simulated and used for the development of adapted algorithms. Synthesized interferograms can be applied, for instance, to investigate the impact of different algorithms on different noise sources separately. The following chapter addresses the main techniques for quantitative phase reconstruction together with the most commonly used image processing methods. Because the reconstruction of the unwrapped phase distribu-

19.1 Basics of Digital Image Processing

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Image pre-processing - Image enhancement - Image restauration - Image coding

Iconic level (Pixels) Image → Image

Image processing

Part C 19.1

Image analysis

Symbolic level

- Feature & pattern recognition - Classification - Measurement

(Objects) Image → - Decisions - Objects - Measurement data

preprocessing is an improvement of the image data by suppressing unwanted disturbances (nonlinearities, noise, geometric distortions, etc.). Image preprocessing operations can be classified into three categories according to the size of the pixel neighborhood that is used for the calculation of its new property:

•

Fig. 19.2 Image processing hierarchy

and optical imaging components the lateral resolution of the sampled image depends not only on the number and density (number per area) of the sampling points of the detector but significantly on the lateral resolution of the entire optical system. Here the Abbe’s resolution limit [19.24] has to be considered. Besides the lateral resolution two further fundamental parameters are important for the characterization of a sensor system: the space–bandwidth product SPB [19.25–27] and the signal-to-noise ratio (SNR). Whereas the SPB distinguishes the information content of an optical image or the degrees of freedom of an optical system, the SNR is an engineering term for the description of the power ratio between a signal (meaningful information) and the background noise caused by different sources of the system. With the quantization the original continuous range of values (e.g., an arbitrary voltage between 0 and 5 V) is transformed to a discrete quantity (e.g., numbers between 0 and 255, i. e., 8 bits). For both procedures, the spatial and the value sampling, the sampling theorem [19.25] has to be taken into account. This important theorem states that the exact reconstruction of a continuous signal from its samples is possible if the signal is bandlimited and the sampling frequency is greater than twice the signal bandwidth. If the image data are available in digitized form, various algorithms dependent on the objective of the processing can be applied to the pixel matrix. In general the processing starts with some preprocessing and is followed by the analysis of the image data with respect to the desired features (Fig. 19.2).

19.1.2 Basic Methods of Digital Image Processing Preprocessing happens at the so-called iconic level of image processing where a pixel with certain properties such as an intensity and/or a color is transformed again into a pixel with altered properties. The aim of

•

•

pixel transformations [19.19–21] Based on a defined transformation rule, the brightness, color or position of the current pixel in the input image is changed in the output image: shading correction, binarization, histogram modification, and geometric transformations such as pixel coordinate transformations. local transformations [19.19–21, 28–30] A small neighborhood of the current pixel in the input image is used to generate a new gray value or color of the same pixel in the output image: linear smoothing operations such as averaging and low-pass filtering; nonlinear smoothing operations such as median and other rank-order operations; edge detection by gradient operators such as the Roberts, the Laplace, and the Sobel; morphological operations referring only to the relative ordering of pixel values such as erosion, dilatation, opening, and closing. global transformations [19.19–21, 31] A global operator affects the entire image. Consequently, the current pixel of the output image is affected by the entirety of pixels of the input image. In contrast to a local transformation a global transformation such as the Fourier transform, cosine transform, Hilbert transform, and Hough transform converts the image into a complete other form of representation. For instance the discrete Fourier transform transfers the original image as a discrete two-dimensional spatial function into its spectrum, i. e., into the spatial frequency domain.

The principal objective of image enhancement is to process a given image so that the resulting image is more suitable then the original image for a specific application. Consequently, image enhancement is in general problem oriented. In our case the main objective of image processing consists of the derivation of dimensional and mechanical quantities from a special class of images, so-called fringe patterns. Image restoration serves the same purpose as image enhancement. However, restoration is to be considered as a process that attempts to reconstruct an image that has been degraded by using some a priori knowledge of the degradation process.

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

of the phase from the fringe-like modulated intensity distribution or the recognition of characteristic fringe patterns. Whereas the phase represents the functional bridge between the observed intensity modulations and the mechanical quantity of interest, the detected and classified fringe pattern is the starting point for the derivation of quality assessments about the object under test. Both methods of quantitative and qualitative analysis of fringe patterns are discussed in the following sections.

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology 19.2.1 Intensity Models in Optical Metrology Fringe patterns generated by holographic interferometry, speckle metrology, structured illumination, and other metrological methods are nothing but the image of the object under test after a characteristic redistribution of the intensity. In the result of this process the original intensity across the object is modified by a harmonic function causing the bright and dark contrasts, known as fringes, by additive and multiplicative random noise, and other influences. The mathematical link between the observed intensity and the quantity to be measured is delivered by an intensity relation. For all techniques discussed herein the primary quantity to be measured is the argument of the harmonic function, which is called the phase of the fringe pattern. The connection between the reconstructed phase and the final quantities such as the displacements or coordinates is given by the socalled basic equations or geometric models. As a matter of principle, the reconstruction of the phase from the observed intensity leads to the solution of an inverse problem. The special features of that task are discussed in Sect. 19.2.4. An intensity model considering the main parameters of the image formation process is quite useful for the processing and analysis of fringe patterns. It is based on the optical laws of interference and imaging. Additionally, the model includes the various disturbances such as background illumination and noise which influence both the accuracy of the reconstructed phase distribution and the choice of the method to be used for its determination. Since it is quite impossible to separate the effects of different kinds of noise in a real image, a physical model describing the image formation process with all its components gives the opportunity to

compute artificial images which approximate step by step the complex structure of a real image. By means of this artificial test environment the performance of algorithms and image processing tools can be studied very effectively. Both error detection and selection of adapted parameters become more convenient. In this section the modeling and simulation is demonstrated specifically for the example of holographic interferograms. Since this image formation process is determined by the coherent superposition of diffusely scattered wavefronts, the resulting intensity distribution is complicated and differs considerably from the theoretically assumed cosine-shaped intensity modulation between bright and dark fringes of constant peak and valley values, respectively. The experience gained in the processing of holographic interferograms can be applied advantageously to other simpler types of fringe patterns such as, for instance, the fringes observed with the method of structured illumination. A general expression for the observed intensity I (x, y) and the phase difference δ(x, y) in holographic interferometry and speckle metrology was introduced by Stetson [19.32]: I (x, y) = I0 (x, y)|M[δ(x, y)]|2

(19.1)

with I0 (x, y) as the basic intensity and M(δ) as the characteristic function which modulates the basic intensity depending on the used registration technique and the object movement during the exposure time tB of the light-sensitive sensor used  tB 2 1  iδt (r,t)   e dt  . |M(δ)| =  tB 2

0

(19.2)

Part C 19.2

Therefore Sect. 19.2 will start with the discussion of a dedicated image model tailored to fringe patterns. On this basis an effective environment can be provided for the investigation and adaptation of different pre- and postprocessing methods. As already mentioned the purpose of image processing in optical metrology consists of the derivation of quantitative data or qualitative assessments from two-dimensional data sets. In this sense image analysis as the process of extraction of meaningful information from images means the reconstruction

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The result of this modulation is an object covered with interference fringes. To find an analytic expression for a concrete movement it is necessary to characterize its time response during tB . To this end the movement is indicated by a time-dependent displacement vector D(r, t) which satisfies a definite temporal function f (t): D(r, t) = d(r) f (t) , a) |M(δ)| 2

(19.3)

where r is a position vector and d(r) considers the vector amplitude of the movement of the measuring point P(r, t). In (19.3) was assumed that the movement can be separated into a time-dependent part of the movement of all points and their spatial position. Under this condition the time-dependent phase difference δt (r, t) can be written as δt (r, t) = δ(r) f (t) . b) |M(δ)| 2

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Fig. 19.3a–f Plot of characteristic functions in optical metrology: (a) double-exposure technique, (b) real-time technique, (c) time-average technique: constant velocity, (d) time-average technique: harmonic vibration, (e) real-time averaging: harmonic vibration, (f) real-time averaging: static displacement and harmonic vibration

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

Consequently for (19.2) it follows that M(δ) =

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(19.5)

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(19.6)

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δ = N2π .

(19.7)

3. Time-averaged technique (object moving with constant velocity). Continuous registration of a moving object and simultaneous reconstruction of all intermediate states yields |M(δ)|2 = sin c2 (δ/2),

sin cx = sin x/x . (19.8)

19.2.2 Modeling of the Image Formation Process in Holographic Interferometry The description and interpretation of an image observed in a holographic interferometer has to consider two classes of effects that influence the image formation process (Fig. 19.5):

• •

the coherent superposition of diffusely reflected wavefronts representing two states of an object with a rough surface and the optical imaging the influence of the optoelectronic processing system

4. Time-averaged technique (harmonic vibration). Therefore the effect of the imaging system consists Continuous registration of an vibrating and simultan- not only of the transformation of the object wavefront eous reconstruction of all intermediate states yields O(ξ, η) into an image A(x, y) but also of the degrada(19.9) tion of the original signal due to nonlinearities L of the |M(δ)|2 = J02 (δ/2) . imaging sensor and different noise components R(x, y). 5. Real-time averaging (harmonic vibration). Registra- Consequently the observed signal A (x, y) can be detion of one state on the sensor and superposition of scribed as its reconstruction with the current wavefront scattered by the vibrating and illuminated object yields (19.13) A (x, y) = L[A(x, y)] + R(x, y) , |M(δ)|2 = 1 − J0 (δ) .

(19.10)

6. Real-time averaging (superposition of static displacement and harmonic vibration). Registration of one state on the sensor and superposition of its reconstruction with the current wavefront scattered by the vibrating and displaced object yields |M(δ)|2 = 1 − cos(δ1 )J0 (δ2 ) .

(19.11)

7. Stroboscopic technique. Registration of two discrete states of a moving object by synchronization of the illumination and its movement yields |M(δ)|2 = cos2 (δ/2),

δ = N2π .

(19.12)

with A(x, y) = H[O(ξ, η)] .

(19.14)

At first the effect of coherent imaging will be investigated and the conditions of fringe pattern formation will be described. The various noise components are discussed later. If linearity and shift invariance can be assumed for the optical system, schematized in Fig. 19.6, the observed image A(x, y) can be interpreted as the convolution of the light O(ξ, η) scattered from the object with the impulse response h(r i − Mr 0 ) of the imaging

Part C 19.2

For the main registration techniques in optical metrology the following well-known characteristic functions can be derived:

Figure 19.3 shows the plots of all these characteristic functions, and Fig. 19.4 shows both a real and a synthetic holographic interferogram made with the double-exposure technique and their intensity profiles along the dotted line. With respect to the reconstruction of the phase distribution δ(r) from the measured intensity the special characteristic function M(δ) has to be taken into account consistently. However, in practice the measured intensity distribution differs due to noise and other influences considerably from these theoretically expected distributions (Fig. 19.4b).

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In optics the impulse response is also known as the point-spread function. For the coordinates in the image and in the object plane the symbols r i = (x, y) and r 0 = (ξ, η) are used, respectively. In comparison with classical interferometry the new quality in holographic

Digital Image Processing for Optical Metrology

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19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

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y

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Fig. 19.6 Schematic arrangement of an holographic inter-

ferometer with the light source Q (illumination direction eQ ), the object with the measuring point P and the displacement vector d, the hologram and the imaging system with the observation point B (observation direction eB )

interferometry consists of the coherent superposition of wavefronts scattered from diffusely reflecting surfaces: the laser light scattered at the rough surface introduces a granular structure known as the speckle effect. The statistical properties of the intensity in the images of coherent illuminated objects have been investigated by Lowenthal and Arsenault [19.33]. For a review of the statistical properties of laser speckle see Goodman [19.34]. Accordingly, the complex amplitude O(r 0 ) of the scattered light at the surface is described by ¯ 0 )ρ(r 0 ) , O(r 0 ) = O(r

(19.16)

¯ 0 ) is the medium amplitude distribution of the where O(r object illumination considering the macroscopic shape of the surface. The complex reflection coefficient ρ(r 0 ) represents the influence of the surface roughness with its random phase variations φ(r 0 ), which can be expressed as a phase function ρ(r 0 ) = ρ0 exp[iφ(r 0 )] ,

(19.17)

where ρ0 is a constant-amplitude factor that may be neglected in the following. The coherent superposition of the light fields coming from different points of the diffusely scattering surface results in a new light field that shows a speckled intensity distribution. Dändliker [19.35] has applied these results to the image formation process in holographic interferometry. Based on an analysis of the averaged interference term it can be shown that macroscopic interference fringes can be observed only if the two random phase distributions ρ1 (r 0 ) and ρ2 (r 0 ) of the both surfaces under

A01 (r i ) = a0 (r i ) exp[iϕ1 (r i )] = O(r 0 ) ⊗ h(r i − Mr 0 )

(19.18)

reference state: A02 (r i ) = A01 (r i + d i ) exp[iδ(r i )] = exp[iδ(r i )][O(r 0 ) ⊗ h(r i + d i − Mr 0 )] , (19.19)

where a0 is the amplitude of both wave fields and ϕ is the phase of the wavefront. In the case of two-beam interference for the intensity in the image point is I (r i ) = (A01 (r i ) + A02 (r i ))(A01 (r i ) + A02 (r i ))∗ = A01 (r i )A∗01 (r i ) + A02 (r i )A∗02 (r i ) + A01 (r i )A∗02 (r i ) + A∗01 (r i )A02 (r i ) , (19.20)

where ∗ denotes the complex conjugate. For lowresolution objects showing a slowly varying medium local intensity distribution, the first term on the righthand side can be defined as the incoherent image of the object in the first state I0 (r i ) = A0 (r i )A∗0 (r i ) ∞ ∞ ¯ 0 )|2 ρ(r 0 )ρ∗ (r 0 ) | O(r = −∞ −∞

¯ 0 )|2 . (19.21) × |h(r i − Mr 0 )|2 dξ dη = | O(r Since A01 and A02 differ only in a microscopic displacement their contributions to the intensity are equal with negligible differences. Referring to (19.18) and (19.19) the interference term is given by A01 (r i )A∗02 (r i ) ∞ ∞ ¯ 0 )|2 exp[iδ(r i )]ρ1 (r 0 )ρ2∗ (r 0 + d 0 ) | O(r = −∞ −∞

× h(r i − Mr 0 )h∗ (r i + d i − Mr 0 ) dξ dη

Part C 19.2

P'

Object

test are correlated. This means that the microstructure of the rough surfaces has to be identical. Consequently, only two states of the same object can be compared interferometrically – a condition that can be satisfied by holographic techniques only. For the formulation of an intensity model it is assumed that both object states differ only by a microscopic displacement d i = d(r i ) that causes a phase difference δ(r i ):

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Noncontact Methods

¯ 0 )|2 exp[iδ(r i )] = | O(r

∞ ∞

ρ1 (r 0 )ρ2∗ (r 0 + d 0 )

−∞ −∞

× h(r i − Mr 0 )h∗ (r i + d i − Mr 0 ) dξ dη = I0 (r i ) exp[iδ(r i )]Ch (d i ) . (19.22)

Part C 19.2

Consequently, with (19.20–19.22) for the intensity of the superposed object fields, one obtains I (r i ) = 2I0 (r i ) + I0 (r i )[exp(−iδ) + exp(iδ)]Ch (d i ) . (19.23)

Equation (19.23) shows that the contribution of the interference term depends essentially on the correlation of the imaged surface roughness Ch given by ∞ ∞ Ch (r i ) =

ρ1 (r 0 )ρ2∗ (r 0 + d i )h(r i − Mr 0 )

−∞ −∞ × h∗ (r i + d i

− Mr 0 ) dξ dη .

(19.24)

Since Ch defines also the so-called speckle size, i. e., the correlation length of the amplitude or intensity in the image [19.35], the interference fringes are only visible as long as the mutual shift of the speckle patterns is smaller than the speckle size. Equation (19.24) requires for nonvanishing interference that the two random phase distributions ρ1 (r 0 ) and ρ2 (r 0 + d i ) are correlated, i. e.,

I

ρ1 (r 0 )ρ2∗ (r 0 + d i ) = 0, at least for two mutual positions r 0 and (r 0 + d i ). Consequently, the two rough surfaces have to be microscopically identical, which means that the object can only be compared with itself and that the surface microstructure must not be changed between the two illuminations. Equation (19.23) can be written as I (r) = 2I0 (r)[1 + Ch (r) cos δ(r)] .

(19.25)

The speckles in the image of the coherently illuminated rough surface cause substantial intensity noise. This influence is considered by a multiplicative noise factor Rs (r). I (r) = 2I0 (r)[1 + Ch (r) cos δ(r)]Rs (r) .

(19.26)

The assumption of multiplicative speckle noise is correct as long as the medium intensity I0 (r) is constant over the speckle correlation length [19.36]. For objects without sharp edges and for sufficiently broad interference fringes this condition is ensured. Other noise components which have to be considered are mainly additive ones such as the time-dependent electronic noise RE (r, t) due to the random signal fluctuations within the electronic components of the photodetector and the image processing equipment [19.37]: I (r, t) = 2I0 (r)[1 + Ch (r) cos δ(r)]RS (r) + RE (r, t) . (19.27)

I

Laser beam

Signal

x

x I

Measured signal

x I

I

Speckles

x

Parasitic interferences

x

Fig. 19.7 The influence of different disturbances on the resulting intensity distribution in holographic interferometry

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

19.2.3 Computer Simulation of Holographic Fringe Patterns The basis for the computer simulation of fringe patterns is (19.27). To simplify the calculation the following assumptions are made:

• •

•

Nonlinearities of the sensor are neglected. This limitation is according to the behavior of modern charge-coupled device (CCD) cameras used in standard interferometric systems. Localization phenomena of the fringes caused by decorrelation of speckle fields [19.35] will not be considered. This assumption is based on the fact that the correlation can be assured by enlarging the speckle size with smaller apertures of the imaging system. However, this results in an increase of speckle noise, of course. Two coherent wave fields are assumed, which differ from each other only by a constant amplitude factor α and a phase difference δ(r).

The two interfering waves can then be expressed as A01 (r) = a1 (r) exp[iϕ1 (r)] , A02 (r) = α A01 (r) exp[iδ(r)] .

(19.28) (19.29)

Since the medium intensity within the image plane corresponds with the incoherent image I0 (r) the intensities of the two object waves can be written as I1 (r) = I0 (r)RS (r) ,

(19.30)

I2 (r) = α I0 (r)RS (r) .

(19.31)

2

For the intensity of the resulting fringe pattern it follows that I2 (r, t) = I0 (r)[1 + α2 + 2α cos δ(r)]RS (r) + RE (r, t) .

(19.32)

Using the definition of the fringe contrast V=

2α Imax − Imin = Imax + Imin 1 + α2

(19.33)

the modulation term in (19.32) can be calculated as √   α = 1/V 1 − 1 − V 2 for α ≤ 1 , (19.34) and (19.32) can be approximated by I (r, t) ≈ I0 (r)[1 + V cos δ(r)]RS (r) + RE (r, t) . (19.32a)

Consequently, for the computer simulation of fringe patterns the following functions have to be calculated:

• • • •

the distribution of the phase difference δ(r) the incoherent image of the illuminated object I0 (r) and the relation α2 between the two intensities the speckle noise RS (r) the electronic noise RE (r)

The calculation of the phase difference δ(r) starts with the modeling of the displacement field d(r) that the surface undergoes between the two states of double exposure (other registration techniques may be treated in a similar way). For this purpose analytical solutions such as the bending of beams and plates or finite element calculations can be used. Assuming an interferometer with defined observation and illumination directions eB (r) and eQ (r), respectively, as given in Fig. 19.6, the phase difference can be calculated pointwise using the well-known basic equation of holographic interferometry [19.39] δ(r) =

2π [eB (r) + eQ (r)]d(r) . λ

(19.35)

Part C 19.2

However, electronic noise is usually only relevant for small signal levels and plays a less important role than speckle noise in coherent optical metrology. A further quantity of influence that can cause considerable disturbances of the theoretically assumed cosine-shaped intensity modulation of the interference fringes is the varying background intensity I0 (r). This intensity variation is also called shading. It changes the peak-to-valley values of the intensity due to the Gaussian intensity profile of the laser beam and its imaging onto the surface, the spatially varying surface characteristics of the object under test, and parasitic interferences such as macroscopic diffraction patterns caused by dust particles on lenses and mirrors. The shading is highly worthy of note with respect to the digital processing of fringe patterns, see Sect. 19.2.4. In comparison to the speckle noise and to the intensity shading the distortions due to the digitization and quantization of the continuous signal are not as relevant. In most practical applications a quantization into 256 grey values and a digitization into an array of 1024 × 1024 pixels is quite sufficient to meet the necessary requirements for a precise phase reconstruction. For phase-shifting evaluation (Sect. 19.3) the resulting error because of quantization into 8 bits was estimated to be 3.59 × 10−4 wavelengths [19.38]. Figure 19.7 illustrates the effect of the main components that disturb the signal.

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Part C 19.2

For the approximation of the incoherent image of the object I0 (r) the projection of the expanded laser beam onto the surface has to be calculated taking into account the geometrical and optical conditions of the illumination system. Speckle noise can be simulated using a simplified model of the formation process of subjective speckle patterns [19.40, 41]. Speckles may be described by adding independent complex amplitudes to yield a circular Gaussian-distributed sum owing to the central limit theorem. A good approach is to follow this definition and to overlay Gaussian-distributed random complex amplitudes A(m, n) at pixel locations (m, n) within the diffraction spot of the imaging system  2   h(k − m, l − n)A(m, n) (19.36) IS (k, l) =  m,n

with A(m, n) = |A(m, n)| exp[iϕ(m, n)] .

(19.37)

The measured intensity is denoted by IS . The pointspread function h(i, j) can be simulated by a rect function. Figure 19.8a shows the result of such a simulation process with the corresponding intensity histogram (Fig. 19.8b). This probability density function is in good agreement with the theoretically expected negative exponential distribution [19.34]. In many practical circumstances, speckles are not resolved completely by the sensor. In these cases the spatial averaging on the target – which is comparable to low-pass filtering – results in a change of the statistics within the pattern. A comparison between Fig. 19.8c and Fig. 19.8e shows that a low-pass-filtered synthetic speckle pattern corresponds even better to a real speckle pattern captured by a CCD camera. Electronic noise RE (r) in photodetectors is a sum of numerous independent random processes obeying different statistical laws, but the central limit theorem of probability theory states that the overall process will be directed by a Gaussian distribution. For the simulation of Gaussian distributed electronic noise the following procedure to transform two uniformly distributed pseudorandom numbers xn and xn+1 into two independent pseudorandom numbers yn and yn+1 of a Gaussian sequence of random numbers can be applied:  yn = − ln(xn ) cos(2πxn+1 ) , (19.38)  yn+1 = − ln(xn ) sin(2πxn+1 ) . (19.39) The complete simulation process is demonstrated by using the example of a cylindrical shell loaded with inner

pressure in Fig. 19.9. The displacement field on the surface was calculated by using the finite element method.

19.2.4 Techniques for the Digital Reconstruction of Phase Distributions from Fringe Patterns Modern methods of optical metrology allow the measurement of the absolute shape as well as the deformation of loaded technical components with high precision over a wide range. The key quantity to be evaluated is the phase of the fringes. As the argument of the characteristic function M(δ) the phase δ carries all the necessary information about dimensional quantities and their derivations. Consequently, over the past 20 years many activities in optical metrology have been directed to the development of automatic techniques for the reconstruction of phases from fringe patterns [19.5–16]. In this section attention is only paid to those basic concepts where principles of digital image processing are relevant

• • • • •

phase-retrieval techniques [19.42] the fringe-tracking or skeleton method [19.43] the Fourier-transform method [19.44] the carrier-frequency method or spatial heterodyning [19.45] the phase-sampling or phase-shifting method [19.46]

All these methods have significant advantages and disadvantages, so the decision for a certain method depends mainly on the special measuring problem and the boundary conditions. For simplification our discussion of these methods is based on the following modification of (19.27): I (x, y, t) = a(x, y, t) + b(x, y) × cos[δ(x, y) + ϕ(x, y, t)] .

(19.40)

In (19.40) the variables a(x, y, t) and b(x, y) consider the additive and multiplicative disturbances, respectively, and ϕ(x, y, t) is an additionally introduced reference phase. The special character of ϕ(x, y, t) is an important feature for the distinction among the different phase-measuring techniques. Since optical sensors are only able to register the intensity I (x, y) as the amount of the square of the complex amplitude A(x, y), interferometric principles are necessary to impress the phase information on the intensity. Consequently, in the evaluation process the phase has to be reconstructed from the measured intensity distribution. The methods of phase retrieval belong in the narrower sense also to the interferometric phase-

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

measurement techniques. However, one should recall that the intensity distribution changes deterministically with the propagation of the light field. The mathematical a) Synthetic speckle pattern

background of the propagation is given by the Maxwell equations [19.47]. Hence, the image registered by the eye or the camera is the result of a complex interaction b) Intensity histogram

Part C 19.2

p (g) 5000

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d) Intensity histogram p (g) 5000

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e) Real speckle pattern

493

f) Intensity histogram p (g) 5000

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Fig. 19.8a–f Real and simulated speckle patterns: (a) synthetic speckle pattern, (b) intensity histogram, (c) spatially averaged synthetic speckle, (d) intensity histogram, (e) real speckle pattern, (f) intensity histogram

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Fig. 19.9a–d Simulation of the image formation process in holographic interferometry using the example of a cylindrical shell loaded with inner pressure: (a) background illumination, (b) interferogram, (c) interferogram with electronic noise, (d) interferogram with speckle noise

b) Interferogram

c) Interferogram with electronic noise

d) Interferogram with speckle noise

Part C 19.2

a) Background illumination

between the amplitude and the phase of the light field. With the propagation of the light field the interaction is changing and consequently a different intensity distribution is generated. The measurement of the intensity distribution in different planes and the knowledge about the light propagation allow the numerical reconstruction of the phase. Particularly the simplicity of the methods and the growing computing power available make this approach attractive for this application. Only a set of intensity images has to be registered with respect to the reconstruction of the phase distribution. However, the mathematical reconstruction requires sophisticated and iterative algorithms such as the Gerchberg–Saxton algorithm [19.42] which allow not always the determination of a unique and precise solution. The precision of interferometric techniques is unrivalled so far. Although phase reconstruction by fringe tracking is generally time consuming and suffers from the nontrivial problem of possible ambiguities resulting from the

loss of directional information in the fringe formation process, it is sometimes the only alternative for fringe evaluation. Its main advantages are that it works in almost each case and that it requires neither any additional equipment such as phase-shifting devices nor additional manipulations in the interference field. The Fourier-transform method (FTE) is applied to the interferogram without any manipulation during the interferometric measurement, too. The digitized intensity distribution is Fourier-transformed, leading to a symmetrical frequency distribution in the spatial frequency domain. After an unsymmetrical filtering including the regime around zero the frequency distribution is transformed by the inverse Fourier transformation, resulting in a complex-valued image. On the basis of this image the phase can be calculated with the arctan function. The disadvantage of the method is the need for individually adapted filters in the spatial frequency domain.

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

• • • • •

the spatial frequency distribution of the fringes over the total frame the fringe contrast the signal-to-noise-ratio (e.g., the speckle index [19.50] or the image fidelity [19.51]) the fringe continuity (analyzed, e.g., with the fringe direction map, see Smoothing of Speckle Noise the linearity of the intensity scale (e.g., saturation effects)

Based on this evaluation the fringe pattern can be accepted for further processing or can be rejected. In the case of rejection it is usually more effective to generate the fringe pattern again under improved experimental conditions than to expend disproportionate image processing effort. Another effect to be considered is due to the finite pixel size of optical sensors such

Preprocessing

Phase reconstruction

Postprocessing

Fig. 19.10 Sequence for the processing of interferograms with respect to the reconstruction of phase distributions

as CCD or complementary metal–oxide–semiconductor (CMOS) devices. The recorded intensities represent spatially averaged values of the intensity across the area of the pixel. To estimate the error to be expected for the reconstructed phase distribution a quality map can be generated that uses the local intensity distribution of the object wave [19.52]. The description of image processing techniques that are useful for the reconstruction of phase distributions from recorded interferograms follows the scheme shown in Fig. 19.10. Because the improvement of the signal-to-noise-ratio is essential with respect to the derivation of reliable phase data, our treatment starts with the discussion of useful preprocessing techniques. Afterwards the basic principles of the aforementioned phase reconstruction techniques are presented. Based on the example of the two major types of fringe evaluation:

• •

evaluation without fringe manipulation (fringe tracking) evaluation with fringe manipulation (temporal phase shifting or stepping)

a more detailed description of relevant image processing methods is given. Here some important postprocessing techniques are presented such as the segmentation and numbering of fringe patterns which are useful for fringe tracking and the unwrapping of mod 2π-phase distributions which is of certain importance for phase sampling. At the end of that chapter the problem of absolute phase measurement is discussed. Methods for Preprocessing of Fringe Patterns Smoothing of Time-Dependent Electronic Noise. The

electronic noise in photodetectors is recognized as a random fluctuation of the measured voltage or current and is caused by the quantum nature of matter. It is a sum of numerous random processes obeying different statistical laws, but the central limit theorem of probability theory states that the overall process will be directed by a Gaussian distribution. Because electronic noise is a time-dependent process its influence on the intensity distribution can be diminished by averaging over a sequence of frames. With respect to this averaging process it is of interest to consider what

Part C 19.2

The most accepted techniques, however, involve calculating the phase δ(x, y) at each point, either by shifting the fringes through known phase increments (the phase-sampling method) or by adding a substantial tilt to the wavefront, causing carrier fringes and Fourier transformation of the resulting pattern (the carrierfrequency method). Two types of phase-measurement interferometry (PMI) can be distinguished as temporal and spatial phase modulation techniques [19.6]. In the first case a temporal phase modulation is used [19.46, 48]. This can be done by stepping the reference phase with defined phase increments and measuring the intensity in consecutive frames (temporal phase stepping) or by integrating the intensity while the phase shift is linearly ramped (temporal phase shifting). In the second case the phase is shifted spatially by adding a substantial tilt to the wavefront (spatial heterodyning) [19.45] or by producing several spatially encoded phase-shifted fringe patterns (spatial phase stepping) by introducing at least three parallel channels into the interferometer, which simultaneously produce separate fringe patterns with the required phase shift [19.49]. In either case, the phase is calculated modulo 2π as the principal value. The result is a so-called sawtooth or wrapped phase image, and phase unwrapping has to be carried out to remove any 2π-phase discontinuities. With respect to the successful application of timeconsuming image processing algorithms to the unprocessed data or to the images that have already been improved by some preprocessing, it has been proved to be useful to assess the quality of the data before they are fed to the image processing system. Some parameters that should be evaluated to test the quality of the fringes are:

495

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Part C 19.2

happens when several Gaussian distributions are added together. This is particularly important in the area of signal averaging, where this property is used to increase the signal-to-noise ratio (SNR). For an intensity signal which consists of a steady signal value plus a random noise contribution with standard deviation σ the average standard deviation after the averaging of n frames is √ σ n (19.41) . σav = n The signal-to-noise ratio after averaging of n frames, superscript (n), in contrast to the signal-to-noise ratio without averaging, superscript (1), is as follows: √ (19.42) SNR(n) = nSNR(1) . Thus, if the number of frames is increased √ from n to (n + m), the SNR is improved by a factor (n + m)/(n). In practical applications of holographic interferometry the electronic noise plays a minor role in comparison to the speckle noise. However, in the case of electronic speckle correlation interferometry the correlogram typically results from the subtraction of two video frames. The amplitudes of the time-dependent noise can be observed in real time and can be in the range of the signal amplitude. Without time averaging the intensity distribution is so noisy that sometimes the skilled eye can observe fringes only. Here special attention has to be paid to electronic noise. As an example, Fig. 19.11 shows a correlogram of a centrally loaded circular disc without and after averaging of 20 frames. a) Without temporal averaging

Between the two images the SNR is increased by about 6 dB: √ n SNR[dB] = 10 lg √ = 10 lg n ≈ 6 dB . (19.43) n Smoothing of Speckle Noise. Fringe patterns obtained

from rough surfaces that are illuminated with coherent light are contaminated with a special kind of noise called speckle. Because of its role in the image formation process, see Sect. 19.2.3, and its function as the carrier of the information to be measured, speckle is unavoidable in coherent optical metrology. However, with respect to the reconstruction of the phase distribution from noisy fringe patterns, it is an unwanted part of the signal that needs to be eliminated. Since speckle is usually modeled as signal-dependent noise with a negative exponential probability distribution [19.34] and high contrast, the reduction of speckle noise needs more effort than for the suppression of other noise components. There have been two main approaches for reducing speckle noise. One approach is to prevent the creation of speckles by introducing incoherency through the sensor itself. A simple method is random spatial sampling using a rotating diffuser in front of the sensor and averaging a number of frames [19.53]. The second approach, which will be considered here, is directed to the processing of the speckled fringe patterns with respect to the reduction of the effects of speckle noise. Various methods for smoothing speckles have been proposed in the past. Since the speckles are noise and the carrier b) With averaging of 20 frames

Fig. 19.11a,b Speckle correlogram of a centrally loaded disc (a) without temporal averaging (b) with averaging of

20 frames

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

I (x, y) = I¯(x, y)RS (x, y) ,

(19.44)

it can be shown that the ensemble average of the n frames of I (x, y) is the maximum-likelihood estia) Rough interferogram

mate of I¯(x, y), which corresponds to the undegraded image [19.55]. If it is possible to receive multiple uncorrelated frames, this technique is a very effective way of improving the SNR in speckled images, and it is used for instance in high-altitude synthetic-aperture radar (SAR) processing. However, in optical metrology the registration of numerous uncorrelated frames is in general not possible. Spatial filtering of speckled images means that the signal processing is limited to a single frame. The standard method of spatial filtering is performed by averaging the intensity values of adjacent pixels. This process of linear low-pass filtering is described by the convolution of the intensity distribution I (x, y) with the b) Intensity plot Intensity 300 250 200 150 100

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Fig. 19.12a–d Smoothing of speckle noise. Application of the mean filter (one time, 3 × 3): (a) rough interferogram, (b) intensity plot, (c) filtered interferogram, (d) intensity plot

Part C 19.2

of the information as well, there is no ideal approach that operates effectively in all cases. Consequently, in general a pragmatic approach has to be chosen. Following Sadjadi [19.54], three general methods can be distinguished: temporal, spatial, and geometric filtering. In temporal filtering, multiple uncorrelated registered frames of the same scene with randomly changing speckle noise are needed. Using the intensity model described previously (Sect. 19.2.2)

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Noncontact Methods

a) Rough holographic interferogram

b) Intensity plot

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Part C 19.2

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200 150 100 50 0 0

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impulse response or convolution kernel h(x, y) of the filter, according to I  (x, y) = I (x, y) ⊗ h(x, y)  = I (x, y)h(x − a, y − b) . a

(19.45)

b

Two types of low-pass filters are commonly used: the box-kernel and the Gaussian-kernel, respectively, given by ⎛ ⎞ ⎛ ⎞ 1 1 1 1 2 1 1 ⎜ ⎜ ⎟ ⎟ h Gauss = h box = ⎝1 1 1⎠ , ⎝ 2 4 2⎠ . 16 1 1 1 1 2 1 (19.46)

If such (3 × 3)-kernels are applied several times to the same image, filters with larger window dimensions are achieved. A very effective method is the connection of elementary (2 × 2)- or (3 × 3)-kernels to a filter cascade [19.5] with (2n × 2n ) and (3n × 3n ) windows, respectively. The implementation of linear low-pass filters is easy and they run very fast on general-purpose processors, but they always result in image blurring which adversely affects the subsequent segmentation procedure. In the case of fringe patterns with high fringe density a specially designed mean filter gives better results [19.56]. This filter compares for all pixels the spatial average M of the grey values in the kernel with the grey value g of the central pixel. The grey value g is only replaced by M if the following rule is

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

499

e) Fringe direction map

Part C 19.2

f) Directionally filtered interferogram

g) Intensity plot

200 150 100 50 0 0

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Fig. 19.13 (cont.)

fulfilled: if |M − g| > T, otherwise g := g

then

g := M, (19.47)

with T as a threshold. This averaging process is usually repeated two or four times. The parameter T ensures that fine structures are preserved while noise is suppressed. Figure 19.12 shows the original and mean filtered holographic interferogram of a centrally loaded plate with the corresponding grey value profile. Investigations made by Guenther et al. [19.57] have shown that nonlinear filters such as square-root and squaring filters give no improvements over linear spatial average filters. Another nonlinear filter that is well known for reducing the so-called salt-and-pepper noise in images is the median filter. This filter belongs to the

class of rank filters [19.58] and has been used successfully for speckle suppression. It operates by moving a rectangular window across the degraded image, replacing the intensity value of the central pixel with the median of the values of all the pixels in the window. The median filter is effective in removing speckle noise and does not blur the fringes as much as the spatial average filter. The disadvantage of median filters is the increased processing time compared to simple low-pass filters. Some more time-effective implementations are described in [19.59, 60]. Methods of image restoration such as Wiener filtering [19.5, 61] can also be applied successfully for noise reduction, provided that some knowledge of the spectral power density of the noise and the signal is available and an additive noise model is assumed. With a logarithmic transformation the multiplicative model

500

Part C

Noncontact Methods

a) Rough interferogram

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Part C 19.2

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Digital Image Processing for Optical Metrology

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19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

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Part C 19.2

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Fig. 19.14a–l The application of low-pass and geometric filtering to a holographic interferogram and a speckle shearogram (the intensity profile is taken from a horizontal line through the middle of the image shearogram): (a) rough interferogram, (b) intensity plot, (c) low-pass-filtered interferogram, (d) intensity plot, (e) geometrically filtered interferogram, (f) intensity plot, (g) speckle shearogram, (h) intensity plot, (i) spatially averaged interferogram, (j) intensity plot, (k) geometrically filtered shearogram, (l) intensity plot  

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a) Original interferogram

b) Intensity plot

Part C 19.2

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Fig. 19.15a–d The application of windowed Fourier transform to a holographic interferogram for fringe smoothing: (a) original interferogram, (b) intensity plot, (c) filtered interferogram, (d) intensity plot

can be converted into an additive model. This technique is known as homomorphic filtering, where log I (x, y) = log I¯(x, y) + log RS (x, y) ,

(19.48)

and has been frequently used for the removal of multiplicative noise [19.62, 63]. Once the noise becomes additive, any standard technique for removing additive noise can be applied. Jain and Christensen [19.55] have compared low-pass, median, spatial average, and Wiener filters in the logarithmic space on a set of test patterns and concluded that the Wiener filter performed slightly better than the rest of the filters. However, these investigations did not show conclusive evidence of superior performances in using homomorphic over conventional filtering.

Directional averaging is useful to protect the fringes from blurring while smoothing. Some directional filters adapted for fringe patterns have been proposed and applied to preprocess holographic interferograms [19.64, 65]. These filters search for local fringe tangent directions, and a so-called fringe direction map is constructed. This direction map contains the relevant fringe directions within the interference patterns. Based on this map the filter mask of the averaging or median filter is controlled in such a way that the pattern is only filtered in the direction of the fringe flow. In Fig. 19.13 a holographic interferogram with high fringe density in the central region is filtered with and without directional control of the filter. The fringe blurring of the directional smoothed pat-

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

I 255

x

Fig. 19.16 Gaussian-modulated intensity distribution with varying fringe amplitude and varying background (bold line: modulated intensity, continuous line: normalized intensity, dashed lines: upper and lower hull)

tern is not as strong as in the case of conventional filtering. A completely different approach for speckle filtering was proposed by Crimmins [19.50, 66]. This so-called geometric filter is more effective in speckle noise suppression than is frame and spatial averaging, without disturbing the fringe pattern adversely [19.40]. The image is viewed as a three-dimensional surface where speckles appear as narrow tall towers, but the signal components to be cleaned from noise appear as relatively broad-based, short towers. The filter applies an iterative convex hull algorithm (the convex hull is the smallest region which contains the object, such that any a) Interferogram

b) Interferogram after binarization (level 128)

Fig. 19.17a,b Binarization of a shaded holographic interferogram. (a) Interferogram (b) interferogram after binarization

(level 128)

Part C 19.2

256

two points of the region can be connected by a straight line, all points of which belong to the region [19.20]), in which the narrow and tall towers are cut down more quickly and effectively. The result is a practical elimination of noise while the fringe pattern is preserved. To compare the performance of the geometric filter with the frequently used spatial low-pass filters, Fig. 19.14 shows the result of speckle smoothing on example of a holographic interferogram (Fig. 19.14a–f) and a speckle shearogram (Fig. 19.14g–l). The different performances of these filter types with respect to blurring effects is visible in the center of the circular fringes. Here the geometric filter delivers the best results. Recently two-dimensional continuous windowed Fourier transforms (WFT) have been applied for filtering different types of fringe patterns [19.67, 68]. The method is very similar to Fourier-transform denoising. The main difference between the two types of transform is the transform basis. The Fourier base is global and orthogonal, while the windowed Fourier base is local and redundant. When the signal and noise are overlapping in spectrum, it is impossible to separate them properly by Fourier-transform denoising. However, since windowed Fourier filtering is localized in both the spatial and frequency domains, it is still possible to separate the signal and noise in the spatial domain even through they have overlapping spectra. The drawback of windowed Fourier transform is the lower resolution of the spectrum. In general, the windowed Fourier transform performs better than the usual Fourier transform in

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a) Background intensity (hologram)

b) Intensity plot Intensity 250 200

Part C 19.2

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Fig. 19.18a–f Shading correction of fringe patterns by background division. (a) Background intensity (hologram) (b) intensity plot (c) interferogram (d) intensity plot (e) normalized interferogram (f) intensity plot

speckle denoising, especially in wrapped phase map filtering. Figure 19.15 shows the same interferogram with high density fringes as already applied in Fig. 19.13 and Fig. 19.14. The result of filtering by using the WFT (Fig. 19.15c) clearly shows less blurring than in the case of applying the simple low-pass filter (Fig. 19.14c,d). Further approaches for speckle filtering are given in [19.11, 69–71]. Shading Correction. The shading correction or normal-

ization of the fringe pattern is an important preprocessing step in the case of intensity-based analysis methods such as fringe tracking or skeletonizing (Sect. 19.2.4). Referring to the intensity model (19.32a) the fringe pattern is still disturbed by local varying fringe amplitudes I1 (x, y) and background modulation I0 (x, y) after electronic and speckle noise filtering I (x, y) = I0 (x, y)[1 + V (x, y) cos δ(x, y)] = I0 (x, y) + I1 (x, y) cos δ(x, y) .

(19.49)

These disturbances are mainly caused by variations of the illumination due to the intensity distribution of the laser beam, inhomogeneities within the intensity distribution of the illumination beam (e.g., diffraction patterns caused by dust particles on lenses or mirrors and distortions due to the projection of the intensity distribution on the object surface), the varying reflectivity of the surface under test, and speckle decorrelations.

Figure 19.16 gives an impression of the influence of shading on the fringe pattern for a centered Gaussian intensity distribution. Simple binarization with fixed thresholds followed by skeletonization will not work in this case (Fig. 19.17). Furthermore it is important to notice that the variation of I1 (x, y) and I0 (x, y) causes a deviation of the fringe peaks from their real positions without shading. In fringe-tracking or skeleton techniques where the fringe centerlines are reconstructed first, this systematic phase error has to be taken into account. It shifts the skeleton lines slightly away from the loci where the phase is an integer multiple of π. A careful investigation of the behavior of the phase error is carried out in [19.72] and a general method for removing or reducing the error is proposed. An important conclusion of these investigations is that the systematic phase error caused by variation of I1 (x, y) and I0 (x, y) oscillates from zero in the middle between bright and dark fringes to its extreme values at the fringe centerlines for every fringe, and this error is not accumulated over the whole field. In other words, the systematic phase errors are relevant only to the local fringe configuration, e.g., the local fringe density, the local gradient of the intensity background, and the local fringe amplitude. Several algorithms have been developed to normalize fringe patterns [19.5, 60, 72–75]. Assuming a multiplicative superposition of the background with the fringe modulation term, the shading correction with

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

c) Interferogram

505

d) Intensity plot Intensity 300 250

Part C 19.2

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Fig. 19.18 (cont.)

background division is the simplest approach: I N (x, y) =

I (x, y) 128 (256 grey levels) . I0 (x, y) (19.50)

To reconstruct the background I0 (x, y) the incoherent image (Fig. 19.18a) of the object or an approximation such as the low-pass or median-filtered fringe pattern can be used [19.58]. With the same assumptions as before the method of homomorphic filtering also delivers good results. Here the Fourier transform is applied to the logarithm of the intensity distribution and the low-frequency components of the spatial frequency spectrum are filtered out using an appropriate function such as the inverted Gaussian low-pass filter. However, both methods remove

the systematic phase error only in the special situation where both the background and the fringe amplitude coincide. A method that approximates the upper and lower hull (Fig. 19.16) is more effective in general. In [19.72, 75], step-like envelopes Imax and Imin are constructed by connecting the local maxima and minima, respectively, of the fringe pattern. With these envelopes a fringe pattern I N (x, y) with constant background I0c and constant amplitude I1c is derived using a twodimensional (2-D) envelope transform Imax (x, y) = I0 (x, y) + I1 (x, y) , Imin (x, y) = I0 (x, y) − I1 (x, y) , [I (x, y) − Imin (x, y)] I N (x, y) = A +B [Imax (x, y) − Imin (x, y)]

Noncontact Methods

= A/2 + B + A/2 cos δ(x, y) = I0c + I1c cos δ(x, y) .

(19.51)

Methods for Digital Phase Reconstruction Phase-Retrieval Technique. If the amplitude and the

phase of a monochromatic wavefront are known at a certain plane, it is possible to calculate the wavefront at a given distance from that plane. The physical basis of that procedure is the propagation of electromagnetic waves described by the Helmholtz equation [19.47]. This allows one, e.g., to focus or defocus an image simply by computer simulation. Modern computers make it possible to run such processes almost in real time. The remaining problem is how to get the complex amplitude of a wave field. Because optoelectronic sensors

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Part C 19.2

The parameter A is a constant equal to twice the fringe amplitude after the transform, i. e., A = 2I1c , and B is a constant equal to the minimum of the intensity after the transform, i. e., B = I0c − I1c . Yu et al. [19.75] improved this method by least-squares fitting the 2-D envelopes from fringe skeletons. Using these nearly correct envelopes for the normalization of the fringe pattern (Fig. 19.19) the systematic phase error is almost completely removed [19.72]. After the fringe pattern is cleaned by removing random noise and the shading correction is performed, various processing methods can be implemented and

applied more easily with respect to the reconstruction of the continuous phase distribution.

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Part C

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Fig. 19.19a,b Two-dimensional-envelope transformed holographic interferogram with nearly correct envelopes. (a) Original interferogram with intensity plot (b) normalized interferogram with intensity plot

Digital Image Processing for Optical Metrology

a)

Δz

Object

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

Intensity patterns recorded at different planes

507

a) Experimental arrangement Diffraction pattern

Light source Object

b)

I0

I1 Phase mask

In

Fig. 19.20 (a) Recording arrangement for phase retrieval, (b) reconstructions of a diffusely transmitting object at

b) Retrieved object amplitude

c) Retrieved object phase

2 mm interval reconstruction planes using multiple intensity measurements. The width of the whole object is 2 mm

such as CCD or CMOS devices are sensitive only to the intensity, the phase information is lost in the recording process. One way to obtain the complex amplitude of a wavefront is to interfere it with a reference wave and to use a detector to record the interference produced by that two waves. Both interferometry and holography are based on that principle. However, during the last 15 years there has been an impressive development of techniques in which holograms are recorded on electronic detectors and digitally reconstructed. This method is known as digital holography [19.76–79] and we will return to this subject later. Configuring a separate reference beam from a single laser source, however, entails additional optical components and reduces the available laser power. Furthermore, configuring and recording the interference pattern formed by the superposition of the object beam and the reference beam involve a tedious and sometimes cumbersome process of optimizing the beam ratio and fringe spacing. Beam-propagation methods that avoid the reference beam are commonly referred to as phase-retrieval methods. They have been utilized in order to reconstruct amplitude and phase from intensity-only patterns. Compared with interferometric approaches the resulting experimental setup is simpler and more robust to external influences. As early as 1972 Gerchberg and Saxton [19.42] proposed an iterative method which allowed the phase information to be obtained if the intensity was known in a certain plane and additionally some information about the object wavefront in

Fig. 19.21a–c Wavefront-modulation-based phase-retrieval approach using a pixelated phase mask. (a) Experimental arrangement (b) retrieved object amplitude (c) retrieved object phase

another plane was given (e.g., a pure-amplitude object or a pure-phase object is investigated). Recently it was shown that, by recording two or more intensity patterns of the object in different planes (Fig. 19.20a) and by application of iterative algorithms [19.80], it was possible to increase the quality of the reconstructed wavefronts considerably. This technique is very promising because it is based on a straightforward procedure. No paraxial approximation is assumed. Consequently, wavefronts with large numerical aperture can be reconstructed. Figure 19.20b shows the experimental results for a diffusely transmitting object. The test object used is a transmitting mask with a ground-glass diffuser attached to it to effect phase randomization. After calculation of the wavefront (amplitude and phase) in the measurement planes, numerical focusing is obtained when propagated back to the image space. The convergence and robustness of the applied iterative methods can be significantly improved if a strong phase modulation is additionally introduced into the object field [19.81]. Thus, in the recorded diffraction pattern, any sampling will have contributions from all points in the object. Distributing information of one object point to all pixels of the sensor minimizes the ef-

Part C 19.2

z0

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Part C 19.2

fect of sensor noise and also eliminates the stagnation. Figure 19.21a illustrates the experimental arrangement. A pixelated phase plate, mounted on a linear stage, is positioned downstream of an extended object. The object can be of transmission or reflection type. Figure 19.21b,c show the results of an experiment with a phase plate consisting of 1200 × 1400 pixels having

a size of 8 μm2 . The object field was generated by illuminating a binary photographic slide with a spherical wavefront emerging from a single-mode fiber. Figure 19.21b,c show the recovered amplitude and phase, respectively, at the object plane after 80 iterations from five recordings. Instead of the static phase plate a spatial light modulator can be adopted.

a) Interferogram

b) Segmented fringe pattern

c) Fringe direction map

d) Improved segmented pattern

e) Skeleton

f) Pseudo-3-D-plot of the phase field

Fig. 19.22a–f Phase reconstruction by skeletonization. (a) Interferogram (b) segmented fringe pattern (c) fringe direction map (d) improved segmented pattern (e) skeleton (f) pseudo-3D-plot of the phase field

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

Fringe-Tracking or Skeleton Method. This method

a) Interferogram

tions. Some of them are tailored to a specific type of fringe pattern and thus may require substantial modifications when applied to other types of fringe patterns. Others fail on interferograms that have sinusoidal intensity modulation or are inefficient if the shape and the frequency of the fringes vary within the pattern. An efficient segmentation algorithm [19.56] that extracts line structures very rapidly from interferograms is described in a special section dedicated to the segmentation problem. A general processing scheme for digitally recorded and stored fringe patterns consists of the following steps [19.5]:

• • • • • • •

b) Mod 2π phase map after applying the RPT technique (phase estimation is wrapped for the purpose of illustration)

improvement of the signal-to-noise-ratio in the fringe pattern by spatial and temporal filtering specification of the region of interest to be analyzed extraction of the raw skeleton by tracking of intensity extrema or pattern segmentation in combination with skeletonization enhancement of the skeleton by linking interrupted lines, removal of artifacts, and adding missing lines (this procedure can be implemented very comfortably by following some simple rules [19.3, 85]) numbering of the fringes with corresponding order numbers reconstruction of the continuous phase distribution by interpolation between skeleton lines [19.86] calculation of the quantity to be measured using the phase values

An example of the fringe-tracking method based on skeletonization is presented in Fig. 19.22. Another powerful approach to reconstruct the phase from a single fringe pattern is the regularized phasetracking technique (RPT) [19.87]. The technique is based on the assumption that the fringe pattern may be considered locally as spatially monochromatic. Consequently, it can be modeled as a cosinusoidal function (19.40), in which the phase term is assumed to be smooth and can be approximated by a plane. The basic function for the RPT technique is defined as [19.88]  Uxy (φ0 , ωx , ω y ) =

Fig. 19.23a,b Phase reconstruction with the RPT technique. (a) Interferogram (b) mod 2π phase map after applying the RPT technique (phase estimation is wrapped for the purpose of illustration)

(ε,η)∈(L xy ∩N)

⎞ {I  (ε, η) − cos[φe (x, y, ε, η)]}2 ⎟ ⎜ × ⎝+{I  (ε, η) − cos[φe (x, y, ε, η)] + α}2 ⎠ (19.52) +λ[φ0 (ε, η) − φe (x, y, ε, η)]2 m(ε, η) ⎛

with the local phase φ0 and the local frequencies ωx , ω y that are estimated in the coordinates (x, y). N is the two-dimensional lattice that has valid fringe data, L xy

Part C 19.2

is based on the assumption that the local extrema of the measured intensity distribution correspond to the maxima and minima of a 2π-periodic function, given by (19.40). The automatic identification of these intensity extrema and the tracking of fringes is perhaps the most obvious approach to fringe pattern analysis since this method is focused on reproducing the manual fringe-counting process. Various techniques for tracking fringe extrema in two-dimensional implementations are known [19.43, 56, 82–86]. They are all based on several pattern segmentation methods that involve fast Fourier transform (FFT), Wavelet transform, adaptive and floating thresholding, and either gradient operators or the piecewise approximation of elementary func-

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Part C 19.2

is a neighborhood region around (x, y) with coordinates (ε, η), m(ε, η) is the indicator field that equals 1 if the site (ε, η) has already been estimated and 0 otherwise, λ is the regularizing parameter that controls (along with the size of L xy ) the smoothness of the detected phase, and α is a constant value used to constrain the phase estimation to a smooth solution. The function φe (x, y, ε, η) = φ0 (x, y) + ωx (x, y)(x − ε) + ω y (x, y)(y − η)

(19.53)

is the plane model used for the phase estimation [19.88]. The fringe pattern I  (ε, η) is a high-pass-filtered version of the fringe pattern given in (19.40), i.e., I  = cos φ. To estimate the phase field using the RPT technique, it is necessary to minimize the basic function (19.52) at each point (x, y) with respect to φ0 , ωx , and ω y . The resultant phase estimation is continuous. The main drawback of the conventional RPT technique is the necessity for fringe pattern normalization, i.e., that I  = cos φ is satisfied (see Shading Correction in Sect. 19.2.4). This means that any deviation from normalized fringes will generate errors in the phase estimation. In [19.89] a further term is introduced into (19.52) that considers the fringe pattern modulation. Based on this approach nonnormalized patterns can be processed successfully. Figure 19.23 shows the result of the processing of a holographic interferogram with the improved RPT technique [19.89]. Fourier-Transform Method. This method is based

on fitting a linear combination of harmonic spatial functions to the measured intensity distribution I (x, y) [19.44, 45, 90]. The admissible spatial frequencies of these harmonic functions are defined by the user via the cutoff frequencies of a bandpass filter in the spatial frequency domain. Neglecting the time dependency and avoiding a reference phase, (19.40) is transformed to I (x, y) = a(x, y) + c(x, y) + c∗ (x, y)

(19.54)

with the substitution c(x, y) = 1/2 b(x, y) eiδ(x,y) .

(19.55)

Here the symbol ∗ denotes the complex conjugation. A two-dimensional Fourier transformation of (19.54) gives I (u, v) = A(u, v) + C(u, v) + C ∗ (u, v)

(19.56)

with (u, v) being the spatial frequencies and A, C, and C ∗ the complex Fourier amplitudes.

Since I (x, y) is a real-valued function, I (u, v) is a Hermitian distribution in the spatial frequency domain: I (u, v) = I ∗ (−u, −v) ,

(19.57)

The real part of I (u, v) is even and the imaginary part is odd. Consequently the amplitude spectrum |I (u, v)| is symmetric with respect to the direct-current (DC) term I (0, 0). Referring to (19.56), A(u, v) represents this zero peak and the low-frequency components which originate from background modulation I0 (x, y). C(u, v) and C ∗ (u, v) carry the same information, as is evident from (19.57). Using an adapted bandpass filter the unwanted additive disturbances A(x, y) can be eliminated together with the mode C(u, v) or C ∗ (u, v). If, for instance, only the mode C(u, v) is preserved, the amplitude spectrum is no longer Hermitian and the inverse Fourier transform returns a complex-valued c(x, y). The phase δ(x, y) can then be calculated with δ(x, y) = arctan

Im c(x, y) . Re c(x, y)

(19.58)

Taking into account the sign of the numerator and the denominator, the principal value of the arctan function having a continuous period of 2π is reconstructed. As a result a mod 2π-wrapped phase profile – the so-called sawtooth map – is received and phase unwrapping is necessary (see Unwrapping of Mod 2π-Phase Distributions in Sect. 19.2.4). The drawback of this method is that the sign ambiguity of holographic interferometry due to the cosine-shaped intensity modulation remains [19.91]: cos(δ) = cos(sδ + N2π), s ∈ {−1, 1} , N integer .

(19.59)

This uncertainty is preserved in (19.57). In the simple Fourier-transform method the distinction between increasing or decreasing phases requires additional knowledge about the displacement of the object. To overcome that problem, Kreis [19.44, 92] has proposed to apply two phase-shifted reconstructions or to make a twofold evaluation with differently oriented bandpass filters. The general processing scheme for the Fourier transform method consists of the following steps:

• •

Fourier transformation of the rough fringe pattern masking the amplitude spectrum by using a bandpass filter to suppress the zero term and one part of the spectrum

Digital Image Processing for Optical Metrology

•

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

applying the inverse Fourier transformation to return a complex-valued function and calculating the phase mod 2π (the so-called sawtooth image)

•

511

unwrapping the sawtooth image to reconstruct the continuous phase distribution

b) Fourier spectrum

c) Filter mask

d) Reconstructed mod 2π fringe pattern

e) Demodulated phase distribution

f) Phase plot

Part C 19.2

a) Interferogram

Fig. 19.24a–f Fourier transform method on example of a centrally loaded circular plate. (a) Interferogram (b) Fourier spectrum (c) filter mask (d) reconstructed mod 2π fringe pattern (e) demodulated phase distribution (f) phase plot

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Noncontact Methods

b) Interferogram with carrier

c) Fourier spectrum

d) Shifted and masked spectrum

e) Wrapped phase

f) Unwrapped phase

g) Fourier spectrum

h) Shifted and masked spectrum

i) Unwrapped phase

Part C 19.2

a) Simulated interferogram

Fig. 19.25a–i Carrier-frequency method applied to a synthetic interferogram. (a) Simulated interferogram (b) interferogram with carrier (c) Fourier spectrum (d) shifted and masked spectrum (e) wrapped phase (f) unwrapped phase (g) Fourier spectrum (h) shifted and masked spectrum (i) unwrapped phase

The complete processing of the centrally loaded plate is shown in Fig. 19.24.

with a fixed spatial carrier frequency f 0 . The recorded intensity distribution is given by

Carrier-Frequency Method. This method [19.45, 93] is based on the introduction of a substantial tilt to the wavefront by tilting the reference wavefront using the reference mirror or by an artificial tilt introduced, e.g., in the computer [19.94]. The measurement parameter is encoded as a deviation from straightness in the fringes of the pattern. Without loss of generality it can be assumed that the carrier fringes are parallel to the y-axis

I (x, y) = a(x, y) + b(x, y) cos[δ(x, y) + 2π f 0 x] = a(x, y) + c(x, y) exp(2πi f 0 ) (19.60) + c∗ (x, y) exp(−2πi f 0 ) , with c(x, y) defined by (19.55). Referring to (19.40), this method can be classified as a spatial phase-shifting technique. Takeda

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

b) Interferogram with spatial carrier

c) Fourier spectrum

d) Masked and shifted fourier spectrum

e) Wrapped phase distribution

f) Unwrapped phase plot

Part C 19.2

a) Interferogram

513

Fig. 19.26a–f Carrier-frequency method using the example of a centrally loaded plate. (a) Interferogram (b) interferogram with spatial carrier (c) Fourier spectrum (d) masked and shifted Fourier spectrum (e) wrapped phase distribution (f) unwrapped phase plot

et al. [19.45] use the FFT algorithm to separate the phase δ(x, y) from the reference phase ϕ(x, y) = 2π f 0 x. To this purpose (19.60) can be

rewritten with the substitution (19.55). After a onedimensional FFT with respect to x and taking into account the Fourier shift theorem, the following

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Noncontact Methods

Table 19.1 Solutions of (19.68) for the cases m = 3 and m = 4 m 3

ϕ0

δ

2π/3

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4

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δ = arctan

√

I3 −I2 3 2I −I 1 2 −I3

Part C 19.2

I2 −I4 I3 −I1

I0 ≈ I0 ≈

holds: I (u, v) = A(u, y) + C(u − f 0 , y) + C ∗ (u + f 0 , y) . (19.61)

Since the spatial variations of a(x, y), b(x, y), and δ(x, y) are slow compared to the spatial carrier frequency f 0 , the Fourier spectra A, C, and C ∗ are well separated by the carrier frequency f 0 , as shown for the example of a simulated interferogram in Fig. 19.25. C and C ∗ are placed symmetrically with respect to the DC term and are centered around u = f 0 and u = − f 0 . The following procedure makes use of either of the two spectra on the carrier. By means of digital filtering, e.g., the sideband C(u − f 0 , y) is filtered and translated by f 0 towards the origin of the frequency axis, to remove the carrier frequency. C ∗ and the term A(u, y) are eliminated by bandpass filters. Consequently C(u, y) is obtained, as shown in Fig. 19.25d. The inverse FFT of C(u, y) returns c(x, y) and the phase can be calculated mod 2π using (19.58). Because the phase is wrapped into the range from −π to π it has to be demodulated by using a phase-unwrapping algorithm. The general processing scheme for the Fourier transform method consists of the following steps:

• • • • •

Fourier transformation of the rough fringe pattern masking the amplitude spectrum by using a bandpass filter to suppress the DC term and one part of the spectrum shifting the relevant part of the spectrum towards the origin of the frequency axis applying the inverse Fourier Transformation to return a complex-valued function and calculating the phase mod 2π (the sawtooth image), unwrapping the sawtooth image to reconstruct the continuous phase distribution

The complete processing of a natural interferogram (a centrally loaded circular plate) is shown in Fig. 19.26. A processing approach that is equivalent to this frequency-domain processing (the Fourier transform

V 1 3 1 4

3 



Ii

V≈

Ii

V≈

i=1 4  i=1



1 (2I −I −I )2 −(I −I )2 2 4 3 1 2√ 3 3I0 (I1 −I3 )2 −(I2 −I4 )2 2I0

method) can be performed in the space signal domain. From (19.61), the spectrum passed by the filter function H(u − f 0 , y) can be written as C(u − f 0 , y) = H(u − f 0 , y)I (u, y) .

(19.62)

Taking the inverse Fourier transform of (19.62), the equivalent processing in the space domain is represented by c(x, y) exp(2πi f 0 x) = [h(x, y) exp(2πi f 0 x)] ⊗ I (x, y) , (19.63) where ⊗ denotes the convolution operation, and h(x, y) is the impulse response defined by the inverse Fourier transform of H(u, y). The sinusoidal convolution of (19.61) was proposed by Mertz [19.95], and its relation to the frequency-domain analysis was described by Womack [19.94]. Sinusoid fitting techniques in which the interferogram is approximated by a pure sinusoid were used by Macy [19.96] and were later improved by Ransom and Kokal [19.97]. Phase-Sampling Method. Phase-sampling or phase-

shifting interferometry is based on the reconstruction of the phase δ(x, y) by sampling a number of fringe patterns differing from each other by various values of a discrete phase ϕ. If ϕ is shifted, for instance, temporally in n steps of ϕ0 , then n intensity values In (x, y) are measured for each point in the fringe pattern: In (x, y) = a(x, y) + b(x, y) cos[δ(x, y) + ϕn ] , (19.64)

with ϕn = (n − 1)ϕ0 , n = 1 . . . m, m ≥ 3, and, e.g., ϕ0 = 2π/m. In general only three intensity measurements are required to calculate the three unknown components in (19.64): a(x, y), b(x, y), and δ(x, y). However, with m > 3 better accuracy can be ensured using a leastsquares fitting technique [19.46, 98, 99]. If the reference phase values ϕn are equidistantly distributed over one or a number of periods, the orthogonality relations of the trigonometric functions provide a useful simplification.

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

Equation (19.40) is first rewritten in the form

I (90°)

515

I (270°)

I (x, y) = K (x, y) + L(x, y) cos ϕ + M(x, y) sin ϕ , (19.65)

–

where

I (180°)

(19.66)

It can be shown on the basis of a least-squares fit that L and M satisfy the following equations in analytical form [19.5]:

I (0°)

–

m 2  In (x, y) cos ϕn , L= m n=1

m 2  In (x, y) sin ϕn . M= m

(19.67)

n=1

A combination of (19.66) and (19.67) delivers the basic equation for the phase-sampling method, where the minus sign is neglected because of the ambiguity of the sign in interferometry, m 

δ (x, y) = arctan

M n=1 = arctan m  L

In (x, y) sin ϕn

Fig. 19.27 Processing scheme in phase-shifting interferometry with

. In (x, y) cos ϕn

n=1

(19.68)

Equation (19.68) is sufficient to determine the phase mod π. In order to compute the phase modulo 2π, the sign of the numerator and denominator must be examined. As result a mod 2π wrapped phase distribution δ (x, y) is measured. The unwrapping or demodulation of this wrapped signal delivers the continuous phase field δ(x, y) = δ (x, y) + N(x, y)2π ,

(19.69)

where N is the integer fringe number. In the same way, expressions for the variables a(x, y) and b(x, y), which give indications for the background intensity I0 (x, y) ≈ a(x, y) and the fringe contrast V (x, y) ≈ b(x, y)/a(x, y), can be derived as a(x, y) = b(x, y) =

1 m 

m 

In (x, y) ,

There are many of solutions for (19.68) [19.48,100]. For m = 3 and m = 4 two well-known solutions are listed in Table 19.1. An example with m = 4 is shown in Fig. 19.27. Another three-frame technique for a π/2-phase shift was published by Wyant [19.101]: m = 3 : ϕ0 = π/2,

(19.70)

δ = arctan

I3 − I2 . I1 − I2

(19.71)

The general processing scheme consists of the following steps

• • •

n=1

L 2 + M2 .

four phase- shifted fringe patterns (ϕ = 90◦ )

Preprocessing: specification of the region of interest to be analyzed and improvement of the signal-tonoise-ratio by spatial filtering Sampling: calculation of the mod 2π phase image according to the equations given in Table 19.1 or (19.71), (19.73), and (19.74) Improvement of the sawtooth image: by using special filters and masking the relevant regions to be unwrapped using the inconsistency check

Part C 19.2

K (x, y) = a(x, y) , L(x, y) = b(x, y) cos δ(x, y) , M(x, y) = −b(x, y) sin δ(x, y) .

516

Part C

Noncontact Methods

• •

Unwrapping: demodulation of the improved sawtooth image to reconstruct the continuous phase distribution Representation: plotting the continuous phase map as a pseudo-three-dimensional (3-D) plot

Part C 19.2

The solutions above require the calibration of the phase-shifter device to ensure a defined amount of phase shift. The technique presented by Carré [19.102] as early as 1966 and improved by Jüptner et al. [19.103] is independent of the amount of phase shift. The solution is based on the fundamental equation (19.64) for the intensity distribution with an unknown value of ϕ0 . In this case at least four interferograms are needed to solve the equation system for the four unknown quantities. The main variables of interest are ϕ0 (x, y) and δ(x, y), given by I1 − I2 + I3 − I4 (19.72) , 2(I2 − I3 ) δ = arctan(I1 − 2I2 + I3 + (I1 − I3 ) cos ϕ0  + 2(I2 − I1 ) cos2 ϕ0 ) 1 − cos2 ϕ0 (I1 − I3 −1 + 2(I2 − I1 ) cos ϕ0 ) . (19.73)

ϕ0 = arccos

The additional phase shift ϕ0 (x, y) is calculated as a function of the point P(x, y). This allows control over the phase shifter as well as the reliability of the evaluation, which might be disturbed by noise. A detailed description of the error sources in PMI taking into account such influences as inaccuracies of the reference phase values, disturbances due to

extraneous fringes, coherent noise, and high spatial frequency noise caused by dust particles is given by Schwider et al. [19.104]. Based on this work Hariharan et al. [19.105] published another five-frame technique which uses π/2 phase shifts to minimize phase-shifter calibration errors: m = 5 : ϕ = −π, −π/2, 0, π/2, π , 2(I2 − I4 ) . δ = arctan 2I3 − I5 − I1

(19.74)

Temporal phase shifting is another method of introducing an additional, known phase change. In this case the intensity is integrated while the phase shift is linearly ramped between (ϕn − Δϕ/2) and (ϕn + Δϕ/2). One frame of integrated recorded intensity data can be written as [19.106] 1 In (x, y) = Δϕ

ϕn +Δϕ/2 

{a(x, y) + b(x, y) cos[δ(x, y) ϕn −Δϕ/2

+ ϕn (t)]} dϕ(t) ,

(19.75)

where ϕn denotes the average value of the phase shift for the n-th intensity measurement. The evaluation of that integral gives In (x, y) = a(x, y) + sinc(Δϕ/2)b(x, y) × cos[δ(x, y) + ϕn ] .

(19.76)

Equation (19.76) shows that the only difference between temporal phase stepping and shifting is a reduction in the fringe visibility by the factor sinc(Δϕn /2).

d Beam splitter Object

CCD Object wave

Laser

Reference wave Lens

Fig. 19.28 Schematic setup for

recording a digital hologram onto a CCD target

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

η

y

ne

ne

la

tp

c bje

O

u' (x', y') y'

x

Digital Holography. The transformation of phase

changes into recordable intensity changes is the basic principle of holography. Because of the high spatial frequency of these intensity fluctuations the registration of a hologram requires a light-sensitive medium with adequate spatial resolution. Therefore special photographic emulsions have dominated holographic technologies for a long period. However, the recording of holograms on electronic sensors and their numerical reconstruction is almost as old as holography itself. First ideas and implementations emerged as early as the 1960s and 1970s [19.111–113]. However, only in the 1990s did the progress in high-resolution camera technology and computer hardware open a real chance to record holograms directly on the CCD target of a camera and to reconstruct the wavefront in a reasonable time [19.114]. However, digital holography is much more than an elegant recording technique. In contrast to conventional approaches digital holography allows the direct calculation of both parts of the complex wavefront, phase and intensity, by the numerical solution of the diffraction problem in the computer. Several advantages for the measurement process result from that new quality. Besides the electronic processing and the direct access to the phase some further benefits recommend digital holography for the solution of numerous imaging, inspection, and measurement problems. Obvious advantages are, for instance, the possibility to correct aberrations numerically, the elegant way in which different object states can be stored independently, the considerable reduction in the number of holograms necessary for quantitative displacement or shape analysis, and the drastic reduction of the size of holographic sensors. In digital speckle-pattern interferometry (DSPI) [19.115] the object is focused onto the target of an electronic sensor. Thus an image-plane hologram is formed as result of the interference with an inline reference wave. In contrast to DSPI, a digital hologram is recorded without imaging. The sensor target records the superposition of the reference and the object wave in the near-field region – a so-called Fresnel hologram [19.116]. The basic optical setup in digital holography for recording holograms is the same as in

h (ξ, η)

u (x, y)

am gr o l o

x'

ne

ge ma

H d

ξ

pla

pla

I

z

d

Fig. 19.29 Schematic setup for Fresnel holography

conventional holography (Fig. 19.28). A laser beam is divided into two coherent beams. One illuminates the object and causes the object wave. The other enters the target directly and forms the reference wave. On this basis very compact solutions are possible. For the description of the principle of digital Fresnel holography a schematized version of the optical setup is used (Fig. 19.29). The object is modeled by a rough plane surface that is located in the (x, y)-plane and illuminated by laser light. The scattered wave field forms the object wave u(x, y). The target of an electronic sensor (e.g., a CCD or CMOS device) used for recording the hologram is located in the (ξ, η)-plane at a distance d from the object. Following the basic principles of holography the hologram h(ξ, η) originates from the interference of the object wave u(ξ, η) and the reference wave r(ξ, η) in the (ξ, η)-plane: h(ξ, η) = |u(ξ, η) + r(ξ, η)|2 = rr ∗ + ru ∗ + ur ∗ + uu ∗ .

(19.77)

The transformation of the intensity distribution into a gray value distribution that is stored in the image ξ u (x, y)

r (x, y)

β

z

Fig. 19.30 Interference between the object and reference wave in the hologram plane

Part C 19.2

With respect to the evaluation both methods are equivalent. A general survey of phase-shifting methods combined with extensive simulations concerning the influence and compensation of disturbances is given by Creath [19.107, 108]. In the papers published by Huntley [19.109] and Surrel [19.110] further information about phase-shifting algorithms can be found.

517

518

Part C

Noncontact Methods

memory of the computer is considered by a characteristic function t of the sensor. This function is in general only approximately linear T = t[h(ξ, η)] .

(19.78)

Part C 19.2

Because the sensor has a limited spatial resolution, the spatial frequencies of the interference fringes in the hologram plane – the so-called microinterferences – have to be considered. The fringe spacing g and the spatial frequency f x , respectively, are determined by the angle β between the object and the reference wave (Fig. 19.30) λ 1 = (19.79) , g= fx 2 sin(β/2) where λ is the wavelength. If we assume that the discrete sensor has a pixel pitch (the distance between adjacent pixels) Δξ the sampling theorem requires at least two pixels per fringe for a correct reconstruction of the periodic function, thus 1 . (19.80) 2Δξ < fx Consequently, we obtain for small angles β: λ β< (19.81) . 2Δξ ξ

a)

u  (x  , y ) = t[h(ξ, η)]c(ξ, η) .

x

P

(19.82)

The hologram t[h(ξη)] diffracts the wave c(ξη) in such a way, that images of the object wave are reconstructed. In general, four terms are reconstructed if the wave u  (x  , y ) propagates in space. An assumed linear characteristic function t(h) = αh + t0 delivers

y' η

r (η,ξ)

Modern high-resolution CCD or CMOS chips have a pitch Δξ of about 4 μm. In that case a maximum angle between the reference and the object wave of only 4◦ is acceptable. One practical consequence of the restricted angle resolution in digital holography is a limitation of the effective object size that can be recorded holographically by an electronic sensor. However, this is only a technical handicap. Larger objects can be placed at a sufficient distance from the hologram or reduced optically by imaging with a negative lens [19.100]. A reduction of the object resolution has to be accepted in this case. Another consequence is the restricted application of off-axis setups that are used to avoid overlapping reconstructions. Therefore inline arrangements are often applied. However, with the lensless Fourier recording setup these problem becomes less serious [19.117]. The reconstruction is done by illumination of the hologram with a so-called reconstruction wave c(ξ, η)

u  = Tc = t(h)c

(19.83)

u  = α(cu 2 + cr 2 + cur ∗ + cru ∗ ) + ct0

(19.84)

and

with two relevant image terms (cur ∗ ) and (cru ∗ ) containing the object wave and its conjugated version, P' respectively. The appearance of the image terms depends on the concrete shape of the reconstruction Image d' Hologram wave c. In general the reference wave c = r or its conplane jugated version c = r ∗ is applied. In the case of the ξ y' b) conjugated reference wave a direct or real image will be reconstructed due to a converging image wave that x' η h (η,ξ) can be imaged on a screen at the place of the original P object. o' Θ However, in digital holography the reconstruction of ρ the object wave in the image plane u  (x  , y ) is done P' by numerical simulation of the physical process, as u' (x', y') shown in Fig. 19.31a. The reconstruction wave with a well-known shape equal to the reference wave r(ξ, η) propagates through the hologram h(ξ, η). Following the Fig. 19.31a,b Reconstruction of a digital hologram. (a) Prin- Huygens principle each point P(ξ, η) on the hologram ciple of wavefront reconstruction (b) light propagation by acts as the origin of a spherical elementary wave. The diffraction (Huygens–Fresnel principle) intensity of these elementary wave is modulated by o

o'

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

−∞...∞

exp(ikρ) × cos θ dξ dη ρ

with ρ(ξ − x  , η − y ) =

 d 2 + (ξ − x  )2 + (η − y )2 (19.86)

O  (x  , y , z

= d)

as the distance between a point in the image plane and a point P(ξ, η, z = 0) in the hologram plane and 2π (19.87) k= λ as the wavenumber. The obliquity factor cos θ represents the cosine of the angle between the outward normal and the vector joining P to O  (Fig. 19.31b). This term is given exactly by d (19.88) cos θ = ρ and therefore (19.85) can be rewritten as  d    u (x , y ) = t[h(ξ, η)]r(ξ, η) iλ

(19.85)

a) Object

b) Hologram

c) Reconstructed intensity I (x',y')

d) Reconstructed phase φ(x',y')

−∞...∞

exp(ikρ) × dξ dη . ρ2

(19.89)

Fig. 19.32a–d Reconstructed intensity and phase of a digital Fresnel hologram. (a) Object (b) hologram (c) reconstructed intensity I (x  , y ) (d) reconstructed phase φ(x  , y )

Part C 19.2

the transparency h(ξ, η). At a given distance d  = d from the hologram a sharp real image of the object can be reconstructed as the superposition of all elementary waves. For the reconstruction of a virtual image d  = −d is used. Consequently, the calculation of the wave field u  (x  , y ) in the image plane starts with the pointwise multiplication of the stored and transformed intensity values t[h(ξ, η)] with a numerical model of the reference wave r(ξ, η). In the case of a normally incident and monochromatic wave of amplitude 1 the reference wave can be modeled by r(ξ, η) = 1. After the multiplication the resulting field in the hologram plane is propagated in free space. At a distance d  the diffracted field u  (x  , y ) can be found by solving the Rayleigh– Sommerfeld diffraction formula, also known as the Huygens–Fresnel principle [19.25]:  1    t[h(ξ, η)]r(ξ, η) u (x , y ) = iλ

519

520

Part C

Noncontact Methods

The numerical reconstruction delivers the complex amplitude of the wavefront. Consequently, the phase distribution φ(x  , y ) and the intensity I (x  , y ) can be calculated directly from the reconstructed complex

function u  (x  , y ) as φ(x  , y ) = arctan

Im|u  (x  , y )| [−π, π] , Re|u  (x  , y )|

b) Pseudo-3-D transformed intensity field

c) Fringe direction map

d) Segmented image

e) Skeleton

f) Interpolated phase samples

Part C 19.2

a) Interferogram

(19.90)

Fig. 19.33a–f Segmented and skeletonized holographic fringe pattern. (a) Interferogram (b) pseudo-3-D transformed intensity field (c) fringe direction map (d) segmented image (e) skeleton (f) interpolated phase samples

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

α = 0° α = 22.5°

Fig. 19.34 Comparison of the grey values between the cen-

tal and the neighboring pixel

I (x  , y ) = u  (x  , y )u ∗ (x  , y ) .

(19.91)

Figure 19.32 shows the results of the numerical reconstruction of the intensity and the phase of a chess knight recorded with a setup as shown in Fig. 19.28. The direct approach to the phase yields several advantages for imaging and metrology applications that are addressed in many publications (see, e.g., [19.100, 118, 119]). Methods for Postprocessing of Fringe Patterns The objective of postprocessing of fringe patterns is the reconstruction of the continuous phase distribution using intensities (grey values) and phase values, respectively, both taken from the preprocessed images. In the following sections those postprocessing problems are discussed that are significant for the fringe-tracking and the phase-sampling method. This means segmentation and fringe numbering for the fringe-tracking method and phase unwrapping for the phase-sampling method.

ΔR (i, j) = [g(i, j) − g(i + R, j + R)], g(i, j) : grey value of the pixel (i, j)

(19.92)

between the current pixel and the eight neighboring ones that are centrosymmetrically located around (i, j) at a distance R is calculated (Fig. 19.34). These differences are compared with a predefined threshold T . To each such neighborhood relation, a neighborhood relation byte is assigned separately to possible ridge and valley candidates using the following rule:

• •

if the ΔR (i, j) is positive and greater than T , then the bit position of the assigned neighbor point in the ridge-candidate relation byte is set to 1; if ΔR (i, j) is negative but absolutely greater than T , then the bit position of the assigned neighbor point in the valley-candidate relation byte is set to 1.

Two hundred and fifty-six codes of neighborhood relations can be derived. Owing to the isotropic character of an interferogram, these 256 configurations can be

Segmentation of Fringe Patterns. Segmentation is

an important processing step in the fringe tracking method (see Fringe Tracking or Skeleton Method in Sect. 19.2.4). The fringe pattern is considered as an array of two-dimensional contour fringes with fixed contour intervals. For cosine-shaped fringe patterns the phase increment between two consecutive contour lines corresponds to the number π. After the processing of the fringe pattern all bright and dark contour fringes are represented by their central lines – the so-called skeleton. To extract these skeleton lines the image has to be segmented into regions with distinguishable properties – the so-called ridges, slopes, and valleys in the grey value mountain (Fig. 19.33b). A direct segmenta-

cos2 x

x

Fig. 19.35 Cosinusoidal intensity modulation function

Part C 19.2

R=3

tion approach is binarization of the preprocessed pattern and its subsequent skeletonization [19.84]. However, for most real fringe patterns it is difficult to determine a global threshold level that delivers good results (Fig. 19.17). Consequently, the intensity distribution has to be normalized by shading correction or adaptive and local thresholds [19.120]. Another segmentation method that works in many cases was developed by Yu et al. [19.121]. This so called 2-D-derivative-signbinary fringe method is based on the fringe direction map and operates without thresholds. In the case of fringe patterns with obvious varying background intensity and fringe amplitude an adapted segmentation algorithms that considers relevant fringe structures delivers good results [19.56]. For each pixel (i, j) the grey value difference ΔR (i, j)

521

522

Part C

Noncontact Methods

Part C 19.2

reduced to 36 rotational invariant patterns. Based on experimental investigations ten significant patterns were selected from these 36 prototypes and proved to be representative for ridge and valley candidates, respectively. The possibility to choose various distances R between the central point and its neighboring points allows the use of this method for a relatively wide range of different fringe patterns. To avoid gaps between neighboring pixels on the discrete plane the same procedure is repeated after rotation around a fitting angle α. In the case of interferograms with varying fringe density the results after the application of different distances R can be merged. Figure 19.33a shows an interferogram with varying fringe density and background and its pseudo-3-D intensity profile (Fig. 19.33b). The segmented image was improved by region growing and binary filtering considering the fringe direction map (Fig. 19.33c,d). Finally, the skeleton (Fig. 19.33e) is derived by line thinning and the continuous phase distribution (Fig. 19.28f) is calculated by interpolation between the numbered skeleton lines.

values of δ(x, y) define the fringe loci on the object surface and can be extracted with intensity-based analysis methods such as skeletonization or temporal and spatial phase-measurement methods (Sect. 19.2.4). However, the reconstruction of the phase δ(x, y) from the intensity distribution I (x, y) by a kind of inversion of (19.40) raises the problem that the cosine is not a one-to-one function, but is even and periodic (Fig. 19.35): cos(δ) = cos(sδ + N2π),

s ∈ {−1, 1},

N ∈Z. (19.93)

Fig. 19.36 Determination of the absolute phase at the point

Consequently the phase distribution reconstructed from a single intensity distribution remains indefinite to an additive integer multiple of 2π and to the sign s. All inversions of expressions such as (19.93) yield an inverse trigonometric function. However, all inverse trigonometric functions are expressed by the arctan function and this function has its principal value between −π/2 and +π/2. If the argument of the arctan function is assembled by a quotient as in the case of phase-measuring interferometry, where the numerator characterizes the sine of the argument δ and the denominator corresponds to the cosine of δ, then the principal value is determined consistently in the interval [−π, +π]. However, a modulo 2π uncertainty as well as the sign ambiguity still remains. This uncertainty is called the fringe-counting problem and has important practical consequences [19.122]: When the phase angle δ extends beyond 2π to N2π – where N denotes the fringe number of the point Px to be measured – the absolute phase value N2π can only be reconstructed if N can be determined unambiguously. Using intensity-based methods this is performed by counting the bright and dark fringes or their skeleton lines along a path starting from a reference point PR with known fringe order NR until the point Px (Fig. 19.36). The resulting relative fringe order between PR and Px consists of an integer part N˜ and a remaining fraction Nˆ = l/D with the quantity D as the line spacing between two bright fringes. For simplification PR is often chosen to be placed in the middle of a bright fringe. In that case NR is an integer and considers the number of fringes between the reference point PR and a point P0 with zero fringe order (such a point should not be contained in the image). Consequently, the absolute phase of the point Px can be written ˆ δ(Px ) = N2π = [NR + N˜ + N]2π . (19.94)

Px under investigation based on the example of the simulated interferogram of a bent beam

If spatial or temporal phase measurement methods are used, the remaining fraction Nˆ can be calculated from

The Numbering of Fringe Patterns. The Fringe-

Counting Problem In optical shape and displacement analysis the fringes to be analyzed show in general a sinusoidal variation in intensity. The evaluation of the acquired intensity distribution across the object with respect to the reconstruction of the primary quantity to be measured – the phase δ(x, y) – and finally the displacement, refractive index, and shape distribution, respectively, is a nontrivial process. As a good approximation (19.40) can be used for the description of the intensity distribution I (x, y) observed in a fringe pattern. Here the intensity is described as a function of the phase angle δ(x, y). The objective in quantitative fringe processing is to extract the phase distribution δ(x, y) from the more or less disturbed intensity distribution I (x, y). Constant D P0

PR

I Px

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

523

a) Opitcal setup for fringe projection

Part C 19.2

III IV I

II

b) Observed fringe pattern

V

c) Intensity profile along a line Grey value 250 200 150 100 50 0 0

50

100

150

200

250 Pixel

Fig. 19.37a–c Outline of the optical setup and classification of different regions for the evaluation of projected fringes. (a) Optical setup for fringe projection (b) observed fringe pattern (c) intensity profile along a line

the directly measured phase δ by solving an equation of the form 1  (19.95) δ (Px ) , Nˆ = 2π where δ is the result of an evaluation according to (19.68). However, this method only gives a direct approach to the principal value of the phase as well. This principal value δ (Px ) corresponds to the fraction of the absolute fringe order Nˆ ˆ δ (Px ) = N2π . (19.96) Consequently, in practice the fringe-counting problem remains since the integer part of N has to be provided by methods known as phase unwrapping. Robinson [19.123] has given an excellent definition of this procedure: “Phase unwrapping is the process by which the absolute value of the phase angle of a continuous function that extends over a range of more than 2π (relative to a predefined starting point) is recovered. This absolute value is lost when the phase term

is wrapped upon itself with a repeat distance of 2π due to the fundamental sinusoidal nature of the wave functions used in the measurement of physical properties.” The removal of all 2π-phase discontinuities to obtain the unwrapped result ˜ y)]2π (19.97) δ(x, y) = δ (x, y) + [NR (x, y) + N(x, is one of the major difficulties and limitations of the phase-sampling procedure. The key to phase unwrapping is the reliable detection of the 2π phase jumps. An important condition is that adjacent phase samples satisfy the relation − π ≤ Δi δ(i, j) < π with Δi δ(i, j) = δ(i, j) − (i − 1, j) .

(19.98)

If the object to be analyzed is not simply connected, i. e., isolated objects or shaded regions occur, then the mod 2π unwrapping procedure fails because of violated neighborhood conditions and unknown phase

524

Part C

Noncontact Methods

jumps greater than π. These difficulties can be explained clearly by using the example of a step-like object to be measured with the projected fringe technique (Fig. 19.37a). That very simply shaped object can be taken to show the problems that are typical in the

Part C 19.2

a) Interferogram

evaluation of more complex objects having shaded regions, unresolved fringes, phase jumps greater than π, etc. Regions I and V on the flat ground as well as the higher area III are normally illuminated and can be observed without problems. However, regions I and III b) Skeleton

c) Associated numbered graph 1 2 3 4 5 6 11 10

9

8

7

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23

6 5

d) Reconstructed continous phase

e) Pseudo-3-D-plot

Fig. 19.38a–e Example of graph based fringe numbering. (a) Interferogram (b) skeleton (c) associated number of graph (d) reconstructed continuous phase (e) pseudo-3-D-plot

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

• • • •

two fringes never cross each other; two adjacent lines represent a phase difference of ±π; a line can end nowhere except at the border of the area; the numbering along closed pathes delivers the same value at the starting and final point.

However, these properties are insufficient to ensure a unique numbering, especially concerning the problem of nonmonotonic changes. Due to the periodicity and the evenness of the cosine function (19.93) it is impossible to determine whether the phase difference between two adjacent fringes is +π or −π. The interference phase δ is only mod 2π definite. Consequently a priori knowledge is necessary to assign the correct fringe number. Additional information concerning the sign of the phase gradient (increasing/decreasing) can be provided by some real-time manipulations in the fringe pattern: an artificial phase shift caused by the displacement

of the reference and the object beam, respectively, or by the change of the wavelength results in a movement of the fringes. If the direction of movement of each fringe is considered, fringe numbering can be performed automatically [19.124]. In some circumstances, this problem can be bypassed by introducing a substantial tilt into the interferogram [19.125]. Thus the fringe number N increases monotonically from one fringe to the next and the fringe numbering can be carried out automatically. However, in general the deformation or the shape of the object to be investigated is more complex and changes in the sign of the phase gradient have to be considered. A way to reconstruct the phase using intensity-based methods is to start from the skeletonized fringe pattern and to number the fringes manually with their relative fringe orders. This way of numbering can be performed on a simple basis. The operator draws a line across several fringes and the system assigns stepwise an increasing or decreasing number to each fringe according to the presumed sign of the phase gradient. By repeating this operation over several clusters of fringes and considering the a priori knowledge about the course of monotony a consistent numbering can be performed. Another so-called graph-based method that automatically computes a consistent numbering of the skeleton lines without a priori knowledge is proposed in [19.126]. To model the problem, a graph of adjacency is built from the skeleton. A graph must be understood here as a structure made of nodes linked by arcs. An arc represents a relationship between two nodes. In the case of fringe patterns, each fringe in the image is associated to a node in the graph, and two nodes are linked together by an arc if and only if both corresponding lines are adjacent. The efficiency of the graph based method is demonstrated by the evaluation of an interferogram with a saddle shown in Fig. 19.38. The original interferogram (Fig. 19.38a) is skeletonized (Fig. 19.38b) and the graph is calculated automatically (Fig. 19.38c). By means of this graph the phase distribution can be reconstructed (Fig. 19.38d). To build the graph, a region-filling algorithm is used. This seems easier and more robust than trying to automatize the manual line drawing method. However, the main drawback of using region filling is that perfectly separated regions are necessary, i. e., closed fringes or fringes ending only at the border of the image. Real interferograms do not always ensure this. Two kinds of problems have to be considered: discontinuities in the skeleton and border effects preventing the fringes from reaching the border of the picture. The first prob-

Part C 19.2

are separated by a shaded area II. On the other side of the rectangular solid the regions III and V are simply connected. However, due to the steep edge, the fringes projected on the surface IV are so dense that they cannot be resolved by the detector. The observed fringe map is shown in Fig. 19.37b. It can be clearly seen that the absolute order of the fringes on top of the step cannot be detected unambiguously without additional information. The advantages of PMI methods in comparison to intensity-based methods are the easier way in which the processing scheme can be automated, the greater accuracy of the results, and the online recognition of points where the monotony of the phase angle is changing. However, both methods suffer from the fringe-counting problem. Methods for Fringe Numbering The phase reconstruction from its contour lines such as skeleton lines is a nontrivial problem due to the ambiguities discussed above and the loss of directional information. Thus it is impossible to determine the slope of the phase at a given point without additional knowledge. The information that can be derived from a single interference pattern may be explained by the comparison with contour lines in a geographical map: the skeleton lines can be considered as contour lines of a phase surface, with the bright fringes corresponding to integer multiples of 2π(N2π) and the dark ones to intermediate values ([2N + 1]π). By analyzing the properties of such contour lines, some important features can be pointed out

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graph and the sign of the phase gradient at one node per branch is necessary. The algorithm does not take into account any external parameter and will assign an arbitrary sign of phase gradient to each branch. The whole numbering process includes the following steps:

δ (x)

4π

• Part C 19.2

2π δ 2 (x)

• •

Pixel

2

Two ways of numbering are possible: the algorithm may either proceed automatically to a consistent but arbitrary result or the algorithm stops at ambiguous points and the operator can decide on the correct sign of phase gradient. In both cases a consistent numbering is delivered.

1

Unwrapping of Mod 2π-Phase Distributions. The key

N

Pixel

Fig. 19.39 One-dimensional (1-D) phase unwrapping scheme

lem can be solved by improving the skeleton (drawing a line or an arc) to make the fringes continuous. The second problem can be managed either by prolonging the fringes to the border, or taking a region of interest that excludes the bordering zone of the skeleton. Once the graph is built, it must be analyzed. Only a limited number of graph configurations are possible. The numbering itself is achieved by enforcing a few rules:

• • • •

select a starting node (if possible, choose a node at the end of a branch); choose a starting number (e.g., N = 0), and a starting sign for the phase gradient; browse the whole graph (for each node, treat all the connected untreated nodes first), and enforce the previously given rules.

to phase unwrapping is the reliable detection of the 2π phase jumps described by (19.68). For noise-free wrapped phase distributions a simple algorithm as illustrated in Fig. 19.39 can be implemented:

• • •

scan line by line across the image; detect the pixels where the phase jumps and consider the direction of the jump; integrate the phase by adding or subtracting 2π at these pixels.

An analysis of the 2-D unwrapping algorithm was provided by Ghiglia et al. [19.127]. According to that analysis the processing scheme for unwrapping the sequence of principal values of the arctan function con-

two adjacent nodes of same gray level will have the same number; two adjacent nodes of differing gray level will have a phase difference ±π; any nonramified path, i. e., a path without branches, made of alternately bright and dark nodes will have a monotonous numbering; two adjacent nodes of same gray level on a nonramified path indicate a change sign of phase gradient.

These rules can easily be applied on a nonramified path. The only extra parameters that are needed to determine the solution completely are the absolute phase and the counting direction at one starting point. The problem becomes more difficult when several branches occur in the graph. Then the absolute phase at one node of the

Fig. 19.40 Phase map of a speckle field with singularities

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

sists of a cascade of three operations: differencing, thresholding, and integrating. On condition that neighboring phase samples satisfy the relations

(19.99)

(19.100)

across the 2-D array both ways of unwrapping along columns or lines and any other combination, such as unwrapping along a spiral starting from a certain point within the pixel matrix, yield identical results. Thus, the process of unwrapping is path independent. Otherwise inconsistent values exist in the wrapped phase field. The checking of wrapped phase distributions for possible inconsistencies is an approved means for the identification of pathological areas and the selection of suitable unwrapping pathes. Unfortunately, in practice the conditions (19.99) and (19.100) are not always satisfied due to the presence of noise, the violation of the sampling theorem, and the influence of the object shape and its deformation, respectively. These phenomena are discussed briefly now. For a physically correct unwrapping it is necessary to distinguish between true mod 2π discontinuities and apparent ones caused by noise or aliasing, which should be ignored. Further phase jumps can be introduced by the object itself due to, for instance, gaps, boundaries, shadows, and discontinuous deformations (e.g., resulting in cracks), which should be considered. Various strategies have been proposed to avoid unwrapping errors in the phase map, but until now there has been no general approach to avoid all types of error without user interaction – especially if objects j +1

j

Δ(i, j) = {[δ(i, j) − δ(i, j + 1)]/2π} , Δ1 i Δ2 Δ4 i +1

(19.101)

where {} denotes rounding to the nearest integer. Possible values of Δ(i, j) are −1, 0 or 1. The sum of all differences along the closed path in the 2 × 2-pixel window S=

4 

Δ(i, j)

(19.102)

i=1

Δ3

Fig. 19.41 Inconsistency check in a 2 × 2-window

gives an indication of the consistency of the wrapped phase in that region. All pixels within the window are consistent if the sum is equal zero. Otherwise the pixels are labeled as inconsistent.

Part C 19.2

− π ≤ Δi δ(i, j) < π with Δi δ(i, j) = δ(i, j) − (i − 1, j) , − π ≤ Δ j δ(i, j) < π with Δ j δ(i, j) = δ(i, j) − δ(i, j − 1) ,

with complex shape undergo noncontinuous deformations. The main sources of error in phase unwrapping can be classified into four classes: noise, undersampling, object discontinuities, and discontinuous phase fields. Some simulations of the influence of these types of error are given in [19.128]. Electronic and speckle noise are the most familiar sources of error in automatic evaluation of interference patterns. In distinction to other evaluation techniques which operate with only one interferogram (e.g., fringe tracking) the stability of the intensity distribution during the reconstruction is essential for phase-sampling methods. This means that the noise configuration must be identical for all considered frames. Consequently, a real-time environment with interferometric stability, in contrast to photographic stability for conventional double-exposure techniques, has to be guaranteed during the reconstruction of the m (m ≥ 3) interferograms. In most practical situations, this condition is not completely satisfied because of the time-dependent character of electronic noise, speckle displacements, and so on. Hence, computed principal phases values are corrupted by noise to such an extent that locally inconsistent regions are caused. Following the statistics of speckle fields it is obvious that dark speckles dominate in the image. At a point where the intensity is zero, the wavefront contains a singularity (Fig. 19.40) and the phase is indeterminate. Around the singularity the phase undergoes a corkscrew-like course [19.129]. As a consequence such pixels generate phase dislocations and inconsistencies [19.130]. A simple method to detect inconsistent regions in the sawtooth image is given by computing the sum of the wrapped phase differences for all 2 × 2-pixel windows along the path depicted in Fig. 19.41 [19.131]. Across the phase map black pixels represent −π and bright π. The difference between two adjacent pixels, (i, j) and (i, j + 1) can be calculated as

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b) Wrapped phase

c) Marked inconsistencies

d) Median filtered wrapped phase

e) Marked inconsistencies

f) Unwrapping using minimum wide-spanning tree

Part C 19.2

a) Interferogram

Fig. 19.42a–f Unwrapping of noisy interferograms. (a) Interferogram (b) wrapped phase (c) marked inconsistencies (d) median filtered wrapped phase (e) marked inconsistencies (f) unwrapping using minimum wide-spanning tree

Huntley [19.131] introduced the name dipoles for those points where the path integral remains nonzero. These dipoles indicate errors in the derived phase map, either a 2π edge detection where there is no

edge or a failure in edge detection where there is an edge. Data in regions well away from these points are influenced by the errors if they are not considered.

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

a) Wrapped phase

Fig. 19.43a,b Directional averaging of wrapped phase images. (a) Wrapped phase (b) smoothed image

The basic assumption for the validity of the phase unwrapping scheme described above is that the phase between any two adjacent pixels does not change by more than π [19.132]. This limitation in the measurement range results from the fact that sampled imaging systems with a limited resolving power are used. There must be at least two pixels per fringe – a condition that limits the maximum spatial frequency of the fringes to half of the sampling frequency (Nyquist frequency) of the sensor recording the interferogram. Fringe frequencies above the Nyquist frequency are aliased to a lower spatial frequency. In such cases the unwrapping algorithm is unable to reconstruct the modified data. If the fringe frequency is higher than the Nyquist frequency the unwrapping algorithm fails. The simplest solution is to improve the sam-

I4 (x, y) − I2 (x, y) = 2b(x, y) sin[δ(x, y)] , I3 (x, y) − I1 (x, y) = 2b(x, y) cos[δ(x, y)] . (19.103) Noisy wrapped phase images can be smoothed by median filtering (Fig. 19.42d) to preserve true phase

Part C 19.2

b) Smoothed image

pling using sensors with higher resolution or to process magnified regions of the interferogram. Discontinuities of the object such as cavities, shaded regions, and irregular borders are also reasons for incorrect unwrapping because the intensity does not change in such regions according to the phase-sampling principle. Consequently, abrupt phase changes exceeding π can appear in the wrapped phase. To avoid such errors the corresponding regions should be excluded from the unwrapping process. Discontinuous deformations due to the nonlinear material behavior under load causing, e.g., cracks also give rise to dislocations in the phase distribution. In the surroundings of the crack the structure of the fringe pattern is influenced in such a way that the fringes are cut and displaced to each other. The noncontinuous behavior of the phase surrounding the discontinuity results in inconsistent areas and consequently in an incorrect unwrapping if straightforward procedures are used. In practice, however, all these influences appear together. Figure 19.42a shows an interferogram of a vibrating industrial component recorded with a doublepulse laser technique. The fringe pattern is corrupted with speckle noise, electronic noise, and specular reflections, which contribute to a distortion and fusion of the fringes. In some sections the signal-to-noise ratio and contrast are very low. Other influences are object structure and irregular boundaries. A conventional unwrapping procedure starting with the column at the right border and following the lines will meet a lot of inconsistent areas (Fig. 19.42c) and will consequently generate streaks with incorrect phase offset over the whole phase map if not considered by an adapted algorithm such as a minimum-wide spanning tree [19.133] (Fig. 19.42f). It is obvious that alternative strategies for a correct unwrapping, including preprocessing of the fringe patterns and smoothing the wrapped phase map, are necessary. A lot of work has already been done in this field but there is no ideal recipe that works satisfactory in all cases [19.128–141]. To preprocess the phaseshifted interferograms the methods described earlier can be applied. An approved and simple means for noise reduction is the smoothing of the intermediate results of the numerator and denominator of (19.68). The special case of m = 4 (Table 19.1) gives, for instance,

529

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Noncontact Methods

b) Calculated background intensity

c) Calculated modulation

d) Interferogram overlayed with the derived mask

Part C 19.2

a) Original fringe pattern

Fig. 19.44a–d Masking of phase-shifted interferograms with the objective of separating areas with relevant and irrelevant data. (a) Original fringe pattern (b) calculated background intensity (c) calculated modulation (d) interferogram overlayed

with the derived mask

discontinuities and to decrease the number of inconsistencies. However, directional averaging is more convenient in this case, as shown for example in Fig. 19.43. A modified rank-order filter designed for electronic speckle pattern interferometry (ESPI) phase data was proposed and compared with some other procedures by Owner-Petersen [19.142]. A useful segmentation step consists of the global partitioning of the interferograms into areas with and without relevant data. According to the phase-sampling principle described above more information than the phase distribution can be reconstructed from the phaseshifted interferograms. Areas with insufficient fringe modulation, disturbing spots, and irregular boundaries, as shown in Fig. 19.44a, give rise to inconsistencies and

should be excluded from further processing by the generation of unwrapping masks. The generation of these masks can be done with help of the computed background intensity and fringe contrast, as illustrated in (19.70) and Table 19.1. The results of both calculations are shown in Fig. 19.44b,c. By a kind of thresholding the binary masks are derived. Both masks can be combined (Fig. 19.44d) to frame the relevant area for further processing. Various noise-immune algorithms have been proposed to cope with the inconsistent points [19.123, 127–146]. Typical examples of these are the regionoriented method proposed by Gierloff [19.143], the cut-line method proposed by Huntley [19.131], the wide spanning tree method published firstly by Judge

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

F(x) = y

(19.104)

is defined as well-posed with a linear operator F ∈ L(X, Y ) in Banach spaces X and Y if the following three Hadamard conditions are satisfied [19.150]:

1. F(x) = y has a solution x ∈ X for all y ∈ Y (existence); 2. this solution x is determined uniquely (uniqueness); 3. the solution x depends continuously on the data y, i. e., the convergence yn − y → 0 of a sequence {yn } = {F(xn )} implies the convergence xn − x → 0 of corresponding solutions (stability). If at least one of the above conditions is violated, then the operator equation is called ill-posed. Simply spoken ill-posedness means that we do not have enough information to solve the problem uniquely. In order to overcome the disadvantages of illposedness in the process of finding an approximate solution to an inverse problem, various regularization techniques are used. Regularizing an inverse problem means that, instead of the original ill-posed problem, a well-posed neighboring problem has to be formulated. The key decision of regularization is to find an admissible compromise between stability and approximation [19.151]. The formulation of a sufficiently stable auxiliary problem means that the original problem has to be changed accordingly radically. As a consequence one cannot expect that the properties of the solution of the auxiliary problem coincide with the properties of the original problem. However, convergence between the regularized and the original solution should be guaranteed if the stochastic character of the experimental data is decreasing. In the case of noisy data the identification of unknown quantities can be considered as an estimation problem. Depending on the linearity or nonlinearity of the operator F, we than have linear and nonlinear regression models, respectively. Consequently, least-square methods play an important role in the solution of inverse problems: F(x) − yε 2 → min

(19.105)

with yε = y + ε. Marroquin et al. applied for the reconstruction of the continuous phase field from noisy mod 2π data the Thikhonov regularization theory [19.152, 153] to find solutions that correspond to minimizers of positivedefinite quadratic cost functionals. This method can be considered as a generalization of the classical least-squares solution by introducing a so-called regularization parameter and a stabilizing functional. A detailed introduction to the theory, algorithms, and software of two-dimensional phase unwrapping is given in the book of Ghiglia and Pritt [19.154]. Further approaches to phase unwrapping using only one interferogram based on an isotropic 2-D Hilbert and

Part C 19.2

et al. [19.133, 136], the pixel-ordering technique proposed by Ettemeyer et al. [19.144], the line-detection method introduced by Andrä et al. [19.135], and Lin et al. [19.145], and the distributed processing method using cellular automata proposed by Ghiglia et al. [19.140] or the neural network approach discussed by Takeda et al. [19.137] and Kreis et al. [19.146]. A common feature of all of these techniques is that the most essential part of the principle is solving a combinatorial optimization problem [19.141] such as the minimization of the overall length of the cut lines using simulated annealing [19.131], minimization of the overall phase changes between neighboring pixels by Prim’s algorithm [19.136], or minimization of the overall smoothness of the phase distribution through the dynamic state changes of cellular automata or neurons [19.137, 140, 146]. Another way to classify the various methods was proposed by Robinson [19.123], who distinguishes between path-dependent and pathindependent methods. To the first class, where the success of the unwrapping process is dependent on the path taken through the wrapped phase distribution, belong such methods as sequential linear scanning (line by line), pixel ordering or random walk. Variations on this method use two or more routes through the data and compare the results to increase the probability of analyzing the data correctly. Path-independent methods such as graph-based techniques and cellular automata integrate the phase along all possible paths between a starting pixel (or reference point) and all pixels to be processed. A further method proposed by Marroquin et al. [19.147–149] was already mentioned earlier with the description of the regularized phase-tracking approach. Here the unwrapping of a phase field from a single intensity measurement is considered as an inverse problem. However, inverse problems usually have some undesirable properties: they are in general ill-posed, ambiguous, and unstable. The concept of well-posedness was introduced by Hadamard [19.150] into the mathematical literature. He defined a Cauchy problem of partial differential equations as well-posed, if and only if, for all Cauchy data there is a uniquely determined solution depending continuously on the data; otherwise the problem is ill-posed. In mathematical notation an operator equation

531

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Noncontact Methods

quadrature transform are described by Larkin [19.155, 156] and Quiroga et al. [19.157–159], respectively.

•

Absolute Phase Measurement and Temporal Phase Unwrapping. In the previous sections it was pointed

Part C 19.2

out that all phase measurement techniques deliver the phase value mod 2π only, and consequently phase unwrapping has to be carried out. However, even a correct phase unwrapping results in a relative phase map with an unknown bias given by the integer NR in (19.94). So, in general it is necessary to measure the absolute phase of each measuring point instead of its relative phase. Problems arising from this fact are, for example,

• •

incorrect numbering of the fringes if no stationary region of the object can be identified to locate the zero-order fringe incorrect calculation of the vector components of 3-D deformations if multiple interferograms with different illumination direction have to be analyzed (in general there is no way to know how the phase at a point in one interferogram relates to the phase at the same point in another) [19.160]

The latter problem was already discussed in Sect. 19.2.4 where we have discussed the fringe counting problem using the example of Fig. 19.37. Various approaches that partially solve the problems of absolute phase measurement are known [19.122], including

• •

• •

a)

h Λ1/2 h1

• h

h = h1 = N1 Δh1 =

b)

δ'1 Λ1 = Nˆ Δh1 2π 2

δ'1 2π

δ'

δ'2 6π

δ'

Λ1/2

h h2

Λ2/2

2π

4π

~ Λ2 δ'2 Λ2 Λ2 + N2 h = h2 – N2 Δh2 – N2 = 2 2π 2 2

Fig. 19.45a,b Schematic description of temporal phase unwrapping by a sequence of synthetic wavelengths: (a) Coarse synthetic wavelength generating only one fringe N˜ 1 = 0, (b) reduced synthetic wavelength generating more fringes, e.g., N˜ 2 = 2

incorrect unwrapping if the object to be analyzed is not simply connected, i. e., isolated objects or shaded regions occur, because of violated neighborhood conditions and unknown phase jumps greater than ±π

the application of a rubber band drawn between the object and a fixed part within the measured scene to determine the fringe number [19.161] the manipulation of components of the optical setup that influence the fringe pattern but not the measuring quantity (e.g., the recording and evaluation of two fringe patterns with two different wavelengths [19.162] or two different illumination directions [19.163]) the use of sensitivity vector variations to provide absolute displacement analysis [19.160] the application of coded light techniques [19.164, 165] for time–space encoding of the projected fringe patterns with respect to the fringe number the combination of fringe patterns with different spatial frequencies [19.166] or different orientations [19.167]

The last approach has been proven to be suitable for a wide range of practical applications. To avoid ambiguity the general principle of these methods consists of the generation of a synthetic wavelength by tuning one of the flexible system parameters (wavelength, angle separation of the light sources, spatial frequency or orientation of the projected fringe pattern, or loading of the object) in such a way that only one fringe period covers the object. An important advantage of this method is the avoidance of the conventional unwrapping process that is based on the evaluation of neighborhood relations by comparing the relative phases of adjacent pixels. However, neighborhood-based procedures are very sensitive to definite signal and object properties such as speckle noise, edges, and isolated regions as discussed above. Therefore several neighborhoodindependent algorithms were developed in the middle of the 1990s. Such procedures are known under the names synthetic wavelength interferometry, absolute interferometry, and wavelength scanning [19.162, 168–173], and hierarchical demodulation [19.174] and temporal

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

ˆ δ(Px ) = N2π = ( N˜ + N)2π    δ (Px ) = + N˜ 2π , 2π

λ1 λ2 λ2 ≈ . (λ1 − λ2 ) Δλ

(19.94a)

(19.106)

For simplification we design the setup such that the observation and illumination direction are coincident. Furthermore, telecentric optical systems are used. Then the distance between consecutive contour lines represents a height difference Δh =

Λ . 2

At the beginning we tune Λ = Λx so that the largest step height h of the object is covered by only one fringe. Consequently, N1 ≤ 1, N˜ 1 = 0, and Nˆ 1 = δ1 /2π (Fig. 19.45a). For the measured height h 1 it holds that δ1 Λ1 (19.109) . 4π This coarse synthetic wavelength ensures the unambiguous demodulation, but as mentioned above the accuracy is comparatively low because of the limited measurement uncertainty ±ε of the wrapped phase δ (Px ): h 1 = N1 Δh 1 =

ˆ = ( Nˆ t ± ε)2π δ (Px ) = N2π

(19.107)

(19.110)

with Nˆ t being the true value. Usually ε is not smaller than 1/10 for fringe tracking and usually better than 1/50 for phase-shifting algorithms. With 8 bit coding and a certain degree of noise we assume that the phase has an uncertainty σδ of about ±2π/64. Consequently, with an increased number of fringes an improved accuracy of the reconstructed phase can be expected. However, this statement is a strong simplification of the complex underlying processes. Therefore the second measurement uses a smaller synthetic wavelength Λ2 that generates a shorter contour interval Δh 2 = Λ2 /2. Thus N2 ≥ 1, N˜ 2 ≥ 1, and Nˆ 2 = δ2 /2π (Fig. 19.45b) with N˜ 2 = 2. For the measured height h 2 it holds that h 2 = N2 Δh 2 =

with N˜  of the set of natural numbers. A synthetic wavelength Λ is generated by the coherent or incoherent superposition of two wavelengths λ1 and λ2 as Λ=

Consequently the complete object height is given by    Λ δ (PX ) Λ ˜ h=N = (19.108) +N . 2 2π 2

δ2 Λ2 Λ2 Λ ˜ 2. + N˜ 2 = h res 2 + N2 4π 2 2 (19.111)

The crucial task consists now in the determination of N˜ 2 . For this purpose the difference between the height h 1 determined in the first step without ambiguity and the remaining fractional part h res 2 of the contour interval Δh 2 is calculated as Δh 12 = h 1 − h res 2 =

δ1 Λ1 δ2 Λ2 Λ2 − = N˜ 2 . 2π 4π 2 (19.112)

Based on this result, N˜ 2 can be determined while calculating how many times the contour interval Δh 2 fits into this difference,   Δh 12 + 0.5 , (19.113) N˜ 2 = round Δh 2

Part C 19.2

phase unwrapping [19.139, 175, 176]. The objective of these algorithms is to extend the limited range of uniqueness (mod 2π) by the generation of an adapted synthetic wavelength and to enable an absolute pixelbased evaluation of the mod 2π-phase distribution. Since the accuracy of the evaluation is limited by using only one synthetic wavelength, adapted sequences of different synthetic wavelength are used. In fact, the generation of only one fringe solves the problem of ambiguity but sufficient accuracy cannot be ensured in general [19.174]. Therefore in practice absolute phase measurement is carried out in a hierarchical or temporal way. At first the synthetic wavelength is chosen to receive only one fringe. Then the flexible parameter is changed step by step to cover the measuring field with an increasing number of fringes to get an improved accuracy. One important advantage of this procedure consists in the method of unwrapping. The classical approach, as described in a previous section, compares pixels in a local neighborhood to identify those points where the phase difference is greater than ±π – the so-called phase jumps. In the temporal version the unwrapping proceeds along the time axis on the pixel itself. Consequently, because the pixel is compared with itself, a lot of problems connected with local unwrapping can be avoided. In the following the principle of temporal phase unwrapping is discussed by using the example of shape measurement [19.174–176]. The same holds for displacement measurement [19.177, 178]. We refer to (19.94) and assume that NR equals zero. Thus for the demodulated phase δ (Px ) it follows that

533

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and finally

Part C 19.2

δ h 2 = Δh 2 2 + N˜ 2 Δh 2 . 2π For the general case we can write   Δh i−1,i + 0.5 , N˜ i = round Δh i h i = Δh i

(19.114)

(19.115)

  h i−1 − h ires δi + 0.5 . + Δh i round 2π Δh i (19.116)

To avoid ambiguity an appropriate synthetic wavelength Λ is necessary. However, a long synthetic wavelength is associated with a small accuracy. Thus the task consists of the construction of a sequence of continuously shrinking synthetic wavelengths which leads to a sufficient high accuracy while keeping the number

of intermediate steps as small as possible. It has been shown [19.174, 176] that the determination of successive synthetic wavelength depends on the measurement uncertainty ε of the phase δ . In every successive step (i) it must be ensured that the corresponding period length Λ(i) of the still ambiguous solution (including its tolerance at the borders) is greater than the remaining tolerance region {Λ(i−1) 4ε} of the previous step (i − 1): Λ(i) (1 − 4ε) ≥ Λ(i−1) 4ε .

(19.117)

This rule implies an upper limit on every successive Λ(k) . Referring to the test object shown in Fig. 19.37, the absolute phase distribution is calculated with the algorithm. The results are presented in Fig. 19.46. Transformation of Phase Data into Displacements and Coordinates. Since the phase is the primary quan-

a) Test object

b) Reconstructed profile

c) Profile for the coarse synthetic wavelength Λ

d) Coarse synthetic wavelength

Abs. fringe order 130 120 110 100 90 80 70 0

50

100

150

200

250

300 Pixels

Fig. 19.46a–h Reconstruction of the profile by a sequence of synthetic wavelengths. (a) Test object (b) reconstructed profile (c) profile for the coarse synthetic wavelength Λ (d) coarse synthetic wavelength (e) profile for the reduced synthetic wavelength (f) reduced synthetic wavelength (g) profile for the finest synthetic wavelength (h) finest synthetic wavelength

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

535

f) Reduced synthetic wavelength

e) Profile for the reduced synthetic wavelength Abs. fringe order 130 120 110

Part C 19.2

100 90 80 70 0

50

100

150

200

250

300 Pixels

g) Profile for the finest synthetic wavelength

h) Finest synthetic wavelength

Abs. fringe order 130 120 110 100 90 80 70 0

50

100

150

200

250

300 Pixels

Fig. 19.46 (cont.)

tity for fringe-based measurement techniques the main effort in laser metrology during the past 20 years has been focused on the development of high-precision phase-measurement techniques and their digital implementation. However, the phase distribution δ(x, y) gives only a first impression of the quantity to be measured. Consequently, the transformation of the phase value into a retraceable measuring unit afflicted with a limited measurement uncertainty is an unavoidable step in the measurement chain. Here the systematic setting and registration of the geometry of the setup, the calibration of the involved components, the estimation of the measurement uncertainty according to the Guide for the Expression of Uncertainty Measurement (GUM) [19.179], and the retracement of the measured data to the standard are important points to be addressed. To calculate, for instance, the components of the 3-D displacement vector d(u, v, w) the sensitivity vectors for the relevant object points have to be specified. This needs as well the Cartesian coordinates of the ob-

ject points as the position of the illumination source and of the sensor. For a successful measurement the following steps have to be carried out [19.180]:

•

• • •

the planning of the experiment, including the choice of the measurement principle, the specification of the loading conditions, the specification of the sensitivity of the optical setup with respect to the measuring quantity, and the selection of the light source the design of the setup with respect to its sensitivity and its robustness concerning the propagation of errors of the input data to the final results the acquisition of the primary phase data representing the shape and displacement values the combination of all data sets acquired with different measuring systems (e.g., the registration of the object shape measured with a topometric system in a holographic setup for the measurement of the displacement data) with respect to the calculation of the displacement field

536

Part C

Noncontact Methods

y

Object Q1

x 0

e Q1

ρ

ε z

Q2 e Q2

y'

Part C 19.2

L

·

ϕ

x'

0' Q4 e Q4

Q3

Hologram H eB

e Q3

Fig. 19.47 Scheme of a holographic interferometer with four illumination directions and one observation direction (the definition of the parameters ε, ρ, ϕ, and L is given in this section)

•

the evaluation of the data with respect to their measurement uncertainty

In the following sections these points are discussed briefly. Planning of The Experiment (a) Basic relations The basic relation of holographic interferometry [19.39] represents the connection between the measured phase difference δ(P) and the displacement vector d = d(u, v, w) of a point P = P(x, y, z): 2π [eB (P) + eQ (P)]d(P) = S(P)d(P) λ (19.118) with δ(P) = N(P)2π ,

δ(P) =

with N as the fringe number, λ as the wavelength of the used laser, eB as the unit vector in observation direction, eQ as the unit vector in illumination direction, and S(P) as the sensitivity vector of the current object point. For the measurement of the three displacement components d(u, v, w) at least three independent equations are necessary. Usually three or more observation and illumination directions, respectively, are chosen. To avoid perspective distortions it is more convenient to use independent illumination directions as illustrated in Fig. 19.47. In this case the following equation system can be derived: Ni (P)λ = [eB (P) + e Qi (P)]d(P), i = 1 . . . n, n ≥ 3 .

(19.119)

Usually a matrix notation of (19.119) is applied Nλ = Gd ,

(19.120)

with the column vector N containing the n fringe numbers Ni and the (n, 3)-matrix G containing the three components of the i-th sensitivity vector Si in each line. Because of measuring errors the solution d = d(u, v, w) of the equation system (19.119) differs from the true value d 0 = d0 (u 0 , v0 , w0 ) by an error vector Δd. Sources of experimental error can be divided into two categories: errors ΔG of the geometry matrix G due to the limited accuracy of the measuring tools for coordinate measurement and errors ΔN of the fringe number vector N due to the limited accuracy of the applied phase measurement technique. All these errors cause an error Δd of the displacement vector. Consequently, we have to write λ(N0 + ΔN) = (G0 + ΔG)(d 0 + Δd) , N 0 λ = G0 d 0 , G = G0 + ΔG , N = N 0 + ΔN , d = d 0 + Δd ,

(19.121) (19.122) (19.123) (19.124) (19.125)

with N0 as the true fringe number vector and G0 as the true geometry matrix. To improve the accuracy of the displacement components, overdetermined systems (n > 3) are used in general. In our case more than three illumination directions e Qi (i = 1 . . . n, n > 3) have to be provided (see Fig. 19.47 with n = 4). Such an equation system can be solved with the least-squares error method using gradd [(Nλ − Gd)T (Nλ − Gd)] = 0 ,

(19.126)

which results in the so-called normal equation system GT Nλ = GT Gd .

(19.127)

The matrix F = GT G

(19.128)

is denoted as the normal matrix with GT as the transpose of G. The importance of the normal matrix F for the construction of the interferometer and the error analysis is explained in detail in that section. (b) Valuable a priori knowledge For an inspection problem it is convenient to start with a certain amount of a priori knowledge concerning the object under test and its loading conditions. This information gives indications for the construction of the interferometer with respect to its sensitivity and for the choice of the loading principle as well as the loading size. If the displacement direction is known, coincident illumination and observation directions can be used and

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

the sensitivity of the interferometer can be controlled in such a way that the amount of the displacement is measured directly d = N(λ/2) .

(19.129)

• • • • •

direct mechanical stressing pressure or vacuum stressing thermal stressing impulse loading vibrational excitation

The choice of a specific loading technique depends mainly on the following criteria:

• • •

the load should simulate the working conditions of the object under test; the load should generate surface displacements within the measuring range (rule of thumb: 100 nm– 50 μm); the load should cause irregular fringe patterns if surface or subsurface flaws occur.

Further boundary conditions for the experiment are determined by the required accuracy of the data, the choice of the laser source, and the distribution of the meas-

Fig. 19.48 Optimized interferometer setup with 90◦ sym-

metry in the orientation of the illumination directions

Part C 19.2

Consequently, some serious problems that will be discussed in the next sections are irrelevant in that very simple case. However, in most inspection tasks a lot of data about the object are available for the experiment, e.g., the shape, the material, and internal structure; the working load, possible deformations, and positions; and types of weak points and faults. Using this knowledge the interferometer can be controlled with respect to its sensitivity and robustness and adapted loading techniques can be applied. Furthermore, model-based simulations as a combination of finite element modeling of surface displacement fields and fringe pattern computations using an analytical model of the image formation (Sect. 19.2.3) can deliver valuable directions for the practical inspection process [19.181]. A tool for the optimal arrangement of objects within the holographic setup that is as simple as it powerful was developed by Abramson as early as the first decade of holographic interferometry [19.182]. Abramson demonstrated the advantage of the so-called holo-diagram by using the example of a steel beam more than 2 m long using a 60 mW He–Ne laser with a coherence length of only 30 cm [19.183]. Concerning the loading technique Birnbaum and Vest [19.184] distinguish in their review of holographic nondestructive evaluation between five basic principles for the solution of different inspection problems

uring points across the object. The accuracy mainly depends on the objective of the experiment. If related mechanical quantities such as strains and stresses have to be calculated by numerical differentiation of the displacements, high accuracy of the phase data has to be guaranteed on a sufficiently narrow mesh [19.185]. This affects the necessary effort for the phase reconstruction considerably. The choice of the laser source [pulsed or continuous-wave (CW) laser] depends on the loading conditions, the quantity to be measured, and the environmental influences. Pulse lasers are better suited for industrial inspection problems than CW lasers because of their high energy and short pulse duration. The distribution of the measuring points is mainly affected by the kind of the inspection problem. In connection with finite element modeling (FEM) simulations the topology of the mesh is strictly predetermined. Otherwise expected stress concentrations and weak points can give orientations. General inspection problems make it necessary to measure 3-D displacements on the surface of 3-D objects. For this purpose at least three independent phase measurements for each measuring point as well as the three coordinates of each point within a predefined coordinate system have to be measured. Additionally, some geometrical parameters of the interferometer must be known such as the central point of the entrance pupil of the observation system to define the observation direction and the locus of the focal point of the illumination system to define the illumination direction. All this has to be done in a concrete interferometer setup. Therefore

537

538

Part C

Noncontact Methods

Part C 19.2

in the next section the physical and mathematical background for a suitable interferometer design is discussed. Design of the Optical Setup (a) Optimization approaches For the calculation of the three displacement components d(u, v, w) at least three phase measurements are required to get an equation system of the type (19.119). According to the theory of linear equation systems these equations have to be linearly independent, which means that independent input data sets have to be measured. The best way to acquire linear independent data is the choice of a interferometer geometry with three orthogonal sensitivity vectors. Such an orthogonal three-leg design ensures minimum propagation of input data errors to the computed results as well as a best possible sensitivity for all displacement components. However, from the practical point of view this kind of geometry is very complex. Thus the task consists of finding a compromise between effort and advantage with respect to the error sensitivity of the interferometer. Such a so-called optimized interferometer is described in the following. A proven method to find a geometry that is characterized by minimum propagation of the experimental errors ΔN and ΔG to the displacement vector is an optimization with help of the perturbation theory of linear equation systems. The condition number κ of the matrix G is a numerical measure for the propagation of errors of input data to the computed results [19.186]:   ΔG ΔN |Δd| (19.130) ≤ κ(G) + , |d| G N where G denotes the norm of the geometry matrix G. Using the spectral norm, the condition number for n ≥ 3 can be calculated with  χmax (F) , κ≥1, κ(G) = GG−1  = χmin (F)

into account the demand for a low complexity of the optical setup [19.187]. In comparison to an orthogonal three-leg setup the complexity of the interferometer is considerably reduced if we use an interferometer geometry where the n illumination sources and observation points, respectively, are placed in one selected plane, as shown in Fig. 19.47. This plane is chosen to be parallel to the (x, y)-plane of the object coordinate system. Such a setup is very often used because of its good conditions for data acquisition. Figure 19.48 shows an interferometer with 90◦ symmetry applied for the investigation of microcomponents [19.188]. First it is shown that this simplification can also result in a robust error behavior if some rules are considered. For the case of n illumination points the greatest possible separation of the sensitivity vectors can be guaranteed with the following simple conditions:

•

All illumination points Qi are arranged within one plane parallel to the (x, y)-plane and placed on a circle with the greatest possible radius ρ (In the case of n observation points all the points Hi where the unit vectors in observation direction eBi break through the sensor plane (e.g., the hologram) are arranged within one plane parallel to the (x, y)-plane, i. e., the sensor plane(s) is/are parallel to the (x, y)-plane.) The center of this circle is the point H where the unit vector in observation direction eB breaks through the sensor plane. (In the case of n observation directions the center of the circle is defined by the point Q where the unit vector in illumination direc-

•

κ 60 50

ρ = 50

40

(19.131)

where χmax and χmin are the maximum and minimum eigenvalues of the normal matrix F, respectively. Since G consists of the n sensitivity vectors Si the number κ(G) also characterizes the influence of the mutual arrangement of these vectors on the error propagation. Consequently, the investigation of the influence of numerous geometrical parameters on the accuracy of the displacement vector can be simply reduced to the analysis of only one quality number. In our sense the objective of the optimization consists of the minimization of κ by variation of the vector orientation, taking

30

ρ = 100

20 10 0

ρ = 200 100

200

300

400

500

600

700

800

900

1000

L (mm)

Fig. 19.49 Dependence of the condition number κ of an optimized interferometer with n = 3 (ϕ = 120%) on the distance L for three different radius ρ (mm), calculated with (19.136)

Digital Image Processing for Optical Metrology

• •

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

◦

ϕ = 360 /n .

Object e–Q

Q1

e–Q

Illumination points

Q3

P

(19.132)

If these conditions are obeyed the norm of all the sensitivity vectors is equal – a necessary criterion of well-conditioned systems. In addition to the angle ϕ only one further parameter is necessary to describe the interferometer – the effective aperture ξ, given by ρ (19.133) ξ = tan ε = . L A further result of this geometry is the simplification of the error analysis. For the object point P0 (0, 0, 0) placed in the origin of the coordinate system the normal matrix F now only has nonzero elements along the main diagonal: ⎞ ⎛ 1 0 0 2 ⎟ ξ2 ⎜ ⎟ ⎜ 0 12 0 (19.134) F=n 2 ⎜  √ 2 ⎟ . ⎠ ξ +1 ⎝ 2 1+ ξ +1 0 0 ξ Equation (19.134) shows that the condition number of the interferometer optimized as described above is independent of the number of equations. This is essential if least-squares error methods are used. Taking into account that the expression   2  (19.135) 1 + ξ 2 + 1 /ξ ≥ 1, ∀ξ with (19.131) we can find the condition number as  √ 1 + ξ2 + 1 (19.136) κ(G) = 2 . ξ The minimum of κ can be derived for ξ → ∞: √ (19.137) κ min (G) = 2 . In practice the aperture ξ can be enlarged by increasing the distance between the source points Qi and the z-axis and/or by decreasing the distance between the object and the sensor. For both cases the separation of the sensitivity vectors is improved. Figure 19.49 illustrates the dependence of the condition number κ on the distance L for three different values of the radius ρ. Inconvenient setups can result in large condition numbers and consequently in a high sensitivity with respect to phase and coordinate measuring errors.

Part C 19.2

tion breaks through the plane spread by the n sensor points Hi .) The sensor point H is arranged in a plane parallel to the (x, y)-plane of the object coordinate system and at an orthogonal distance L. The angle ϕ between two adjacent lines connecting the source points Q i with the sensor point H is

e–Q –

r

y

Q2

Hologram

x OKS

r–B

z y

IKS

539

x r–Q z

e–B Q4 e–Q

Observation point B

SKS

e–j

Cameratarget

e–i

(i, j )

e–a

Fig. 19.50 Experimental setup for the investigation of the displace-

ment field of an aluminum cylinder loaded by inner pressure using different illumination directions eQ1 , . . . , eQ4 (IKS – illumination coordinate system; OKS – object coordinate system; SKS – sensor coordinate system)

Equation (19.136) gives an orientation for the quality of the interferometer with regard to the error propagation, but the results are only valid for the origin P0 (0, 0, 0) of the object coordinate system. In practice the measurement of point fields of extended objects is needed. Here the optimization should be performed using generalized condition numbers applied to the point field [19.189]. The same results can be derived for n observation directions and one illumination direction because of the symmetry of (19.118) concerning the observation and illumination direction. (b) The influence of the sensitivity Equation (19.136) shows that the measured phase difference δ(x, y) is equivalent to the component of the displacement vector d S in direction of the sensitivity vector S. However, this direction varies with the location P(x, y, z) where the phase δ(x, y) is measured. Unfortunately this dependence is neglected very often by assuming a constant sensitivity across the surface of the object. In this way the measurement is simplified considerably because the influence of the geometry of the object is not considered. As shown in the following experiment this simplified approach causes serious errors [19.190].

540

Part C

Noncontact Methods

a)

b)

c)

d)

e)

Part C 19.2 Fig. 19.51 Cylinder and the corresponding interferograms for the illumination directions eQ1 , . . . , eQ4

The experiment is performed with a cylinder of aluminium (diameter 100 mm, length 200 mm) that is fixed at the bottom (Fig. 19.50). A load is applied by changing the inner pressure of a gas. The coordinates, measured with the fringe projection technique [19.191], are used for the determination of all point-dependent sensitivity vectors. For comparison purposes a constant sensitivity vector is assigned to all object points, too. For every illumination direction the double-exposed hologram is made by using only one hologram plate with mutually incoherently adjusted reference beams. Figure 19.51 shows the corresponding interferograms with a pressure difference of 1.2 × 105 Pa. The phase distribution δ(x, y) was reconstructed with the phaseshifting technique (Sect. 19.2.4). a)

Based on the measured geometry and interference phase δ(P) the equation system (19.119) is solved for all points P(x, y, z). The result is a data field containing each point P and its corresponding displacement vector d(u, v, w) (Fig. 19.52). The comparison between the calculation with and without consideration of a locally varying sensitivity is shown in Fig. 19.53. The evaluation for the radial component along the cross section of the cylinder provides similar results in both cases. However, the tangential component shows obvious deviations such as its amount and its sign, both of which are in conflict with the expected rotational symmetry. Also the axial component along the longitudinal axis differs with increasing zvalues.

b)

c)

y

y 150

150

100

z

w

60 75

50

0.002 50

0.001

40 30

0 60

70 100

x

120 –10

80

100

120

0

x

Fig. 19.52a–c Displacement distribution on a cylindrical surface. (a) Measured geometry data of the cylinder, (b) displacement field considering the locally varying sensitivity across the surface of the cylinder (|dS | ≈ 1.5 μm) in mm, (c) deformation of the cylinder calculated by FEM simulation (all measures in mm)

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

a) Radial component (μm) over cylinder coordinates ϕ (°)

a) Radial component (μm) over cylinder coordinates ϕ (°) 2

1.5

1.5

1

1

0.5

0.5

0

20

40

60

80 ϕ (deg)

b) Tangential component (μm) over cylinder coordinates ϕ (°)

0

1

0.5

0.5

0

0

–0.5

–0.5

0

20

40

60

80 ϕ (deg)

c) Axial component (μm) over cylinder coordinates z (mm)

–1

2

1.5

1.5

1

1

0.5

0.5

0

50

100

150

200 (mm)

Fig. 19.53a–c Comparison between the calculation of the displacement components with and without the consideration of a locally dependent sensitivity (dashed line: constant sensitivity, solid line variable sensitivity). (a) Radial component (μm) over cylindrical coordinates ϕ (◦ ) (b) tangential component (μm) over cylindrical coordinates ϕ (◦ ) (c) axial component (μm) over cylindrical coordinates z (mm)

The reason for this is the sensitivity of holographic interferometry to out-of-plane components. In-plane components are measured with considerable lower sensitivity. Consequently, the measurement of the in-plane components such as the tangential and axial one in

40

60

80 ϕ (deg)

0

20

40

60

80 ϕ (deg)

c) Axial component (μm) over cylinder coordinates z (mm)

2

0

20

b) Tangential component (μm) over cylinder coordinates ϕ (°)

1

–1

0

0

0

50

100

150

200 (mm)

Fig. 19.54a–c Computed (dashed line) and measured

(solid line) displacement components for a cylinder loaded with inner pressure. (a) Radial component (μm) over cylindrical coordinates ϕ (◦ ) (b) tangential component (μm) over cylindrical coordinates ϕ (◦ ) (c) axial component (μm) over cylindrical coordinates z (mm)

the case of the cylinder are measured less sensitively and robustly. Measuring errors of the primary quantities, the interference phase, and the coordinates affect the in-plane components more than the out-of-plane component. The conclusion we can draw from this experiment is that the geometry and consequently the local variation of the sensitivity vector should be con-

Part C 19.2

2

0

541

542

Part C

Noncontact Methods

Part C 19.2

sidered if in-plane components have to be measured on extended objects. For the evaluation of the results a comparison between the results derived by FEM simulation and the experimentally measured data is useful. Figure 19.52c shows the deformed shell due to the inner pressure as a cut through the FEM mesh. The simulation returns a radial displacement that is constant because of the rotational symmetry along the cross section (Fig. 19.54a). As expected, the tangential component is zero (Fig. 19.54b). The axial displacement increases almost linearly along the longitudinal axis. As reasons for the differences between simulation and measurement the following sources of error can be listed: rigid-body displacements of the cylinder caused by variations of the inner pressure, possible phase offsets due to inaccurate location of the zero-order fringe during the evaluation of the fringe pattern, and simplifications concerning the geometry and the boundary conditions for the simulation. Nevertheless, it can be stated that the consideration of the locally variable sensitivity for the in-plane components contributes to a considerable improvement of the measured displacements. Acquisition of the Data In holographic displacement analysis the objective of the measurement is the determination of vector displacements of a point field. To this purpose the three Cartesian coordinates of all relevant object points and some other geometrical parameters of the setup have to be registered in a predefined coordinate system to determine the sensitivity vectors. In practice the measurement of these coordinates is performed using coherent or noncoherent shape measurement techniques [19.119, 190, 192]. Consequently, the conventional time-consuming and inaccurate manual measurement process is replaced by high-precision phasemeasuring techniques. In this sense interferometric displacement analysis including optical shape measurement gets a uniform basis in the application of modern phase-measurement principles (Sect. 19.2). Evaluation of the Data If all input data are measured, the displacement components can be computed by solving the equation system (19.119). In this section our interest is focused on the evaluation of these components with respect to their accuracy. Based on the methods of perturbation theory of linear algebraic systems as discussed above (Design of the optical setup) we described a convenient procedure for the optimization of the interferometer. For the evaluation of the accuracy of data these methods are un-

suitable because they give only an orientation for the maximum possible error. More-powerful results can be derived using the statistical error analysis [19.193]. In addition to the estimation of the measurement uncertainty this method also delivers an elegant approach to the understanding of the influence of the geometry of the interferometer on the propagation of errors in the input data to the calculated displacement vector. For the derivation of the basic error relation a separate consideration of the two types of error (faulty fringe orders and faulty geometric data) is preferred. This method is possible because the effects of these two types of input error are additive to first order. In [19.194] a general equation is derived taking into account both errors simultaneously and is applied to the optimized interferometer. The derivation is based on four basic assumptions, which correspond to the physical conditions of the measuring process 1. The measurement of the vectors N and G is repeated very often. So we get ΔN as a random vector and ΔG as a random matrix. 2. All errors ΔNi are uncorrelated, normally distributed, random variables with zero mean and variance σ N2 i . Therefore E[ΔN] = 0

(19.138)

with E[x] as the expectation value of the random variable x. 3. The errors of the measured Cartesian coordinates Δxi , Δyi , and Δz i are also uncorrelated, normally distributed, random variables with zero mean. This assumption does not imply that the components Δgij (i = 1 . . . n, j = 1, 2, 3) of ΔG have the same properties. It is only assumed that E[ΔG] = 0 .

(19.139)

4. ΔN and ΔG are uncorrelated: E[ΔNΔG] = E[ΔN]E[ΔG] = 0 ,

(19.140)

E[G−1 ΔN] = 0 .

(19.141)

A simple transformation of (19.121) delivers the relation for the error Δd:   (19.142) Δd = λG−1 ΔN − ΔG(G0 )−1 N 0 . If we consider the approximation G−1 ΔG ≈ G−1 0 ΔG ,

(19.143)

Digital Image Processing for Optical Metrology

19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology

543

f u , f υ, f w 9

fu , fυ

8

3

7

2.5

fu , fυ

6

2

5 4

1

3

0.5

2

0

0

fw

1 100

200

300

400

500

600

3 700

800

900

5

1000

7

9

L (mm)

Fig. 19.55 Dependence of the error factors of the in-plane components f u , f v and the out-off-plane component f w on the orthogonal distance L for the object point P0 (0, 0, 0) and a radius ρ = 100 mm

and apply (19.141–19.142) we get E[Δd] = 0 .

(19.144)

The objective of the analysis is to find a statistical estimate of the expected error Δd in terms of ΔN and ΔG. For this purpose the covariance matrix  of d is used [19.193]  = E[(d − E[d])(d − E[d])T ] .

(19.145)

Considering (19.144) this relation can be simplified to  = E[ΔdΔd T ] .

(19.146)

Finally with (19.142) a general expression taking into account both types of error can be derived [19.194]:  = λ2 E[G−1 ΔNΔN T (G−1 )T ]   + E G−1 ΔGd 0 d T0 ΔGT (G−1 )T .

(19.148)

The expression μ = E[ΔNΔN T ] is the covariance matrix of the fringe number vector N. If we assume that the standard deviations σ Ni are identical for all measured fringe numbers Ni we have μ = σN2 I

(19.149)

with I as the identity matrix and finally N = λ2 σN2 F−1 .

13

n

15

1

5

3

7

9

11

13

15

m

Fig. 19.56 Dependence of the error factors f u and f v for in-plane components on the number n of equations

(b) Errors of geometric data ΔG Here it is supposed that ΔN = 0. With this assumption, and taking into account the approximation (19.143), (19.147) is reduced to  −1 T  T T (19.151) G ≈ G−1 0 E ΔGd 0 d 0 ΔG (G0 ) . If the conditions that all elements Δgij of ΔG are uncorrelated and that the standard deviations σSx , σSy and σSz of the three components of the sensitivity vector are identical for all sensitivity vectors is fulfilled we can use the approximation [19.5]   2 2   2 2 2 2 u 0 + σSy v0 + σSz w0 I E ΔGd 0 d T0 ΔGT ≈ σSx (19.152)

and (19.151) can be rewritten as   2 2 2 2 2 2 −1 G ≈ σSx u 0 + σSy v0 + σSz w0 F .

(19.153)

(19.147)

Based on (19.147) three cases can be discussed. (a) Errors of fringe orders ΔN In this case only phase measuring errors are considered: ΔG = 0. Consequently, (19.147) is reduced to N = λ2 G−1 E[ΔNΔN T ](G−1 )T .

11

(19.150)

Za (%) 60 50 40 30 20 10 0

1

3

5

7

9

11

13

15 (m)

Fig. 19.57 Percentage increment in accuracy Z a (%) for (n + m) instead of n equations (n = 3)

Part C 19.2

1.5

544

Part C

Noncontact Methods

a) Interferogram of the loaded car brake

b) Pseudo-3-D-displacement plot

Part C 19.2 W Y

c) Deformed and non-deformed object

d) 3-D-displacement field

2 x10

2.5 x10–3

1.6 x10–3

2.25 x10–3

1.2 x10–3

2 x10–3

–3

8 x10

1.75 x10–3

4 x10–4

1.5 x10–3

0

1.25 x10–3

–4 x10–4

1 x10–3

–8 x10–4

7.5 x10–4

–1.2 x10–3

5 x10–4

–4

–1.6 x10–3 –3

–2 x10

X

Y

X

2.5 x10–4

Z

0

Y Z

X

Fig. 19.58a–d Results of holographic interferometric displacement analysis based on measured shape data to consider the variation of the sensitivity across the object (scales in mm). (a) Interferogram of the loaded car brake (b) pseudo-3-D displacement plot (c) deformed and nondeformed object (d) 3-D displacement field

(c) Errors of fringe orders and geometric data On the condition that both types of error are additive to first order we get a formula to calculate the errors of the displacement components dependent on the phase measurement and shape measurement errors:  ≈ N + G   2 2 2 2 2 2 −1 ≈ λ2 σN2 + σSx u 0 + σSy v0 + σSz w0 F = CNG F−1 .

(19.154)

The main diagonal elements of the covariance matrix  are the variances σu2 , σv2 , and σw2 of the three displacement components. Therefore we call the square roots of the three main diagonal elements f¯ij (i = j = 1, 2, 3)

of the inverse normal matrix F−1 the error factors. As weights these factors control the propagation of the input errors (considered in CNG ) to the standard deviations of the displacement components  σu , f u = f¯11 ≈ √ CNG  σv , f v = f¯22 ≈ √ CNG  σw . (19.155) f w = f¯33 ≈ √ CNG Because the elements f¯ij are determined by the geometry of the setup, the error factors f u , f v , and f w

Digital Image Processing for Optical Metrology

19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology

σw

fw

This result can be applied to the optimized interferometer described in earlier section. In the case of a 360◦ /n symmetry the normal matrix F has only nonzero elements along the main diagonal for the object point P0 (0, 0, 0) (19.134). This provides very simple formulas for the calculation of the error factors  2(ξ 2 + 1) , (19.157) f u (P0 ) = nξ 2  2(ξ 2 + 1) f v (P0 ) = , (19.158) nξ 2  f w (P0 ) =

ξ 2 +1 n

 . 1 + ξ2 + 1

(19.159)

Using these equations it is possible to compare the different behavior of the in-plane components u, v and the out-off plane component w with respect to the error propagation dependent on both of the parameters ρ and L which control the aperture of the interferometer. Such a comparison is shown in Fig. 19.55. The better sensitivity of the method for the out-of-plane component represented by an out-of-plane sensitivity vector (19.119) leads to a lower error sensitivity too. With increasing distance L this quality is changing slowly compared to the increasing error sensitivity for the inplane components. The consideration of this behavior is very important for the investigation of large objects where large distances between the object and the hologram are applied.

Using (19.157–19.159), the dependence of the measuring accuracy on the number n of equations can be investigated. If (n + m) instead of n observation or illumination directions are used, (19.54) is valid for all error factors 1 f u,v,w ∝ √ , n  n (19.160) f u,v,w (n + m) = f u,v,w (n) . n +m Figure 19.56 shows the decreasing trend of the in-plane error factors for increasing number m of additional measurements. However, it is obvious that, the more equations are used, the lower the additional gain in accuracy. The percentage increment in accuracy Z a (%) for (n + m) instead of n equations can be estimated by (19.161) to be    n (19.161) . Z a (%) = 100 1 − n +m In accordance with Dhir and Sikora [19.195], the largest increment is already obtained if only one additional equation is used (n = 3, m = 1). This characteristic behavior of Z a is illustrated in Fig. 19.57. Using these results the experiment can be better planned and controlled with respect to the accuracy of the measured displacement components. Displacement Calculation and Data Presentation If all input data are measured the displacement components can be computed by solving the equation system (19.119). It should be considered, however, that the data acquired with holographic interferometry and optical shape measurement have to be transformed into a joint coordinate system. This means that, for both data sets, a registration problem has to be solved [19.196]. Figure 19.58 shows the results of the investigation of a car brake with respect to the response of an operational load. The shape was measured with fringe projection [19.192] and considered for the calculation of the position-dependent sensitivity vector. Finally, the computed displacement data are mapped onto a FEM mesh to present the response of the object and to provide valuable boundary conditions for the FEM calculation.

19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology Modern optical measurement techniques such as fringe projection, moiré techniques, and holographic and

speckle metrology can be applied not only for highprecision quantitative analysis of displacement and

Part C 19.3

represent the influence of the geometry on the measuring accuracy. Consequently, we have an additional criterion for the assessment of the interferometer geometry. However, in contrast to the condition number κ, this criterion is more significant. It delivers an approach to the separate evaluation of the error influence for each of the three displacement components. For the estimation of the standard deviations of the displacement components it follows that ⎛ ⎞ ⎛ ⎞ fu σu  ⎜ ⎟ ⎜ ⎟ (19.156) ⎝ σv ⎠ = CNG ⎝ f v ⎠ .

545

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Part C 19.3

strain fields but also for nondestructive inspection of simple objects and complex structures with regard to surface and internal flaws. These flaws are recognized by the evaluation of the resulting fringe patterns with respect to characteristic pattern irregularities such as ‘bull’s eye’ fringes, distorted fringes, locally compressed fringes, and dissected and mutually displaced fringes. Based on practical experience and knowledge about the material behavior as well as on the boundary conditions of the experiment, the flaw can be classified as, e.g., a void, debond, delamination, weak area, crack, or similar imperfection. a) Set-up for holographic nondestructive testing interferometry Load Object

Hologram CCD camera Laser Image processing system

b) Example of an interferogram with a flaw (subsurface void) indicating pattern

The methods have been successfully applied for industrially relevant testing problems in different fields: quality control of circuit boards and electronic modules, inspection of satellite fuel tanks and pressure vessels, tire testing, glass- and carbon-fibre-reinforced material testing, investigation of turbine blades, automobile engine and car body inspection, and building and artwork inspection (see, e.g., [19.184]). However, automatic evaluation does not work perfectly in many cases, since the recognition and classification of faultindicating patterns in a fringe patterns is often more difficult than the quantitative phase reconstruction applied in displacement and shape measurement. The task consists not only of the automatic recognition of complex patterns within noisy images but also of the systematic evaluation of these patterns with respect to their cause. First results were reported by Glünder et al. [19.17] in 1982. In this work a hybrid optoelectronic processor was used for fast and effective data reduction. A complete digital system for the analysis of misbrazing in brazed cooling panels was proposed by Robinson [19.82] in 1983. When the plate was slowly pressurized, misbrazing was observed as a closed-ring fringe pattern before the standard deformation pattern appears. However, the published evaluation procedure was very time consuming. Another approach [19.197] was proposed that applies parallel hardware. In less than 1 min the skeleton of the fringe pattern was derived and two line features (density and curvature) were determined to recognize flaw-induced patterns. Statistical methods have been applied to the fringe system since the beginning of automatic evaluation in order to detect inhomogeneities [19.198]. The idea was to detect anomalously higher and lower fringe densities in either the spatial or frequency domain. This works quite well for simple geometries. However, when the geometry of

eQ Prism P (x, y)

Fig. 19.59a,b Optical nondestructive testing by holographic interferometry (HNDT). (a) Setup for holographic nondestructive testing interferometry (b) example of an interferogram with a flaw (subsurface void-indicating pattern)

α

P (x+Δx, y) Object

eB

Lens CCD target

Fig. 19.60 Experimental setup for speckle shearography

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19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology

a) Shearogram

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b) Demodulated phase image

Part C 19.3

Fig. 19.61a,b Shearogram of an aircraft sandwich plate loaded by a light flash. The so-called butterfly patterns are indications of subsurface voids caused by local debonding. (a) Shearogram (b) demodulated phase image

the component is complex, the fringe system becomes complex and will often contain inhomogeneities even in the absence of defects. So these methods fail for widespread application of holographic nondestructive testing (HNDT). The fact that skilled operators can detect fault-indicating fringe pattern changes even in complex structures leads to the approach of using processing techniques which apply knowledge to the evaluation of the fringes. The most obvious deficit in all published procedures is the insufficient capability for flaw classification. Two modern approaches have good potential to overcome this unsatisfactory situation. The first one applies knowledge-based systems or neural networks to “learn” different kinds of flaws from simulated or practical examples [19.199–201]. The second approach combines model-based simulation methods and practical measurements [19.202–206]. After a short introduction in the technology of optical nondestructive testing (ONDT) two promising approaches are discussed: a knowledge-based [19.205] approach and an iterative approach that combines an experimental and a simulative part [19.204, 206].

19.3.1 The Technology of Optical Nondestructive Testing (ONDT) The basic idea in optical nondestructive testing is that a component with a fault will react in a different manner compared to a sound structure, since the stiffness or the heat conductivity will be modified locally

or globally [19.207]. The response of the object on a suitable load results in characteristic fringe patterns when optical methods such as holographic interferometry are applied (Fig. 19.59a). The fringe pattern such as that shown in Fig. 19.59b can be evaluated to determine the displacement and the deformation. However, in ONDT the objective is not to evaluate surface deformations quantitatively but to determine unacceptable local deformations identified by inhomogeneities in the fringe pattern. The task consists of the detection of fault-indicating fringe irregularities and in their classification with respect to the fault that caused them. However, not all inhomogeneities indicate a defect. Consequently, experience and a priori knowledge about the object and its response to artificial and working loads, respectively, is very important for ONDT. Speckle shearography (Chap. 23) is a very promising technology for industrial inspection. It delivers a direct approach to derivations of the displacement components by shearing of two wavefronts coming from the same object (Fig. 19.60). Consequently one point in the image plane receives contributions from two different points P(x, y) and P (x + Δx, y) on the object (here a lateral shear Δx was used as an example). Due to a deformation both points are displaced relatively, which results in a relative phase difference δ(P, P ) [19.208]: δ(P, P ) = 2π/λ[(∂w/∂x)(1 + cos α) + (∂u/∂x) sin α]Δx

(19.162)

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with α (Fig. 19.60) as the angle between the illumination and the observation direction. Shearography is a very robust technique since both sheared wavefronts act as mutual references. Consequently the disturbing influence of rigid-body motions in

HNDT is removed. Figure 19.61 shows a shearogram of an aircraft sandwich panel that was loaded by a light flash to show some internal faults (debonding) [19.209]. Some recent results are published in [19.210, 211].

Part C 19.3 Fig. 19.62 Various fringe patterns caused by surface and subsurface faults

Digital Image Processing for Optical Metrology

19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology

549

a) Object

Sensor including detector

Preprocessing

Feature extraction

Evaluation

Illumination

Object

Sensor including detectors

Preprocessing

Feature extraction

Evaluation

b)

Actuators

Fig. 19.63a,b Different image processing strategies: (a) Passive approach: classical data-driven bottom-up strategy, (b) active approach: combination with the expectation-driven top-down strategy by driving a feedback loop between

the components responsible for image analysis and image acquisition

19.3.2 Direct and Indirect Problems The problems in optical metrology are very similar to those of computer vision. Both disciplines process an image as input and use methods of image analysis to derive a symbolic description of the image content or to reconstruct various physical quantities from the acquired intensity distribution. The well-known paradigm of Marr [19.212], according to which computer vision is the development of procedures for the solution of the inverse task of the image formation process, describes nothing else but the task to conclude from the effect (e.g. the intensity I (i, j) in the pixel (i, j)) to its cause (e.g. the displacement of the related object point P(x, y, z)). In other words, an inverse problem has to be solved. From the point of view of a mathematician the concept of an inverse problem has a certain degree of ambiguity which is well illustrated by a frequently quoted statement of Keller [19.213]: “We call two problems inverses of one another if the formulation of each involves all or part of the solution of the other. Often for historical reasons, one of the two problems has been studied extensively for some time, while the other has never been studied and is not so well understood. In such cases, the former is called direct problem, while the latter is the inverse problem.” Both problems are related by a kind of duality in the sense that one problem can be derived from the other by exchanging the role of the data and that of the unknown: the data of one problem are the unknowns of the other, and vice versa. As a consequence of this duality it may seem arbitrary to decide which is the direct and what is the

inverse problem. For physicists and engineers, however, the situation is quite different because the two problems are not on the same level [19.214]: one of them, and precisely that called the direct problem, is considered to be more fundamental than the other and, for this reason, is also better investigated. Consequently, the historical reasons mentioned by Keller are basically physical reasons. Processes with a well-defined causality such as the process of image formation are called direct problems. Direct problems need information about all quantities which influence the unknown effect. Moreover, the internal structure of causality, all initial and boundary conditions, and all geometrical details have to be formulated mathematically [19.151]. To them belong well-known initial and boundary value problems which are usually expressed by ordinary and partial differential equations. Such direct problems have some excellent properties which make them so attractive for physicists: if reality and mathematical description fit sufficiently well, the direct problem is expected to be uniquely solvable. Furthermore, it is in general stable, i. e., small changes of the initial or boundary conditions cause also small effects only. Unfortunately, numerous problems in physics and engineering deal with unknown but nonobservable values. If the causal connections are investigated backwards we come to the concept of inverse problems. Based on indirect measurements, i. e., the observation of effects caused by the quantity we are looking for, one can try to identify the missing parameters. Such problems, which are also called identification

Part C 19.3

Illumination

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Part C 19.3

problems, are well known in optical metrology: the recognition and interpretation of subsurface flaws using ONDT and the reconstruction of phase distributions from the observed intensity. Inverse problems usually have some undesirable properties: they are in general ill-posed, ambiguous, and unstable. The concept of well-posedness was discussed in Sect. 19.2.4 in the context of phase unwrapping. For the solution of inverse/ill-posed problems it is important to apply a maximum amount of a priori knowledge or predictions about the physical quantities to be determined. The question of whether the measured data contain enough information to determine the unknown quantity uniquely has to be answered. In the case of direct problems, where the data result from the integration of unknown components, data smoothing happens. Consider the example of the response of a fault under the surface to an applied load. The effect can only be measured at the surface after its propagation through the material. Consequently, the direct problem is a problem directed towards a loss of information and its solution defines a transition from a physical quantity with a certain information content to another quantity with a smaller information content. This implies that the solution is much smoother than the corresponding object. Consequently, the information about certain properties of the defect gets lost and therefore very different causes may give almost the same effect after integration. A well-known Experiment

Analysis

Simulation

Loading

Hypothesis

Measurement

Simulation

Interferogram

Analysis

Interferogram

Feature extraction

Result

Feature extraction

Features

Features

Fig. 19.64 Implementation of the recognition by synthesis algo-

rithm in HNDT: feedback loop and combination of a simulation loop for running the direct problem and a measurement loop for running the indirect problem

example for specialists in HNDT is the ambiguous relation between an observed effect – the fringe pattern appearing on the surface – and its corresponding cause under the surface – the material fault. The response of the fault to the applied load is smoothed since only the displacement on the surface gives rise to the observed fringe pattern. On the one hand these fringe pattern show a limited topology because of the mentioned integration process [19.205, 206]. On the other hand, a large variety of different fringe patterns can be observed depending on the structural properties of the object under test, the existing faults, and the applied loading (Fig. 19.62). Therefore, the practical experience shows that fixed recognition strategies based on known flaw–fringe pattern relations are successful only if the boundary conditions of the test procedure are limited in an inadmissible way. Since such conditions cannot be guaranteed, generally more-flexible recognition strategies have to be developed. In order to overcome the disadvantages of illposedness in the process of finding an approximate solution to an inverse problem, various techniques of regularization are used. Regularizing an inverse problem means that, instead of the ill-posed original problem, a well-posed neighboring problem has to be formulated. The key decision of regularization is to find an admissible compromise between stability and approximation [19.151]. The formulation of a sufficiently stable auxiliary problem means that the original problem has to be changed accordingly radically. Most of the classical regularization procedures [19.152,153] refer to a spatial neighborhood and are applied in passive methods of image analysis. In contrast to these passive methods, active approaches in optical metrology adopt another way to handle the difficult regularization problem. This is assured by formulating an adequately stable auxiliary problem by forming a temporal neighborhood and adding systematically more a priori and experimental knowledge about the object under test into the evaluation process. A practical way to do that is the implementation of a feedback loop including the image formation process to create an expectation-controlled data input, (Fig. 19.63b). In the next section we turn our attention to a more detailed description of two modern approaches for the solution of the identification problem in HNDT [19.206]. One approach uses an active strategy for the classification of fault-indicating fringe patterns that can be described by the term recognition by synthesis. The other approach is based on a mathematical

Digital Image Processing for Optical Metrology

19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology

a) Real interferogram of the loaded tank with fault indicating pattern

551

b) Simulated interferogram of the loaded tank with fault indicating pattern

Part C 19.3

c) FE-model of the satellite tank

d) Calculated displacement field of the tank loaded with inner pressure

Fig. 19.65a–d Recognition by synthesis: Some processing steps on example of a model of a satellite tank with local wall weakening. (a) Real interferogram of the loaded tank with fault-indicating pattern (b) simulated interferogram of the loaded tank with fault-indicating pattern (c) finite element (FE) model of the satellite tank (d) calculated displacement field of the tank loaded with inner pressure

modeling of the fringe pattern formation process and introduces valuable a priori knowledge about the appearance of material faults in fringe patterns into the evaluation.

19.3.3 Fault Detection in HNDT Using an Active Recognition Strategy Following the current trend in image analysis, more flexibility in the processing strategy is obtained by combining the classical data-driven bottom-up strategy (Fig. 19.63a), with the so-called expectation-driven top-down strategy [19.215] (Fig. 19.63b). The former strategy has been proven to be very efficient but extra

effort must be paid to obtain a high image quality and in most cases a priori knowledge has to be added by operator interaction to derive an unambiguous solution. The image formation process is considered as a rather fixed/passive data source and is not actively involved in the evaluation process. In contrast to this approach the second strategy includes the image formation as an active component in the evaluation process. Dependent on the complexity of the problem and the state of evaluation, new data sets are actively produced by driving a feedback loop between the system components that are responsible for data generation (light sources, sensors and actuators) and those that are responsible for data pro-

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Noncontact Methods

a) Eye

b) Groove

c) Bend

d) Displacement

e) Compression

Part C 19.3 Fig. 19.66a–e Basic fringe pattern types in HNDT. (a) Eye (b) groove (c) bend (d) displacement (e) compression

a) 3-dimensional representation of the function together with several contour lines

b) Gradients and tangents in chosen points on the contour lines

cessing and analysis (computers). Support from other sensors at different positions, recordings from different time instances or the exploitation of different physical sensor principles, i. e., multisensor approaches, are considered in this concept [19.13, 216, 217]. The strategy that supervises the image analysis is connected to and controlled by the information gathered by sensors and by a knowledge base. This database contains the relevant knowledge about the application, the scene which has been imaged, and the image acquisition/processing methods. Fault Recognition by Synthesis of the Direct and the Indirect Approaches The active approach discussed here combines knowledge about the pattern formation process with the possibilities of modern computer-aided engineering (CAE) tools [19.218, 219]. The basic idea and the implemented processing scheme of this method is shown in Fig. 19.64. At the beginning an interferometric measurement (e.g., made with holographic interferometry, EPSI, or shearography) creates an interferogram containing information about the object under test. Figure 19.65a shows the holographic interferogram of Fig. 19.67a,b The Gaussian function 2

2

f (x, y) =

√1 2π

exp −x 2−y as an example of phase difference. (a) Threedimensional representation of the function together with several contour lines (b) gradients and tangents at chosen points on the contour lines

Digital Image Processing for Optical Metrology

19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology

a) Node

tions. The active approach also considers the generation of a sequence of interferograms caused by a continuous increasing/decreasing of the load and/or a change of the kind of load (thermal load, pressure load, etc.). As a result of this iterative approach an improved hypothesis containing the desired information such as the type of the fault and its criticality, dimension, and location is derived. The feedback loop is finished if the comparison between the fringe patterns delivered by the simulation loop (the direct problem) and the measurement loop (the indirect problem) are in sufficient agreement. The searched fault parameters are defined by the input data of the finite element model of the final interferogram synthesis. The fault detected in the given example was a weakening of the inner part of the satellite tank wall with the parameters: spheric shape, depth 0.8 mm, and diameter 5 mm. The detection certainty could be improved considerably by an active loading strategy [19.221,222]. b) Saddle

υ

υ

u

c) Spiral

u

d) Centre

υ

υ

u

Fig. 19.68a–d Selected critical points for linear systems. (a) Node (b) saddle (c) spiral (d) center

u

Part C 19.3

a model of a satellite tank made from aluminum and loaded with inner pressure [19.220]. A first analysis (e.g., by skeletonization or phase shifting) and feature extraction allows a hypothesis about the object and its fault to be made. This hypothesis serves as a basis for a finite element model of the object, including the fault and considering the experimental conditions (Fig. 19.65c,d). With the known circumstances of the experimental setup a synthetic interferogram of the object is generated (Fig. 19.65b) followed by a second analysis and feature extraction. All properties (patterns, phase values, and relevant features) of both interferograms are compared to decide if the supposed hypothesis is correct [19.221, 222]. If the difference between the calculated and the measured pattern is too large, an iterative process is started to improve the conformance. In this process the hypothesis is modified in combination with a manipulation of the load (strength and/or type) or other known boundary condi-

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a) Centre

Part C 19.3 b) Saddle

Fig. 19.69a,b Centers and saddle points in fringe patterns. (a) Center (b) saddle

Fault Recognition by Using A Priori Knowledge about the Pattern Formation Process The main problem in HNDT was already discussed: the causality problem. Faults located under the surface can only be recognized due to their response on the surface caused by a load applied to the specimen. However, as shown in Fig. 19.62, this connection causes the difficulty of the great variety of fringe patterns that correspond to a limited class of typical material faults. Thus, the question remains of whether there is a limited number of so-called basic fringe patterns that are representative of the main types of material faults. A positive answer would be a great advantage for all knowledgebased approaches because the target group of relevant patterns to be detected could be reduced considerably. This hypothesis and its proof is discussed below.

Despite the huge range of fringe patterns appearing an experienced human interpreter is able to detect material faults on the basis of fringe pattern irregularities caused by a local deformation of the surface. Both extended practical testing and systematic numerical investigations lead to the hypothesis that the appearance of faults in interferograms can be reduced to a finite set of typical fringe irregularities [19.199, 203]. Five so-called basic patterns are considered (Fig. 19.66). Two of these are characterized by topological changes of the regular fringe pattern whereas the others show rather quantitative geometrical variations. The first basic pattern, which is known as the bull’s eye, can be observed very often. It is marked by concentric fringes. The second one is the so-called groove. It typically shows systematic distortions of neighboring fringes, which is connected to a local change of fringe curvature. A bend is defined by a local, sudden change of the direction of some fringes. Locally ending and laterally displaced fringes are called a displacement. A compression is characterized by a change of the spatial frequency of the fringes. The proof that no other fault-indicating irregularities can appear in holographic interferograms is based on both a description of fringe patterns by systems of ordinary differential equations and the physical relationship between object loading and the resulting displacement [19.205, 223]. The observed intensity I (x, y) in the pixel (x, y) of a digitized double-exposure interferogram can be described by (19.40). Such a cosine-modulated intensity distribution appears as a high-contrast pattern of bright and dark fringes. Bright regions correspond to intensity maxima, whereas dark fringes can be observed in places of intensity minima. After reducing the fringe pattern to a map of dark and bright lines the resulting lines can be considered as contour lines of the phase difference δ(x, y) (Fig. 19.67). For any point (x, y) situated on one of these lines it holds that δ(x, y) = Nπ, with an even number N for the bright lines and an odd number N for dark lines. A contour line consists of all those points for which the phase difference has the same constant value Nπ. The properties of these contour lines were already mentioned before. Two of them follow from their definition:

• •

different contour lines have no point (x, y) in common, exactly one contour line passes through each point (x, y).

Digital Image Processing for Optical Metrology

19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology

h(t) = δ(x(t), y(t)) .

(19.163)

Some more calculations (for details see [19.223]) bring us to the position that the description of a contour line crossing the point (x0 , y0 ) and satisfying δ(x0 , y0 ) = c corresponds to the solutions x, y of the initial-value problem x  (t) = −δ y (x(t), y(t)) , y (t) = δx (x(t), y(t)) , δ(x0 , y0 ) = c .

equations (19.164) are linear functions of x and y, so that we get x  (t) = ax(t) + by(t) , y (t) = cx(t) + dy(t)

with constants a, b, c, and d. Solutions are given by the eigenvalues of the coefficient matrix. It is assumed that no eigenvalue is zero. Different cases have to be distinguished

• • •

(19.164)

This initial-value problem is characterized by three important properties. Firstly, the equation system (19.164) is a two-dimensional system. Secondly, the right-hand sides of the equations do not depend explicitly on the parameter t. Therefore (19.164) form a plane, autonomous system of differential equations. Systems of this kind are well understood in the theory of ordinary differential equations. Their solutions can be represented as orbits in the plane, where an orbit is the set of points traced out by the solution (x(t), y(t)) if the parameter t varies. Additionally, (19.164) is a Hamiltonian system with the Hamiltonian function δ(x(t), y(t)) because both differential equations of (19.164) are obtained from the same scalar function δ(x, y) as partial derivatives in an antisymmetric manner. Consequently, the initial-value problem (19.164) representing the contour lines of the phase δ is a plane, autonomous Hamiltonian system. Based on this assumption, interesting conclusions can be drawn about the appearance of interference patterns by investigating solutions of ordinary differential equations of the type (19.164). With respect to the understanding of the behavior of contour lines we are interested in whether the gradient of the phase δ disappears. Due to the structure of the initial-value problem (19.164) the right-hand sides of both differential equations are the zero and we get so-called critical points. A critical point can be considered as an orbit which is degenerated to a single point. It is useful to determine the topology of the orbits in the neighborhood of such critical points. In the case of linear systems, the right-hand sides of both differential

(19.165)

•

If both eigenvalues are real and different and if they have the same sign then the orbits describe a socalled node (Fig. 19.68a); If both real eigenvalues have opposite sign then the critical point is called a saddle point (Fig. 19.68b); If the eigenvalues are a complex-conjugate pair α ± iβ with α = 0 and β = 0 then the orbits in the neighborhood of the critical describe a spiral point (Fig. 19.68c); If both eigenvalues are pure imaginary and form a complex-conjugate pair ±iβ with β = 0 then the orbits are closed lines and the critical point is a center (Fig. 19.68d).

For the Hamiltonian system (19.164) it can be shown that critical points are centers, spiral or saddle points. Furthermore, it can be proven that in autonomous Hamiltonian systems spiral points cannot occur and nondegenerated critical points are either saddle points, corresponding to a saddle of the phase difference δ, or centers corresponding to a maximum or minimum of the phase difference δ [19.223]. Consequently, spirals and nodes do not occur as contour lines in fringe patterns. A center is defined as a family of closed orbits which contains the critical point (Fig. 19.69a). This situation is caused by a minimum or a maximum of the phase difference δ. Saddle points are characterized by two dominant orbits which converge to the saddle point and gather all other orbits (Fig. 19.69b). The hypothesis that the number of fault-indicating irregularities in fringe patterns is limited, i. e., that each detected material fault can be classified as one of the mentioned basic patterns (or as a combination of them) can be supported on the basis of the above described considerations. For this purpose, two types of surfaces have to be distinguished: continuous and noncontinuous object surfaces. The analysis results in the following statements:

Part C 19.3

If the phase difference δ is a differentiable function and if its gradient is not zero then the contour line δ−1 (Nπ) can be represented as a curve in the (x, y)-plane. The task to determine contour lines of any differentiable function δ leads to a description of the problem by ordinary differential equations. To this end we assume that a pair of functions x, y exists that delivers a parametric description of the considered contour line

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b) Skeleton

c) Direction field with marked critical regions

d) Detected pattern

Part C 19.3

a) Interferogram (crack)

Fig. 19.70 Detection of a fault indicating patterns on example of a crack

•

•

In the case of continuous surfaces only centers may be caused by material faults occurring under the surface. In the fringe pattern they appear as eyes or grooves depending on the surface topology and the amount of the surface displacement; In the case of macroscopically noncontinuous surfaces where, e.g., cracks or edges occur, on both sides of the discontinuity all considerations regarding the first case are valid. At the border line fringes seem to be displaced against each other. The fringe pattern looks like a displacement.

Based on this knowledge the final procedure of processing the fringe patterns with respect to the detection and classification of fault indicating patterns is divided into three steps: Step 1: Preprocessing Here the basis is the rough interferogram and the objective consists in the reduction of the compex pattern

to a fringe skeleton, its vectorization and transformation to a condensed data list containing relevant metrical and topological information about the skeleton lines. Step 2: Model based pattern recognition (1st classification step) Here the basis is the derived data list and the binary skeleton image. The objective of step 2 consists in the extraction of relevant features from the data list with respect to the pre-classification of fault indicating patterns. The step results in detected regions pre-classified for fault presence. Step 3: Knowledge based flaw classification (2nd classification step) Here the basis is the skeleton image with preclassified regions indicating flaw presence. The objective of step 3 consists in the knowledge assisted fine classification taking into account additional object knowledge and results in a hypothesis about the fault type An important step is the preprocessing of the origi-

Digital Image Processing for Optical Metrology

results in a skeleton of the pattern, Fig. 19.70b . The preclassification - that means the recognition of the fault indicating pattern - is running automatically. The recognition ability is demonstrated on example of the class displacement, Fig. 19.66d. The highlighted square indicates where the computer has found a fault indicating pattern based on the evaluation of such features as the direction and the density of the fringes, Fig. 19.70c,d.

References 19.1

19.2

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19.4

19.5 19.6 19.7

19.8

19.9

19.10

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19.168 H. Tiziani: Optische Verfahren zur Abstands- und Topografiebestimmung, Informationstechnik 1, 5– 14 (1991) 19.169 T. Pfeifer, J. Thiel: Absolutinterferometrie mit durchstimmbaren Halbleiterlasern, Tech. Mess. 60(5), 185–191 (1993) 19.170 P.J. de Groot: Extending the unambiguous range of two-color interferometers, Appl. Opt. 33(25), 5948–5953 (1994) 19.171 M. Takeda, H. Yamamoto: Fourier-transform speckle profilometry: three-dimensional shape measurement of diffuse objects with large height steps and/or spatially isolated surfaces, Appl. Opt. 33, 7829–7837 (1994) 19.172 Y. Zou, G. Pedrini, H. Tiziani: Surface contouring in a video frame by changing the wavelength of a diode laser, Opt. Eng. 35(4), 1074–1079 (1996) 19.173 S. Kuwamura, I. Yamaguchi: Wavelength scanning profilometry for real-time surface shape measurement, Appl. Opt. 36(19), 4473–4482 (1997) 19.174 W. Nadeborn, P. Andrä, W. Osten: A robust procedure for absolute phase measurement, Opt. Laser. Eng. 24, 245–260 (1996) 19.175 H.O. Saldner, J.M. Huntley: Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector, Opt. Eng. 36, 610–615 (1997) 19.176 C. Wagner, W. Osten, S. Seebacher: Direct shape measurement by digital wavefront reconstruction and multi-wavelength contouring, Opt. Eng. 39(1), 79–85 (2000) 19.177 R. Cusack, H.M. Huntley, H.T. Goldrein: Improved noise-immune phase unwrapping algorithm, Appl. Opt. 35, 781–789 (1995) 19.178 G. Pedrini, I. Alexeenko, W. Osten, H.J. Tiziani: Temporal phase unwrapping of digital hologram sequences, Appl. Opt. 42, 5846–5854 (2005) 19.179 GUM: Guide for the Expression of Uncertainty in Measurement (ISO, Genf 1995) 19.180 W. Osten, W. Jüptner: Measurement of displacement vector fields of extended objects, Opt. Laser Eng. 24, 261–285 (1996) 19.181 R.J. Pryputniewicz: Experiment and FEM modeling, Fringe’93, Proc. of the 2nd Intern. Workshop on Automatic Processing of Fringe Patterns, ed. by W. Jüptner, W. Osten (Akademie Verlag, Bremen, Berlin 1993) pp. 257–275 19.182 N. Abramson: The holo-diagram: a practical device fo making and evaluating holograms, Appl. Opt. 8, 1235–1240 (1969) 19.183 N. Abramson: Sandwich hologram interferometry. 4. Holographic studies of two milling machines, Appl. Opt. 18, 2521 (1977) 19.184 G. Birnbaum, C.M. Vest: Holographic nondestructive evaluation: status and future, Int. Adv. Nondestruct. Test 9, 257–282 (1983) 19.185 J.E. Sollid, K.A. Stetson: Strains from holographic data, Exp. Mech. 18, 208–216 (1978)

19.186 J. Stoer: Einführung in die numerische Mathematik I (Springer Verlag, Berlin, Heidelberg, New York 1972) 19.187 W. Osten: Theory and practice in optimization of holographic interferometers, Proc. SPIE 473, 52–55 (1984) 19.188 S. Seebacher, W. Osten, W. Jüptner: 3DDeformation analysis of microcomponents using digital holography, Proc. SPIE 3098, 382–391 (1997) 19.189 W. Osten, F. Häusler: Zur Optimierung holografischer Interferometer, Preprint P-Mech-08/81 19.190 W. Osten: The application of optical shape measurement for the nondestructive evaluation of complex objects, Opt. Eng. 39(1), 232–243 (2000) 19.191 V. Srinivasan, H.C. Liu, M. Halioua: Automated phase-measuring profilometry of 3-D diffuse objects, Appl. Opt. 23, 3105–3108 (1984) 19.192 H. Tiziani: Optical techniques for shape measurement, Fringe’93, Proc. of the 2nd Intern. Workshop on Automatic Processing of Fringe Patterns, ed. by W. Jüptner, W. Osten (Akademie Verlag, Berlin, Bremen 1993) pp. 165–174 19.193 D. Nobis, C.M. Vest: Statistical analysis of errors in holographic interferometry, Appl. Opt. 17, 2198– 2204 (1978) 19.194 W. Osten: Some considerations on the statistical error analysis in holographic interferometry with application to an optimized interferometer, Opt. Acta 32(7), 827–838 (1985) 19.195 S.K. Dhir, J.P. Sikora: An improved method for obtaining the general displacement field from a holographic interferogram, Exp. Mech. 12, 323– 327 (1972) 19.196 P. Andrä, A. Beeck, W. Jüptner, W. Nadeborn, W. Osten: Combination of optically measured coordinates and displacements for quantitative investigation of complex objects, Proc. SPIE 2782, 200–210 (1996) 19.197 W. Osten, J. Saedler, W. Wilhelmi: Fast evaluation of fringe patterns with a digital image processing system, Laser Mag. 2, 58–66 (1987), (in German) 19.198 W. Jüptner: Nondestructive testing with interferometry, Proc. Fringe’93, ed. by W. Jüptner, W. Osten (Akademie Verlag, Berlin 1993) pp. 315–324 19.199 W. Osten, W. Jüptner, U. Mieth: Knowledge assisted evaluation of fringe patterns for automatic fault detection, Proc. SPIE 2004, 256–268 (1993) 19.200 W. Jüptner, T. Kreis, U. Mieth, W. Osten: Application of neural networks and knowledge based systems for automatic identification of fault indicating fringe patterns, Proc. SPIE Interferometry’94, Vol. 2342 (1994) pp. 16–26 19.201 T. Kreis, W. Jüptner, R. Biedermann: Neural network approach to nondestructive testing, Appl. Opt. 34(8), 1407–1415 (1995) 19.202 T. Bischof, W. Jüptner: Determination of the adhesive load by holographic interferometry using the

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of Visual Information (Freeman, San Francisco 1982) J.B. Keller: Inverse Problems, Am. Math. Mon. 83, 107–118 (1976) M. Bertero, P. Boccacci: Introduction to Inverse Problems in Imaging (IOP, Bristol 1998) C.E. Liedtke: Intelligent approaches in image analysis,. In: Laser Interferometry IX: Techniques and Analysis, Proc. SPIE, Vol. 3478, ed. by M. Kujawinska, G. Brown, M. Takeda (SPIE, Bellingham 1998) pp. 2–10 W. Osten: Active optical metrology – a definition by examples, Proc. SPIE 3478, 11–25 (1998) W. Osten, W. Jüptner: New light sources and sensors for active optical 3D-inspection, Proc. SPIE 3897, 314–327 (1999) W. Osten, F. Elandaloussi, W. Jüptner: Recognition by synthesis – a new approach for the recognition of material faults in HNDE, Proc. SPIE 2861, 220–224 (1996) F. Elandaloussi, W. Osten, W. Jüptner: Automatic flaw detection using recognition by synthesis: Practical results, Proc. SPIE 3479, 228–234 (1998) T. Merz, F. Elandaloussi, W. Osten, D. Paulus: Active approach for holographic nondestructive testing of satellite fuel tanks, Proc. SPIE 3824, 8–19 (1999) F. Elandaloussi: Erkennung durch Synthese – Modellgestützte Detektion und Analyse von Materialfehlern an technischen Objekten. Ph.D. Thesis (Univ. Bremen, Bremen 2002) F. Elandaloussi, W. Osten, W. Jüptner: Detektion und Analyse von Materialfehlern nach dem Prinzip Erkennung durch Synthese, Tech. Mess. 5, 227–235 (2002) U. Mieth: Erscheinungsbild von Materialfehlern in holografischen Interferogrammen, Series Strahltechnik, Vol. 11 (BIAS Verlag, Bremen 1998), (On the appearance of material faults in holographic interferograms.)

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result of FEM-calculations, Proc. SPIE 1508, 90–95 (1991) U. Mieth, W. Osten, W. Jüptner: Numerical investigations on the appearance of material flaws in holographic interference patterns, Proc. International Symposium on Laser Application in Precision Measurement (Akademie Verlag, Berlin, Balatonfüred 1996) pp. 218–225 W. Osten, F. Elandaloussi, W. Jüptner: Recognition by synthesis – a new approach for the recognition of material faults in HNDE, Proc. SPIE 2861, 220–224 (1996) U. Mieth, W. Osten, W. Jüptner: Investigations on the appearance of material faults in holographic interferograms, Proc. Fringe 2001 (Elsevier, 2001) pp. 163–172 W. Osten, F. Elandaloussi, U. Mieth: Trends for the solution of identifikation problems in holographic non-destructive testing (HNDT), Proc. SPIE 4900, 1187–1196 (2002) W. Jüptner, T. Kreis: Holographic NDT and visual inspection in production line application, Proc. SPIE 604, 30–36 (1986) Y.Y. Hung: Displacement and strain measurement. In: Speckle Metrology, ed. by R.K. Erf (Academic, New York 1987) pp. 51–71 M. Kalms, W. Osten: Mobile shearography system fort the inspection of aircraft and automotive components, Opt. Eng. 42(5), 1188–1196 (2003) V. Tornari, A. Bonarou, P. Castellini, E. Esposito, W. Osten, M. Kalms, N. Smyrnakis, S. Stasinopulos: Laser based systems for the structural diagnostic of artworks: an application to XVII century Byzantine icons, Proc. SPIE 4402, 172–183 (2001) M. Y. Y. Hung, Y. S. Chen, S. P. Ng, M. S. Shepard; Y. Hou, J. R. Lhota: Review and comparison of shearography and pulsed thermography for adhesive bond evaluation, Opt. Eng. 46(5), 051007-1-16 (2007) D. Marr: Vision: A Computational Investigation into the Human Representation and Processing

References

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Digital Image 20. Digital Image Correlation for Shape and Deformation Measurements

Michael A. Sutton

response in a heterogeneous weld zone. Sections 20.11 and 20.12 present applications using 3-D image correlation to quantify material response during quasistatic and dynamic tension–torsion loading, respectively, of an edge-cracked specimen. Section 20.13 presents closing remarks regarding the developments.

20.1 Background ......................................... 566 20.1.1 Two-Dimensional Digital Image Correlation (2-D DIC) ................... 567 20.1.2 Three-Dimensional Digital Image Correlation (3-D DIC) ................... 568 20.2 Essential Concepts in Digital Image Correlation .................. 568 20.3 Pinhole Projection Imaging Model ......... 20.3.1 Image Distortion ........................ 20.3.2 Camera Calibration for Pinhole Model Parameter Estimation........ 20.3.3 Image-Based Objective Function for Camera Calibration ................

569 571 572 572

20.4 Image Digitization ................................ 573 20.4.1 Discussion ................................. 573 20.5 Intensity Interpolation.......................... 573 20.5.1 Discussion ................................. 575 20.6 Subset-Based Image Displacements ....... 575 20.6.1 Discussion ................................. 577 20.7 Pattern Development and Application .... 577 20.7.1 Discussion ................................. 579 20.8 Two-Dimensional Image Correlation (2-D DIC) .............................................. 20.8.1 2-D Camera Calibration with Image-Based Optimization .. 20.8.2 Object Displacement and Strain Measurements ........... 20.8.3 Discussion .................................

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The essential concepts underlying the use of two-dimensional (2-D) digital image correlation for deformation measurements and threedimensional (3-D) digital image correlation for shape and deformation measurements on curved or planar specimens are presented. Twodimensional digital image correlation measures full-field surface displacements with accuracy on the order of ±0.01 pixels on nominally planar specimens undergoing arbitrary in-plane rotations and/or deformations. Three-dimensional digital image correlation measures the complete 3-D surface displacement field on curved or planar specimens, with accuracy on the order of ±0.01 pixels for the in-plane components and Z/50 000 in the out-of-plane component, where Z is the distance from the object to the camera, for typical stereo-camera arrangements. Accurate surface strains can be extracted from the measured displacement data for specimens ranging in size from many meters to microns and under a wide range of mechanical loading and environmental conditions, using a wide range of imaging systems including optical, scanning electron microscopy, and atomic force microscopy. In Sect. 20.2, the essential concepts underlying both 2-D DIC and 3-D DIC are presented. Section 20.3 introduces the pinhole imaging model and calibration procedures. Sections 20.4 and 20.5 describe the image digitization and image reconstruction procedures, respectively, for accurate, subpixel displacement measurement. Section 20.6 presents the basics for subset-based, image pattern matching. Section 20.7 provides a range of methods for applying random texture to a surface. Sections 20.8 and 20.9 provide the basics for calibration and deformation measurements in 2-D DIC and 3-D DIC applications, respectively. Section 20.10 presents an example using 2-D image correlation to extract the local stress–strain

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20.9 Three-Dimensional Digital Image Correlation ...................... 20.9.1 Camera Calibration ..................... 20.9.2 Image Correlation for Cross-Camera Matching.......... 20.9.3 3-D Position Measurement with Unconstrained Matching .................................. 20.9.4 3-D Position Measurement with Constrained Cross-Camera Image Matching ......................... 20.9.5 Discussion .................................

Part C 20.1

20.10 Two-Dimensional Application: Heterogeneous Material Property Measurements ........................ 20.10.1 Experimental Setup and Specimen Geometry ............. 20.10.2 Single-Camera Imaging System ...................................... 20.10.3 Camera Calibration ..................... 20.10.4 Experimental Results .................. 20.10.5 Discussion .................................

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20.11 Three-Dimensional Application: Tension Torsion Loading of Flawed Specimen ...... 20.11.1 Experimental Setup with 3-D Imaging System ............ 20.11.2 Experimental Procedures............. 20.11.3 Calibration of Camera System for Tension–Torsion Experiments . 20.11.4 Experimental Results .................. 20.11.5 Discussion ................................. 20.12 Three-Dimensional Measurements – Impact Tension Torsion Loading of Single-Edge-Cracked Specimen .......... 20.12.1 Experimental Setup .................... 20.12.2 3-D High-Speed Imaging System.. 20.12.3 Camera Calibration ..................... 20.12.4 Experimental Results .................. 20.12.5 Discussion .................................

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593 593 593 594 595 595

20.13 Closing Remarks ................................... 597

586 587 587 587

References .................................................. 598

Data extraction through image analysis is one of the most dynamic areas in advanced, noncontacting measurement system development. A specific area in this field, digital image correlation, has seen explosive growth in the past two decades. In recent years, the method has been modified and extended to encompass a breathtaking number of novel measurement systems. Today, the technology is being used to measure (a) 3-D surface shape and deformations using a variety of illumination sources for a wide range of material systems, with size scales ranging from tens of meters to the microscale, (b) 2-D surface deformations at the nanoscale using atomic force microscopy and scanning electron microscopy, (c) interior deformation measure-

ments through volumetric imaging of biological and porous materials using technology such as computeraided tomography, and (d) dynamic/impact behavior of materials using high-speed camera systems. With increasing processor speed and improved computational software, the method has been extended for use in areas such as automatic inspection, system control, and real-time structural assessment. Given the continuing growth in computer technology, as well as recent advances in biosensors and nanoscience that have given rise to a host of new areas of inquiry, further developments and adaptations utilizing digital image correlation and its derivatives are assured.

20.14 Further Reading ................................... 597

20.1 Background As used in this article, the term digital image correlation refers to the class of noncontacting methods that acquire images of an object, store images in digital form, and perform image analysis to extract full-field shape and deformation measurements. Within the broad field of image analysis, digital image correlation (i. e., matching is performed using image

correlation metrics) is generally considered a subset of digital image registration techniques. Digital image correlation (i. e., matching) has been performed with many types of object-based patterns, including lines, grids, dots, and random arrays. One of the most commonly used approaches employs random patterns and compares subregions through-

Digital Image Correlation for Shape and Deformation Measurements

out the image to obtain a full field of measurements.

20.1.1 Two-Dimensional Digital Image Correlation (2-D DIC)

turers) have become a viable option for appropriate applications. Today, CMOS imagers offer more chip functions (e.g., pixel level accessibility to data), lower chip-level power dissipation, and smaller system size. However, these functions are balanced by a reduced image quality and, in some cases, reduced flexibility. For these reasons, CMOS cameras are well suited to high-volume, space-constrained applications where image quality is not paramount. It should be noted that high-quality consumer-grade cameras have been used successfully for imaging and post-experiment image analysis. Disadvantages for most such cameras include (a) on-camera manipulation of digital data prior to storage, resulting in loss of information and the potential for increased measurement error, (b) fixed camera settings, resulting in reduced adaptability to meet changing experimental conditions, and (c) the general bulkiness of the cameras and accessories. Typical hardware for modern 2-D image correlation systems includes (a) a CCD (or CMOS) camera with 12.7 mm2 sensor format, (b) 1280 × 1024 pixels in the sensor array, (c) 8–10 bit intensity resolution for each pixel, (d) a computer system with digital image acquisition components, (e) a sturdy tripod with mounting head, (f) high-quality lenses with focal lengths ranging from 17 to 200 mm, and (g) lighting (e.g., fiber optic illuminators). Estimated hardware costs at the time of writing for scientific-quality components are US $700 for a Nikon lens, US $700 for the uniform, high-intensity light source, US$ 700 for a rugged tripod with mounting head, US $4–6k for a scientific-grade CCD camera (1280 × 1024 spatial resolution, 8 bit intensity resolution, frame rate of 12–30 fps), and US $3k for a computer with appropriate data acquisition capability. Image analysis software can be used consistently and repeatedly to obtain surface deformations with (a) an accuracy of ±0.01 pixels or better for in-plane displacement components and (b) point-to-point accuracy of ±100 με for the in-plane surface strains εxx , ε yy , and εxy . This accuracy has been established using the VIC-2-D software (Correlated Solutions, Inc.; www.correlatedsolutions.com) based on a wide range of experimental and simulation studies. The accuracy of other software has not been established by the author. The accuracy noted previously is achievable even when the object is subjected to in-plane rigid-body rotations from +180◦ to −180◦ and arbitrary amounts of in-plane rigid-body translations since these motions do not corrupt the strain measurements. Commercial 2-D DIC software packages with advanced data processing

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Part C 20.1

One of the earliest papers to propose the use of computer-based image acquisition and deformation measurements in material systems was by Peters and Ranson in 1981 [20.1]. Using the fact that changes in images can be described by the same continuum concepts that govern the deformation of small areas on a surface, an approach was proposed to relate measurable image deformations to object deformations. For use in the field of experimental mechanics, these original concepts have been refined and incorporated into numerical algorithms [20.2–9] to extract object deformations from an image sequence. The resulting algorithms and software have been used successfully to obtain surface deformations in a wide variety of applications [20.10–23]. A digital camera typically is used to obtain digitized images for detailed analysis. Analog cameras perform image digitization by completing M row scans of the image, transferring the analog signal for each row to an analog-to-digital (A/D) board for conversion into N digital values having 8–12 bits of resolution. Though analog imaging systems can be used for 2-D or 3-D DIC, they have two disadvantages related to the rowby-row scanning process. First, shifts in the digitized images are introduced due to synchronization errors between row scans. Second, electronic noise added to the analog intensity data during scanning, transfer, and A/D conversion will introduce unwanted variations into the digitized images. For image-correlation-based measurement purposes, modern scientific-grade digital cameras are generally used to (a) obtain high-quality images on the sensor plane, (b) perform onboard digitization of the intensity at each sensor location, and (c) transfer the digital data to a storage location. Both the charge-coupled device (CCD) camera and complementary metal–oxide–semiconductor (CMOS) camera have been used successfully in a wide range of image applications. CCDs have been the dominant solid-state imagers since the 1970s, primarily because they gave far superior images with the fabrication technology available at that time. As manufacturing technology has advanced, CMOS image sensors (which have requirements on uniformity and reduced feature size that could not be obtained previously by silicon wafer manufac-

20.1 Background

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features and an array of image acquisition options cost on the order of US $10k.

20.1.2 Three-Dimensional Digital Image Correlation (3-D DIC)

Part C 20.2

Stereo-vision principles developed for robotics, photogrammetry, and other shape and motion measurement applications were modified and successfully implemented to develop and apply a two-camera stereo vision system to accurately measure the full three-dimensional shape and deformation of a curved or planar object, even when the object undergoes large out-of-plane rotation and displacement. It has been shown both theoretically and experimentally [20.24] that relatively small out-of-plane motions will change the magnification and introduce errors in the measured in-plane displacement field when using a single camera and 2-D DIC, a limitation that is eliminated by 3-D DIC. The 3-D DIC method has been validated for a wide range of mechanics-based measurements in both large and small structures [20.25–31], even as innovations have been proposed and developed [20.32–35]. It is important to note that 3-D DIC for accurate shape and deformation measurements is far more complex to perform than 2-D computer vision due to (a) the sophisticated stereo-calibration procedures that are essential for easy-to-use accurate 3-D shape and deformation measurements, (b) the requirement for accurate crosscamera image correlation, and (c) the importance of image modeling for multicamera arrangements to increase the accuracy of the measurements. Typical hardware for a modern 3-D DIC system consists of (a) two CCD cameras with 8–10 bit quantization of the incident light (intensity resolution); stereo-camera systems typically using a matched pair of cameras, (b) computer system with digital image acquisition components capable of simultaneously recording images from both cameras, (c) two sturdy tripods with mounting heads, (d) a rigid bar for mount-

ing the two cameras and minimizing the potential for relative motion between the cameras during an experiment, (e) two high-quality lenses (17–105 mm are standard), and (f) lighting (e.g., two 250 W halogen spotlights). Estimated hardware costs at the time of this writing for high-quality components are US $1400 for two Schneider lens, US $1400 for two uniform, highintensity light sources, US $1000 for a rugged tripod with mounting head and rigid mounting bar, US $3000 for rotation and translation stages for camera attachment to the mounting bar, US $1000 for a calibration grid; $8–12k for two scientific-grade CCD cameras with 1392 × 1040 spatial resolution, 8 bit intensity resolution and a framing rate of 12–30 fps, and US $3000 for a computer with appropriate data acquisition capability. Image analysis software for stereo vision can be used consistently and repeatedly to (a) calibrate the camera system, (b) perform experiments and simultaneously acquire stereo image pairs, and (c) analyze several sets of stereo image pairs to obtain surface deformations. Results from multiple experiments indicate (a) an accuracy of ±0.015 pixels or better for crosscamera image correlation to identify the same image point, (b) point-to-point accuracy of ±100 με for the in-plane surface strains εxx , ε yy , and εxy . This accuracy is achievable even when the object is subjected to large rigid-body rotations and arbitrary amounts of rigid-body translations since these motions do not corrupt the strain measurements. This accuracy has been established using VIC 2-D (Correlated Solutions, Inc.; www.correlatedsolutions.com) based on a wide range of experimental and simulation studies. The accuracy of other software has not been established by the author. Due to the complexity of the calibration, image acquisition and image analysis processes for stereovision, commercial 3-D DIC software packages with advanced data processing features and an array of image acquisition and calibration options are on the order of US $ 40k.

20.2 Essential Concepts in Digital Image Correlation Two key assumptions are generally employed to convert images into experimental measurements of object shape, displacements, and strains. First, it is assumed that there is a direct correspondence between the motions of points in the image and motions of points on the object. As long as this assumption holds, im-

age motions can be used to quantify the displacement of points on the object. Specifically, such assumption provides for a one-to-one correspondence between (a) image points and object points in 2-D computer vision and (b) image points in each camera view and the common object point in 3-D computer vision. Since

Digital Image Correlation for Shape and Deformation Measurements

ation in contrast can be obtained either by applying a high-contrast random pattern (e.g., painting, adhering, surface machining) or it may occur due to the natural surface characteristics of the material. The object profile, or 3-D shape, of an object also can be determined using digital image correlation by projecting a random pattern onto the specimen and using one of several optical arrangements. Situations where the assumptions break down include the following examples:

• •

correlation of subsets across a discontinuity (e.g., crack, hole) in the material loss of contrast during loading process due to one of many factors: – Debonding or delamination of applied pattern, resulting in either loss of pattern or loss of correspondence between object motion and image motions as debonded pattern does not follow object motions – Change in diffuse reflectivity of surface during loading, resulting in loss of contrast in recorded images

20.3 Pinhole Projection Imaging Model A common model for the imaging process is the pinhole camera. Figure 20.1 shows a schematic of a pinhole camera with pinhole O, focal length f , and image plane center C, imaging a general object point B. As shown in Fig. 20.1, the intensity distribution emanating from the vicinity of object point B is imaged onto the retinal/image plane in the vicinity of point P. Without loss of generality, it is convenient to define a set of coordinate systems to represent the imaging process. The first coordinate system is the world coordinate system (WCS), with axes (X W , YW , Z W ). In practical applications, the WCS is often used to define specific object positions during the calibration process. For example, if a two-dimensional planar grid is used for calibration, the grid lines may be defined to be the X W - and YW -axes, respectively, with the Z W -axis being perpendicular to the planar grid. The second coordinate system typically defined is the camera coordinate system (CCS) located at the pinhole O. In many cases, the Z C -axis is aligned with the optical axis through points O–C, while X C and YC are oriented to align with the camera sensor axes. The third set of axes defines the sensor coordinate system (SCS) with (X S , YS ) aligned with the

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row and column directions of the sensor plane; the third sensor coordinate, Z S , is rarely used since 2-D coordinates are sufficient to describe the pixel locations. The SCS has units for (X S , YS ) in terms of pixels, where a pixel defines positions/sensor locations. Using the terminology shown in Figs. 20.2, 20.3, and 20.4, the pixel location (10, 15) is the 16th sensor location on the 11th row of the sensor plane. The SCS is located in the retinal or image plane of the pinhole camera. The fourth system is the object coordinate system (OCS), which can be located arbitrarily. In Fig. 20.1, the system is located at the intersection of the optical axis and the object. In vector form, the location of B is written as r B in the WCS, the location of the origin of the WCS in the CCS is defined to be t O , and the position of B in the CCS is defined to be SB . The vector components for each term can be written as follows: SB = (X B , YB , Z B ) , t O = (tx , t y , tz ) , r B = (X W , YW , Z W ) ,

(20.1)

Part C 20.3

an object is generally approximated as a continuous medium, the correspondence between image and object points ensures that continuum concepts are applicable to describe the relationship between points in an image subset as the object deforms. As will be shown in Sect. 20.3, mapping from the sensor plane of a single camera using a standard pinhole imaging model is a one-to-many transformation into the object space, since the Z distance along the optical axis to the object point is not known using only the sensor position data from a single camera; additional cameras viewing the same point are used to extract the Z location. This does not affect the applicability of continuum concepts to map undeformed image points into deformed image points. Second, it is assumed that each subregion has adequate contrast (spatial variation in light intensity) so that accurate matching can be performed to define local image motions. Together with the first assumption, accurate matching can be improved by allowing each image subregion to deform using an appropriate functional form (e.g., affine, quadratic) and hence increase accuracy in the measured motions. The required vari-

20.3 Pinhole Projection Imaging Model

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World coordinate system

Ys

Xw rB

Sensor plane

Zw

B

Xs

Yw

Sensor coordinate Ys system

tO

sB Object coordinate system

Xs C

P

Grid standard

Part C 20.3

Image/retinal plane

f

Fig. 20.2 Grid standard as viewed in sensor plane. Note the location of fiducial marks in bottom left, top left and top right of grid. The marks are used to extract estimates for rigid body motion of grid and identify matching grid intersection positions in remaining calibration images

O Camera coordinate system

Fig. 20.1 Schematic of pinhole camera model imaging region

around 3D point B onto retinal plane around image point P

and the components can be related as ⎛

⎞ ⎛ XB R11 R12 R13 ⎜ ⎟ ⎜ ⎜ YB ⎟ ⎜ R21 R22 R23 ⎜ ⎟=⎜ ⎝ Z B ⎠ ⎝ R31 R32 R33 1 0 0 0

⎞⎛ ⎞ tx XW ⎟⎜ ⎟ t y ⎟ ⎜ YW ⎟ ⎟⎜ ⎟, tz ⎠ ⎝ Z W ⎠ 1

(20.2)

1

where Rij is the rotation matrix between the WCS and the CCS. Equation (20.2) is the form commonly used in the computer-science community to describe rigid-body motion. Since the distance between the retinal plane and the CCS is f , the prospective projection into the retinal plane of point B is given in the CCS by (X, Y, f ) and can be written in the form ⎛ ⎞ ⎛ 0 X f/Z B 0 ⎜ ⎟ ⎜ f/Z B 0 ⎜Y ⎟ ⎜ 0 ⎜ ⎟=⎜ ⎝f⎠ ⎝ 0 0 f/Z B 0 0 0 1

⎞⎛ ⎞ 0 XB ⎟⎜ ⎟ 0⎟ ⎜ YB ⎟ ⎟ ⎜ ⎟ . (20.3) 0⎠ ⎝ Z B ⎠ 1

1

In (20.3), perspective is defined with a matrix that is a function of the position, Z B , so that the solution process is a nonlinear one. A more appropriate notation allows one to separate the matrix form from Z B and is

written using homogeneous coordinates in the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ XB X f 0 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜Y ⎟ Λ ⎝ Y ⎠ = ⎝ 0 f 0 0⎠ ⎜ B ⎟ , (20.4) ⎝ ZB⎠ 1 0 0 1 0 1 where Λ is a scale factor used to eliminate Z B from the matrix in (20.3). For projective transformation, the factor Λ = Z B is defined by the last equation in (20.4). Perspective geometry and related concepts are used in many computer vision textbooks to develop fundamental mathematical formulations. Interested readers are referred to the computer vision texts in the general references for more information [Faugeras, 1993; Faugeras et al., 2001; Hartley, 2003]. Defining (λx , λ y ) as the scale factors relating sensor coordinates (pixels) to distance in the retinal (image) plane and letting the offset distance between the origin of the sensor coordinate system and the intersection with the optical axis be C, with position (C x , C y ) in pixels, we can write the following transformation. If the row and column directions in the sensor plane are not orthogonal, a skew factor s is required. Also, the trivial identity defined by the last row and column of the previous transformation has been removed ⎞⎛ ⎞ ⎛ ⎞ ⎛ λx s C x X XS ⎟⎜ ⎟ ⎜ ⎟ ⎜ (20.5) ⎝ YS ⎠ = ⎝ 0 λ y C y ⎠ ⎝ Y ⎠ . 1 0 0 1 1 Assuming that the rows and columns in the sensor plane are orthogonal (s = 0), the sensor coordinates can be

Digital Image Correlation for Shape and Deformation Measurements

Xw

Zw Yw

3

Optic axis Image plane 1

t3 Xs

2

t2 t1

3

Ys

Fig. 20.3 Imaging of grid standards from general, 3D positions and orientations in sensor plane of camera system. Sensor locations of each grid origin denoted by ti . World coordinate system oftentimes assumed to be located at origin of grid 1 and aligned with grid directions

(20.7)

20.3.1 Image Distortion

written in scalar form as ⎞ ⎛ X W +R12 YW +R13 Z W +tx X S = C x + f λx RR11 31 X W +R32 YW +R33 Z W +tz ⎠. ⎝ R X W +R22 YW +R23 Z W +t y YS = C y + f λ y R21 31 X W +R32 YW +R33 Z W +tz

Since the imaging process for physically realizable imaging systems results in deviations of the actual image location from the pinhole projection model predictions of the image location given by (20.1)–(20.6), improved models have been developed based upon knowledge of the source of these errors. In a general sense, it can be assumed that there exists a vector distortion function, D(X S , YS ), which represents the deviations from the actual image posi-

(20.6)

Inspection of (20.6) indicates that there are six independent parameters relating the orientation of the WCS and the CCS: three angular measures and three rigid(0,0)

(0,1)

(0,2)

Ys

(0,0)

220

120

60

(0,2)

(1,0)

120

50

120

(1,2)

(2,0)

60

120

220

(2,2)

(2,0)

(2,1)

(2,2)

Xs

Fig. 20.4 Schematic of digitization and storage of discretely-sampled image of continuous intensity distribution, I (X S , YS )

571

Part C 20.3

body displacements. Since these parameters define the external orientation and position of the camera, they are known as the extrinsic parameters. Thus, any reorientation of the camera will alter one or more of these parameters. It is worth noting that the rotation matrix R can be written in several ways. One minimalist approach is to define R in terms of Euler angles. Here, three successive rotations about specific axes are used to describe the orientation of an object. Among the many Euler angle formulations, the combination z–y–x is presented in this work. Assume that the first rotation, θz , is about z W , that the second rotation angle, θ y , is about the rotated yW axis, and that the last rotation angle, θx , is about the twice-rotated xW axis, then the rotation matrix can be written as the multiplication of three successive rotations to give ⎞ ⎛ ⎞ ⎛ 1 0 0 R11 R12 R13 ⎟ ⎜ ⎟ ⎜ ⎝ R21 R22 R23 ⎠ = ⎝0 cos θx sin θx ⎠ R31 R32 R33 0 − sin θx cos θx ⎞⎛ ⎛ ⎞ cos θz sin θz 0 cos θ y 0 − sin θ y ⎟⎜ ⎜ ⎟ ·⎝ 0 1 0 ⎠ ⎝− sin θz cos θz 0⎠ . sin θ y 0 cos θ y 0 0 1

1

2

20.3 Pinhole Projection Imaging Model

572

Part C

Noncontact Methods

Part C 20.3

tions relative to the predictions of the pinhole model, where the goal is to define D(X S , YS ) and obtain more accurate image locations. Though the general distortion measurement problem has been discussed recently in the literature [20.36], there are well-established approaches based on existing parametric distortion models that are adequate for most practical applications. Lens distortions have several mathematical forms, depending upon the type being modeled. For typical lens systems, radial lens distortion at any point P in the sensor plane is modeled using a vector function. Defining the vector difference between undistorted sensor plane position (X S , YS ) and distorted sensor plane position (X S , YS ) by the vector D, one can write D(P) = ku r 3 (P)er  3/2 = ku (X S (P) − C x )2 + (YS (P) − C y )2 er , (20.8)

where ku is the radial distortion coefficient, r is the radial distance in the image plane of a point P measured relative to (C x , C y ) and er is the radial unit vector in the image plane. The value of D at any point (X S , YS ) is the radial distortion vector. Thus, the distortion-corrected position is determined by the equation

  X S , YS = (X S + Dx (X S , YS ), YS + D y (X S , YS )) . (20.9)

The notation used in this study defining the uncorrected position (X S , YS ), i. e., the measured position, is different from previous articles [20.37, 38] by the author. The reader is cautioned to take care when comparing formulations. Of course, one could define the distortion vector as the difference between distorted and undistorted sensor plane positions. In this case, typicially one would modify (20.9) and subtract both Dx and D y from the distorted positions. Since the digitized intensity pattern is known at integer pixel locations, the distortion-corrected positions typically form a nonuniform grid at noninteger positions. In the following sections, (20.9) will be used to model distortion, though the approach developed will be applicable for multiparameter distortion models.

20.3.2 Camera Calibration for Pinhole Model Parameter Estimation As shown in (20.1)–(20.8), there are a total of 11 independent parameters to be determined during the calibration process: six extrinsic parameters defining the orientation and position of the grid (tx , t y , tz , θx , θ y , θz ) and five intrinsic parameters (C x , C y , f λx , f λ y , ku ). If

all 11 parameters are known, (20.6) shows that the sensor position of a known 3-D point is uniquely determined. The 3-D position corresponding to a known sensor location in a single camera cannot be uniquely determined. However, the 3-D position corresponding to sensor positions in two or more cameras is overdetermined and can be optimally estimated. The process for optimally estimating all 11 parameters for a camera is known as camera calibration. A typical camera calibration procedure uses a grid standard that has a predetermined grid spacing. Figure 20.2 shows a line grid with fiducial marks. During the calibration process, the grid is translated and rotated, and images are acquired at each position. To obtain initial estimates for the position and orientation of the calibration grid in each image, the deformed positions of intersection points are generally determined by locating and identifying landmarks (e.g., fiducial points) in the grid standard; such marks are shown in Fig. 20.2. By locating these marks (at least three non-collinear points are required) in each deformed image, the extrinsic parameters for each view (orientation and translation) are estimated and used to give initial locations for corresponding grid points throughout the view. Other grid standards have been used successfully for calibration, including two-dimensional dot patterns. The procedures outlined above remain valid for performing the calibration process, with thresholding and accurate centroid algorithms used to extract the center-point location for each dot.

20.3.3 Image-Based Objective Function for Camera Calibration Defining an image-based objective function in the form N VIEWS N PTS

ij 2 ij E= X S measured − X S model i=1

j=1

2 

ij ij + YS measured − YS model ij

ij

(20.10)

the measured values for (X S , YS ) are the j = 1, 2, . . ., NPTS pixel locations from all grid intersection points that are extracted by analyzing all i = 1, 2, . . ., NVIEWS ij ij images of the grid. The model values, (X S , YS ), for the NPTS pixel locations for all grid intersection points are obtained using (20.6). Figure 20.3 shows schematically the rotated positions of a line grid during the calibration process. Thus, if one assumes that images of a calibration grid are acquired with NPTS in each grid, then there are 2 × NPTS × NVIEWS terms in (20.10) relating measured to

Digital Image Correlation for Shape and Deformation Measurements

Though several approaches have been used to perform the nonlinear optimization of (20.10) for camera calibration, a common method requires a linearization of the functional in terms of the required parameters. Once the linearization has been completed, the optimization process results in a linear set of equations that are solved for the remaining unknowns. The solution to the linear equations is then used to obtain initial estimates for the parameters. The initial estimates are used to perform nonlinear optimization on (20.10) and complete the calibration process. The nonlinear optimization is performed using approaches such as steepest descent, Newton– Raphson or a combined method such as Levenberg– Marquardt.

20.4 Image Digitization Consider again the optical arrangement shown in Fig. 20.1, which represents a pinhole projection imaging model for a general 3-D point, B. Letting the intensity distribution recorded on the sensor plane be defined as I (X S , YS ), where (X S , YS ) are sensor coordinates in the image plane, Fig. 20.4 shows a representative image as it is appears after being discretely sampled by a 3 × 3 sensor array. Each sensor performs an averaging process over the exposure time to convert the incident radiation into an integer value. The data for each sensor is then stored at a location within a digital array. Figures 20.4 and 20.5 show the common scheme for identifying a pixel location, where the X S axis is located along a row and YS is located along a column within the intensity array. Typically, a row for an analog camera corresponds to the direction of scanning, while the column direction corresponds to the raster direction between row scans. Depending upon the number of quantization levels in the camera, each intensity value is stored with N bits of intensity resolution. Many cameras store each intensity value in one byte, resulting in 8 bits of resolution so that the total range is 0–255 for each intensity level; this situation is shown in Fig. 20.5. The array of digital values obtained by sampling a continuous intensity pattern is the primary data used

in the process of extracting the deformation field. In this regard, Figures 20.5a and 20.5b show a 2-D gray level representation and a three-dimensional representation of the intensity variation across the field, respectively, of the digitized data for a 10 × 10 array of intensity values. It is important to note that the digitized image intensity field is the primary experimental data for 2-D and 3-D DIC.

20.4.1 Discussion If electronic or active cooling of a camera’s sensor plane is performed to reduce thermal noise in the recorded intensity values, higher intensity resolution is achievable, with cameras having 10 bit (0–1024 range) or 12 bit (0–4096 range) resolution being commercially available. To use the added bits effectively in digital image correlation, it is important that the range of intensity values encoded in a pattern be maximized. For example, if two cameras record a pattern with all values from 0.2 to 0.9 of the range, then the pattern with gray levels from 200 to 900 in a 12 bit camera is certainly preferable to a pattern with a range from 50 to 225 in an 8 bit camera (as long as the signal-to-noise ratio is similar in both cameras).

20.5 Intensity Interpolation To obtain surface deformations from digital images, subregions from digital images are compared. Defin-

ing one of the images to be the reference image (i. e., the image of the object when it is considered to be

573

Part C 20.5

predicted 2-D image locations. Minimization of (20.10) must provide sufficient information to determine (a) the orientation and position of the grid in various locations and (b) the five intrinsic camera parameters. The presence of rigid-body rotation (Rij = Iij , the identity tensor) and image distortion (ku = 0) requires a nonlinear optimization process. Thus, if the grid spacing is known, a maximum of [6 × NVIEWS + 5] parameters are obtained. The process described here is often referred to as bundle adjustment, since all motions of the grid and the intrinsic parameters are bundled together and optimally estimated. In the vision literature, bundle adjustment generally includes re-estimation of the grid point positions. The procedure described herein is a reduced form, with the grid point separation specified.

20.5 Intensity Interpolation

574

Part C

Noncontact Methods

a)

b)

Ys

Ys (pixels)

Xs (pixels)

Xs

Gray level

200 100 0

0 2

0 4

2 4

6 6

8 8

Part C 20.5

10

10

d)

c) Ys (pixels)

Xs (pixels)

Gray level

200 100 0

0 2

0 4

2 4

6 6

8 8 10

10

Ys (pixels)

Xs (pixels)

Gray level

200 100 0

0 2

0 4

2 4

6 6

8 8 10

10

Fig. 20.5 (a) Discrete gray level representation of intensity values for a 10 by 10 array of sensors. (b) Three-dimensional representation for 10 by 10 array of intensity values. Height represents gray level value at pixel locations. (c) Bi-linear representation for intensity values within a 10 by 10 pixel area. Height represents gray level value at pixel locations. (d) Cubic spline representation for intensity values within a 10 by 10 pixel area. Height represents gray level values at pixel locations

in the undeformed or initial configuration), general deformations of the object will introduce noninteger displacements of the corresponding positions in the reference image. Thus, accurate subpixel estimates for local object motions require that intensity values at integer positions in the reference image are registered with intensity values in the deformed images at noninteger positions. To obtain noninteger estimates for intensity values, the discrete intensity pattern recorded for each deformed image is converted into a continuous functional representation. Figure 20.5c,d shows bilinear and cubic spline representations, respectively, for the same 10 × 10 intensity value shown in Fig. 20.5a and Fig. 20.5b.

It is noted that accurate subpixel measurement of surface deformations requires accurate reconstruction of the object’s textured pattern. Reconstruction of this type requires oversampling of the textured pattern. Since sensor arrays have a fixed number of pixels (e.g., 1024 × 1024), control of feature size during pattern application is important to ensure that (a) each feature is properly sampled for accurate reconstruction and (b) excessive oversampling is not performed, since this will result in a requirement for larger subset sizes during matching and a reduction in the spatial resolution that is achievable when performing 2-D or 3-D DIC. With regard to quantization during digitization of the intensity pattern, the majority of digital cameras

Digital Image Correlation for Shape and Deformation Measurements

record the intensity values using 8 bits (0–255) with Gaussian noise in the intensity values having a standard deviation on the order of 1.6 gray levels. Increasing the number of bits to 10 or 12 has the potential to improve the accuracy of the measurements, especially for small-deformation problems. However, experimentally observed improvements when increasing the number of quantization levels appear to be less than those theoretically anticipated due to a variety of factors present during normal laboratory measurements (e.g., the effects of image distortion, vibrations, lighting, out-of-plane motion for 2-D systems, gray level variability, nonuniform sensor response (photometry)).

Higher-order interpolation functions, such as cubic b-splines, have been shown [20.39] to give better ac-

curacy, providing a significant advantage when the displacement gradients are small. Pixel intensity noise can be determined by acquiring and storing M consecutive images without varying the system configuration. These images can be used to obtain the variation in intensity at representative pixel locations. The author has obtained variations of ±1.6 and ±2.5 gray levels with 8 bit scientific-grade CCD and CMOS cameras operating under standard laboratory conditions, respectively. Average feature size in a pattern can be determined by performing autocorrelation on selected rows and columns from the image, and defining the feature size to be the pixel width when the autocorrelation function is 0.50. The average feature size should be on the order of 4 × 4 pixels to ensure a minimum of oversampling of the pattern.

20.6 Subset-Based Image Displacements Image-plane deformations are extracted through image comparison, where it is common to select an image of the object and designate it to be the reference image. All additional images are designated as deformed images. Image correlation is performed by comparing small subsets from the digitized textured pattern in the reference image to subsets from each of the deformed images. Since the matching process is performed to locate the corresponding position of each reference image subset within each deformed image, the optimization for accurate matching is performed locally (i. e., the points to be matched do not occupy the entire sensor plane). Figure 20.6 shows a typical combination of subsets within the reference image that are arranged in a serpentine pattern for the search process. Here, a total of m × n subsets are chosen, with an X S -spacing between the subsets of ΔX S and a YS -spacing between subsets of ΔYS . The comparison process performed for each of the m × n subsets shown in Fig. 20.6 determines a full field of image displacements for each case. As noted previously, continuum-based principles are used to develop a mathematically sound approach for performing subset-based image comparisons. Figure 20.7 shows a schematic of a local image deformation process that transforms a region from the undeformed image into a new position in the deformed image. For each small subset centered at (X PS , YSP ) in the image of the undeformed object (hereafter des-

ignated the undeformed image), the intensity values for points P and Q located at positions (X PS , YSP ) and (X PS + ΔX S , YSP + ΔYS ) respectively, can be written in terms of subset-based coordinates γS = X S − X PS and K ΔXs

Xs

Sensor plane in reference image

Ys

1

2

n

3

2n

n+2

n+2

n+1

ΔYs 2n+1

L

2n+2

2n+3

575

Part C 20.6

20.5.1 Discussion

20.6 Subset-Based Image Displacements

3n

3n+1

mn

Fig. 20.6 Typical serpentine process for selection of pixel subsets

in the designated reference image for image correlation. Sensor plane has K × L discrete sensing elements. Appropriate subset size is based on spatial frequency of pattern being imaged. Pixel distance between adjoining correlation subsets is given by ΔX s or ΔYs depending upon the image location of a subset

576

Part C

Noncontact Methods

ηS = YS − YSP : I (P) = I (Q) =



Xs



I X PS , YSP ,

Q I X PS + γSQ , YSP + ηS . p X S + ΔX S

Sensor plane

Reference subset

(20.11) p YS + ΔYS

Part C 20.6

If the values for and are integer pixel values, then no interpolation of the deformed image is required. No interpolation is required in the reference image since all intensity values are located at integer positions. Since most applications use subsets with integer pixel values in the reference image, matching is generally performed using an N × M rectangular subset from the digitized intensity pattern in both the reference and deformed images. As shown in Fig. 20.7, image points P and Q are transformed into image locations p and q, respectively, which are not at integer pixel locations. Assuming that the intensity pattern recorded after deformation is related to the undeformed pattern through a continuous image deformation field and defining the center point of each subset to be P with coordinates (X P , Y P ), then the displacement vector field for a general point in the subset can be written in the form {u(γS , ηS ), v(γS , ηS )}, and the resulting intensity value at an arbitrary point q in the deformed image can be written

Ys

Q P

(u(Q), υ(Q))

q (u(P), υ(P))

p

Deformed subset

Fig. 20.7 Mapping of sensor positions P and Q in ref-

erence subset to p and q in deformed subset. Shape of deformed subset can be modeled using higher order shape functions

in the undeformed subset to corresponding values of intensity in the deformed region. In most of the results to be discussed in the following sections, a normalized cross-correlation function is used to optimally match subsets and obtain best estimates for the parameters; it is defined as

(20.12)

1.0 − CCl (c0 , c1 , . . . , cW )  i=N j=M

j I γSi , ηS = 1.0 − i=1 j=1 



j j j  I  γSi + u γSi , ηS , ηS + v γSi , ηS  i=N j=M

i=N j=M j I 2 γSi , ηS I 2 i=1 j=1 i=1 j=1 

i



j j j  1/2 γS + u γSi , ηS , ηS + v γSi , ηS . (20.13)

where the second form in (20.12) represents a firstorder Taylor’s series approximation for the deformation field. The form of the displacement function for each subset can be selected by the investigator. If a linear variation in displacement is assumed within the subset, then the subset displacement functions can be written u(γS , ηS ) ≈ c0 + c1 γS + c2 ηS and v(γS , ηS ) ≈ c3 + c4 γS + c5 ηS . In this form, c0 and c3 are the translations of the image subset center P. Assuming that each subset is sufficiently small, the displacement gradients are approximately constant within a subset so that it undergoes rigid-body rotation and uniform strain, resulting in a parallelogram shape. The coefficients ci for each subset are determined by optimally matching the intensity values at each point

The values for c0 , c1 , . . . , cW that minimize (20.13) represent the best estimates of the subset’s displacement field. It is noted that the quantity (1.00 − CCl ) is zero for a perfect match and unity for orthogonal (completely mismatched) patterns, with intensity interpolation of the deformed intensity pattern (Fig. 20.5c,d) required for optimum subpixel displacement accuracy. As with the camera calibration problem, a wide range of optimization methods has been used successfully to obtain the optimal value for (1.00 − CCl ), including Newton–Raphson [20.30, 31], coarse– fine [20.25–29], and recently the Levenberg–Marquardt method. Currently, analyses performed using the Levenberg–Marquardt method indicate that this method is as fast as the Newton–Raphson approach and has

I  (q)   P

Q Q  Q = I X PS + γSQ + u γSQ , ηQ S , YS + ηS + v γS , ηS    ∂u ∂u ∼ ΔηS , ΔγS + = I X PS + u(0, 0) + 1 + ∂γS ∂ηS    ∂v ∂v ΔγS + 1 + ΔηS , YSP + v(0, 0) + ∂γS ∂ηS

Digital Image Correlation for Shape and Deformation Measurements

P P'

Q

R

R'

20.7 Pattern Development and Application

577

Fig. 20.8 Method to estimate local displacement vector for a point Q and the local rotation of sub-region in vicinity of Q. Pixel locations of P, Q and R in undeformed state are used to locate corresponding rotated and translated points P  , Q  and R in deformed state

Q'

1. input initial estimates for the subset parameters (see Sect. 20.6.1), 2. allow the gradients to be nonzero and perform a full, six-parameter search process to minimize (20.12). The values of u, v and displacement gradients that minimize the term (1 − CCl ) are the optimal estimates for the displacements and displacement gradients, 3. using results from the previous subset as the initial estimate for the subset parameters, repeat step 2 for the next subset, 4. repeat steps 2 and 3 until data is obtained throughout the region of interest.

20.6.1 Discussion In the presence of noise in the gray values (a typical CCD pixel has variability in the measured intensity on the order of 1–2 gray levels out of 256), the magnitude of the normalized correlation value, 1 − CCl , for a signature pattern provides a quantitative measure of confidence in the match between deformed and undeformed subsets. A value below 0.001 is generally considered a reasonable match, with values > 0.01 generally considered to be a lower quality of match. It is important to emphasize that the two requirements

outlined in Sect. 20.2 are assumed to hold. If the pattern does not contain adequate contrast, then there is nonuniqueness in the measurements. In this case, the magnitude of (1 − CCl ) is not a good indicator of the accuracy of the image-based motion measurements. Since a nonlinear search process is required to match subregions in an image, initial estimates for some of the parameters are often required. To obtain initial estimates, the user can visually identify three non-collinear points in both the undeformed and deformed images. This is shown schematically in Fig. 20.8. For the linear mapping model, the correspondences can be used to estimate c0 –c5 and initiate the optimization process. Since the intensity values are recorded at integer locations in uncorrected coordinates, correlation is typically performed in uncorrected coordinates. The computed distortion vector, D(X S , YS ), can be used to determine the corrected positions and corrected displacements. The normalized cross-correlation shown in (20.13) is relatively insensitive to changes in lighting. Previous studies by the author have shown that (20.13) can be used to obtain optimal affine transformation parameters with background lighting changes of up to 30–40%. Furthermore, the normalized correlation coefficient obtained during image matching can be used to identify situations where imaging comparisons are not as good and the accuracy of the estimated displacements may be degraded.

20.7 Pattern Development and Application To achieve a usable random pattern of the required size, several approaches have been developed. In each case, the surface must be properly prepared so that the pat-

tern will deform/move with the material system being studied. This process may include degreasing, etching, polishing, and coating of the specimen surface.

Part C 20.7

better convergence characteristics. The optimization procedure can be described as

578

Part C

Noncontact Methods

One common method applies a thin layer of white spray paint to the surface; white epoxy spray paint has been used when plastic deformation is very large and enamel paint has been used for plastic strains less that 30–40%. A high-quality random pattern is obtained using several different techniques

1. a light over-misting of black paint with an airbrush system using fine-tipped attachments to obtain a pattern with size ≈ 50 μm and larger, 2. a light coating of unfiltered Xerox toner powder onto the wet painted surface for a pattern with size ≈ 30–50 μm,

Part C 20.7

1 mm

Xerox toner powder embedded in enamel paint ~ 160 pix/mm) on aluminum (Λ =

40 mm

Viewed through red filter 40 mm

Viewed through green filter

Enamel spray paint pattern on composite specimen ~ 10 pix/mm) (Λ =

Adhesive-backed vinyl with two distributions of ~ 10 pix/mm used speckles. Green speckles at 0.8 mm (Λgreen = ~ 2 pix/mm) for measurements). Red speckles at 5 mm (Λred = 100 μm

100 μm 4 micron and 20 micron photolithography patterns on polished aluminum specimens ~ 20 pix/μm for both images) (Λ =

Fig. 20.9 Speckle patterns and approximate scale factors for images

Digital Image Correlation for Shape and Deformation Measurements

Sample patterns manufactured in different ways are shown in Fig. 20.9.

20.7.1 Discussion Regardless of the method employed to construct a random pattern, each pattern should be imaged with the setup used for the experiment. The digitized image should then be processed to obtain both (a) a histogram for the distribution of intensity values in the image and (b) an estimate for the size of the speckles to determine the average feature size in the pattern. Optimal patterns have a relatively broad, flat histogram over the entire range of the intensity pattern; for

an 8 bit digitizer, a range from 0 to 255 with a uniform distribution across the field being ideal, but seldom seen in practical applications. Intensity ranges from 50 to 220 are considered to be good. Optimal patterns have an average pattern size determined by the width of the autocorrelation function peak at 0.50 that is in the range of 3–7 pixels for modest levels of oversampling. If the speckle size is too small for accurate matching, then a higher magnification can be used to improve pattern sampling. If this is not possible for the application, the specimen should be repatterned. As an example, Fig. 20.9 shows digitized images obtained using a scientific grade camera when viewing a red-green speckle pattern with two different size distributions using (a) a red filter and (b) a green filter. The digitized grayscale images would have an autocorrelation pattern with two different peaks, each of which can be used to determine the magnification factor for the size of speckle that provides optimal spatial resolution and pattern density. If the speckle size is too large for accurate matching with an N × N subset, then there are two options. First, the subset size can be increased to achieve accurate matching. The data will be accurate, but there will be a reduction in spatial resolution due to the increased subset size. Second, a lower magnification can be used to decrease speckle size, if this is appropriate for the application. If neither choice is acceptable, then the surface should be repatterned. Intensity values of 0 are rare due to the presence of thermally emitted photons. Intensity values at 255 are an indication that intensity resolution may be lost due to sensor saturation. At saturation, all higher intensity values will be given the value of 255, resulting in an inability to discern any image texture above this value. Saturation must be avoided to ensure accurate image matching.

20.8 Two-Dimensional Image Correlation (2-D DIC) In addition to the need for a high-contrast textured pattern for accurate subset-based matching, the main requirement for the successful use of 2-D DIC is for the deformations of the planar object to be in a plane parallel to the sensor plane. Assuming the world coordinate system has an origin at the camera pinhole,is aligned with the rows and columns of the sensor plane and image distortions are negligible, then the projection

equations in (20.6) can be simplified and written X S = (Λx )X W + C x , YS = (Λ y )YW + C y ,

(20.14)

where Λx = ( f/Z)λx and Λ y = ( f/Z)λ y . Since f/Z is a constant for plane-to-plane pinhole projection, Λx and Λ y can be viewed as scale or magnification factors with

579

Part C 20.8

3. a light coating of filtered Xerox toner powder onto a wet painted surface for a pattern with size ≈ 15 μm, 4. a splatter-paint procedure using a brush filled with black paint to obtain a large pattern with size from 250 μm to several millimeters, 5. a vinyl sheet with an imprinted speckle pattern that has two types of adhesive, one to hold the sheet during placement and a microencapsulated epoxy that is released during rolling of the sheet for permanent installation. Patterns from 0.25 mm to several millimeters have been printed and used, 6. indelible ink marker for manually staining lightcolored polymer materials. Patterns from 0.25 mm to several millimeters have been manufactured and used, 7. chemical etching of metallic materials. Patterns ranging from 1 μm (viewed in a scanning electron microscope) to 200 μm have been manufactured, depending upon grain size and material composition, 8. photolithography to develop patterns ranging from 1 μm to 20 μm on polished metallic specimens.

20.8 Two-Dimensional Image Correlation (2-D DIC)

580

Part C

Noncontact Methods

dimension of (pixels/unit length) on the object. Thus, the relationship between world coordinates and sensor coordinates in the presence of image distortion has five independent parameters: Λx , Λ y , C x , C y , and ku .

20.8.1 2-D Camera Calibration with Image-Based Optimization

Part C 20.8

The procedures outlined in Sects. 20.3.1–20.3.3 can be used to determine Λx , Λ y , C x , C y , and ku optimally. As shown in Fig. 20.2 and Fig. 20.3, a grid pattern with known spacing is translated and rotated in the plane of the object and image-based optimization is performed to complete the calibration process. If the distortions are small, then calibration is not required for C x , C y , and ku . For this case, estimates for Λx and Λ y can be obtained by imaging a measurement standard attached to the surface of the planar object. By imaging the measurement standard (e.g., a ruler or a grid), with approximate alignment along the row and column directions in the sensor plane, the images can be analyzed to obtain the pixel positions of several of the marks having a known spacing along both the horizontal and row directions. Using these values, Λx and Λ y can be determined with an accuracy of 0.2% in most cases.

20.8.2 Object Displacement and Strain Measurements The procedures outlined in Sect. 20.7 are used to apply an appropriate textured pattern to the specimen surface for the magnification of interest. As discussed in Sects. 20.5 and 20.6, 2-D DIC with intensity interpolation is performed using an appropriate subset size for the pattern to obtain a dense set of displacement components for each deformed image of interest. Object displacements are obtained in metric units by using the multiplicative scale factors in (20.14) to convert measured image plane motions into object motions. Strains are obtained using the dense set of center-point displacements and a method such as local finite difference or local smoothing of displacement data.

20.8.3 Discussion Regarding the use of (20.13), two points should be noted. First, the use of zero gradients for all subsets to obtain a displacement field is computationally efficient. Furthermore, for small-strain applications, the choice of zero displacement gradients has no discernible bias and

gives reasonable accuracy for the underlying displacement field. However, for strain or rotation >0.03, the choice of zero gradients will cause errors in the measured displacements that will increase rapidly. Secondly, recent studies [20.40] have shown that subset-level quadratic shape functions yield fewer systematic errors than linear shape functions for the same level of random errors in the intensity pattern. Though quadratic shape functions do not directly lead to improvements in the first-order gradient estimates in all cases, they do improve gradient estimates when lowerorder shape functions are incapable of representing the displacement field. At the time of writing, the speed of the correlation process using normalized cross-correlation or any reasonable metric for matching with efficient software implementations is >5000 subsets/s (29 × 29 subsets, cubic spline intensity pattern interpolation, affine subset shape function, PC with 2 GHz processor). Thus, the time required to perform the correlation process is not a significant factor in the measurement process. The gradient terms determined for each small subset (e.g., 29 × 29) during the matching process (e.g., c1 , c2 , c4 , and c5 in (20.13)) may have considerable variability, ranging up to values of 5 × 10−3 , even in a uniformstrain region. Such variability is due to a combination of factors, including interpolation error, subset pattern anisotropy, limited contrast (range in quantization levels), and noise in the intensity pattern. Due to the variability expected in these values for practical applications, the gradient terms are generally used to improve estimates of the local sensor displacement vector (u l (P), vl (P)), but are not used for estimating the actual surface deformations. By obtaining the centerpoint displacement vector for a dense set of positions in the image plane, a full field of displacements is generated with optimal accuracy. Typical standard deviations in the displacement field for a well-designed experiment are ±5 × 10−3 pixels for each displacement component, with values on the order of ±0.015 pixels observed in less optimal cases. Finally, after acquiring a dense set of displacement data in the image plane, the data field is converted into displacement gradients. A robust method involves local smoothing using one of the many techniques available [20.8, 41, 42] to reduce noise in the measured displacement fields and provide a functional form for estimating the local displacement gradients. A local approach used effectively by the author to obtain local deformations involves a local least-squares fit-

Digital Image Correlation for Shape and Deformation Measurements

ting procedure. For example, if one uses a quadratic functional form to fit the displacements in a region around a point P, which corresponds to the nearest ten neighbors in all directions, we have obtained standard deviations in the gradients on the order of ±50 × 10−6 on a consistent basis for high-quality random patterns

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using (a) 8 bit digitizers, (b) spline interpolation, and (c) affine and quadratic shape functions at the subset level. The accuracy estimate is valid for a region with smooth variations in displacement, and not for regions where step changes or sharp gradients in deformation are present.

20.9 Three-Dimensional Digital Image Correlation

World coordinate system

Sensor 1 coordinate system

views for the same object point can be used to determine an accurate estimate for the three-dimensional position of the common object location.

20.9.1 Camera Calibration There are two approaches that are commonly used to calibrate cameras in a stereo-vision system. The first approach calibrates each camera separately and the second approach calibrates the camera system, often known as a stereo rig. In both cases, the first part of the camera calibration process is generally the same. Each camera is modeled using the pinhole projection model with distortion correction outlined in Sect. 20.3. Both

B

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Fig. 20.10 Schematic of two camera stereo-vision system. World coordinate system arbitrarily located, though generally aligned with initial calibration grid with origin at grid intersection point. Epipolar lines constrain search region within images to locations along line. Corresponding image points for object point B in camera 1 and camera 2 are P1 and P2 , respectively

Part C 20.9

Single-camera 2-D DIC systems are limited to planar specimens that experience little or no out-of-plane motion. Both of these limitations can be overcome by the use of two or more cameras observing the surface from different directions. Figure 20.10 shows schematically a two-camera stereo-vision arrangement. The remainder of the developments for stereo-vision systems will focus on a generic two-camera system, though the concepts can be generalized readily to multicamera systems. Three-dimensional image correlation is based on a simple binocular vision concept that utilizes (20.1)– (20.9) to model each of the cameras. Once the cameras are calibrated, the sensor plane locations in the two

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cameras view the same grid standard and images are acquired simultaneously by both cameras as the grid is moved to NVIEWS locations. The process is described in Sects. 20.3.1–20.3.3 and shown graphically in Fig. 20.3. After the images are obtained, the grid locations within the sensor plane of each camera are extracted and stored in files for calibration. Each image of the grid/standard must remain within the depth of field for the camera(s) being calibrated. Defocused images can introduce errors in the calibration process that will degrade the quality of the calibration process and hence decrease accuracy during the measurement phase.

Part C 20.9

Independent Camera Calibration for Stereo Systems If each camera is calibrated independently, the error function defined in (20.10) is used to obtain 6NVIEWS extrinsic parameters for each grid location and five intrinsic camera parameters. Once each camera is calibrated, the six extrinsic parameters for the first grid for both cameras and the five intrinsic calibration parameters for both cameras can be used to determine the three-dimensional positions of points in a WCS, as shown schematically in Fig. 20.11. Stereo-System Calibration If both cameras are calibrated as a stereo rig, then the cameras are assumed to be rigidly linked throughout the experimental process. In this case, the procedure for calibration is slightly different. Assuming that camera 1 in Fig. 20.10 is the reference camera, and considering the entire stereo rig as a single system, the intrinsic unknowns to be determined are

B (xs2, ys2)

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P2

These sixteen parameters, including the relative orientation and position of the two sensors, are considered intrinsic since the cameras remain in fixed positions relative to each other during the experiment. Thus, the extrinsic unknowns are (a) six rigid-body motion parameters relating the world system for the first grid standard to the pinhole system for camera 1 and (b) 6(NVIEWS − 1) extrinsic parameters relating the remaining grid positions to the world coordinate system.

20.9.2 Image Correlation for Cross-Camera Matching Once the cameras are calibrated, the process for determining three-dimensional positions requires that the sensor location (X S , YS ) for the same object point be identified in both cameras. The 2-D image correlation process is used to identify the location of corresponding subregions in the images obtained by the two cameras. To identify the image positions that correspond to the same object point, one of the images (e.g., camera 1) is selected and image correlation is performed to identify the corresponding location in the other sensor plane of the remaining camera (e.g., camera 2). The process described above is repeated to obtain a dense set of corresponding subset locations throughout the sensor plane of camera 2. Typically, the image matching process uses the search procedure shown in Fig. 20.6 and the initial parameter estimation procedure in Fig. 20.8. Since the correlation process is performed on image subsets, subset matching can either be constrained to include geometric, stereo-camera constraints or unconstrained.

Xs2

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1. five intrinsic camera parameters for camera 1 2. five intrinsic camera parameters for camera 2 3. six rigid-body motion parameters relating the pinhole camera system for camera 1 to the pinhole camera system for camera 2

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O1

Fig. 20.11 Schematic of optimal 3D location with two skew rays

passing through matched subset centers in a stereo-rig system

Unconstrained Cross-Camera Image Matching Unconstrained, subset-based image correlation matches subregions in separate cameras by using a 2-D, subsetlevel displacement function appropriate for the situation. The matching procedure is identical to the process described in Sect. 20.6 for two-dimensional digital image correlation. In this form, the matching process does not employ either of the pinhole perspective models to transform 3-D positions into corresponding image locations in the cameras. The subset-level displacement

Digital Image Correlation for Shape and Deformation Measurements

function typically is chosen so that the effect of pinhole perspective distortion can be accounted for during the matching process. Advantages of this approach include (a) the use of established 2-D image correlation concepts, (b) improved speed in subset matching by eliminating the need for implementing constraints during matching, (c) elimination of potential positional errors due to inaccuracies in the pinhole model, and (d) the ability to match subsets without the need to describe the local object geometry. The primary disadvantage of the approach is a complete lack of physical connection to a common object region during the matching process.

After calibration has been completed, the image acquisition process is synchronized so that both cameras acquire images simultaneously during the experiment. After acquiring N pairs of images during the loading process, the 3-D measurement process is as follows. Initial shape 1. A master camera is selected (e.g., camera 1). 2. Subsets are selected in camera 1 and image correlation is performed to locate the matching positions in camera 2. 3. The process shown in Fig. 20.11 is used to determine the optimal 3-D position of the common object point for each matching pair of subsets. 4. The resulting dense set of 3-D points represents the initial object shape. Object deformations: 1. The same master camera and the same subsets used to determine the initial 3-D profile are selected in camera 1. 2. For a given subset in camera 1, image correlation is performed to locate the matching position in the deformed image for camera 1. 3. For the same subset in camera 1, image correlation is performed to locate the matching position in the deformed image for camera 2. 4. Rays are projected from camera 1 pinhole through the center of the deformed subset in camera 1 and from the camera 2 pinhole through the center of the matching, deformed subset in camera 2. The process shown schematically in Fig. 20.11 is used to determine the 3-D position of the common object

point in the deformed state for each matching pair of subsets. – Object point B is determined by minimizing the difference between image-correlation-based sensor positions and model-based sensor positions that are a function of known camera parameters and the to-be-determined (X B , YB , Z B ). 5. Repeating process for both the undeformed 3-D positions and the deformed position for matching subsets throughout the image, the initial and deformed configurations are determined. 6. Subtracting the deformed and undeformed 3-D positions for points spanning the field of view, a dense set of object displacement values are obtained and represent the 3-D displacement field. 7. The process outlined in 1–4 is repeated for each set of deformed images, with master camera subsets providing the basis for the matching process. It is noted that camera 2 can be defined as the master camera and the process described above repeated. In this case, two dense sets of data again are determined over the same object domain.

20.9.4 3-D Position Measurement with Constrained Cross-Camera Image Matching In the absence of camera distortion it can be shown that the vector connecting two pinhole locations (known as as the baseline vector for the stereo camera system), and the two rays that define a common 3-D object point, form a plane O1 –O2 –B. Furthermore, inspection of Fig. 20.10 shows that 1. the projection of the ray O1 –B onto the camera 2 sensor plane intersects the ray O2 –B at the projection of B onto the camera 2 sensor plane, 2. the projection of the ray O2 –B onto the camera 1 sensor plane intersects the ray O1 –B at the projection of B onto the camera 1 sensor plane. These observations can be written mathematically and describe the epipolar constraint equations. They can be used to limit the search regions to locations along the constraint line when locating a common point in both cameras. A form of constrained, cross-camera matching using planar object surface patches was employed in previous work [20.28–31] and is described as follows.

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Part C 20.9

20.9.3 3-D Position Measurement with Unconstrained Matching

20.9 Three-Dimensional Digital Image Correlation

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Initial shape measurement: 1. A master camera is selected (e.g., camera 1). 2. A subset is selected from the undeformed image in camera 1 and a ray is projected through each pixel in the subset as a function of unknown object coordinates (X B1 , YB1 , Z B1 ). 3. The local region on the object is approximated as a small plane. As shown in Fig. 20.12, the equation for the plane can be written X B1 n xB1 + YB1 n yB1 + (Z B1 − Z 0 )n zB1 = 0 , (20.15)

Part C 20.9

where (n xB1 , n yB1 , n zB1 ) is the unit normal to the plane, and (n xB1 )2 + (n yB1 )2 + (n zB1 )2 = 1, Z 0 is the position where the optical axis intersects the object plane. 4. Each pixel is virtually projected onto this object plane. 5. Assuming initial values for the location and orientation of the plane, the pixel locations are then projected onto the sensor plane of the undeformed image in camera 2 and corrected for distortion. 6. Intensity values from camera 1 are matched to the intensity values at the projected locations in camera 2 using a cross-correlation metric (see Sect. 20.3.3). n (O, O,Z0)

7. The process is repeated by updating all five parameters for the plane (the 3-D location of the center of the plane and two orientation angles for the normal) until convergence. 8. The results are the 3-D position of the object point. 9. The process is repeated until a dense set of 3-D positions is obtained. This is the initial object shape. Object deformations: 1. The same master camera and the same subsets used to determine the initial 3-D profile are selected in camera 1. 2. For a given subset in the undeformed image for camera 1, the process outlined above in steps 2–8 is repeated using a deformed image for camera 2. 3. The result is the 3-D deformed position of each object point C. 4. The displacement components for object point C are obtained by subtracting the undeformed position from the deformed position. 5. The process outlined above is repeated for each set of deformed images, with master camera subsets providing the basis for the matching process. The process described above can be reversed, with camera 2 defined as the master camera and the process described above repeated. In this case, two dense sets of data again are determined over the same object domain.

20.9.5 Discussion r

B (XB , YB ,ZB )

Z0 P1

Camera 1 ray O1

Fig. 20.12 Projection of ray through image point P1 and intersection with local tangent plane for object surface near point B. The vector n is perpendicular to local plane. The vector r is as shown, from intersection of optic axis with extended tangent plane for vicinity of B and the object point B

Calibration of stereo-camera systems is the most important aspect in the development and use of a multicamera measurement system. Once the measurement system has been configured and calibrated, image acquisition and storage is procedurally similar to 2-D image correlation. Provided that the cameras in a stereo-vision system are solidly mounted to a rigid bar so that the baseline vector and the relative orientation of the two cameras are fixed, calibration can be performed in a laboratory environment and then moved to the site of the experiment. As long as the intrinsic camera parameters (e.g., focal length, magnification, lens position) are held constant and the relative position of the two cameras is maintained, the experiment can be performed without requiring additional calibration procedures. This process has been used by the author on multiple occasions, including measurements on the crown of a large jetliner, where calibration could only be performed on the ground and the system moved to the top of the aircraft for measurements.

Digital Image Correlation for Shape and Deformation Measurements

Long-focal-length lenses are commonly used when conditions require that the imaging system maintain a large standoff distance from the object. In general, such lenses provide a large depth of field and relatively low image distortion, while requiring higherintensity lighting to maintain sufficient illumination for accurate measurements. A possible disadvantage when using such lenses is the relative insensitivity to model parameters (e.g., lens center), resulting in large variability for these parameters. Shorter-focal-length lenses are used to image objects while minimizing standoff distance. These lenses generally have lower depth of field and higher radial lens distortion, while minimizing the lighting required for accurate measurements. There are few studies focused on quantifying measurement accuracy in a stereo-vision setup. However, evidence from a wide range of experimental studies can be used to provide reasonable estimates. For example, results indicate that in-plane measurement accuracy is similar to that achievable in 2-D computer vision, which is ≈ 1/50-th of a pixel for in-plane components of displacement (i. e., Λx /100 in physical dimensions) and ≈ Z/50 000 for out-ofplane displacement accuracy for modest pan angles, decreasing in accuracy as the pan angle is reduced.

20.10 Two-Dimensional Application: Heterogeneous Material Property Measurements The goal of this study is to estimate the local stress– strain response of the material in a series of friction stir welds (FSWs) produced by varying specific weld parameters. In this study, results obtained using 2-D DIC to quantify the local deformations in welds designated slow, medium, and fast in correspondence with the increasing weld speed are presented. Though the FSW process results in a heterogeneous material system, the primary macroscopic effect is to change the local yield stress. Therefore, a single specimen tensile test was developed to estimate the local uniaxial stress–strain response in the FSW region for all three weld types using 2-D digital image correlation to determine the local strain field throughout the FSW zone.

20.10.1 Experimental Setup and Specimen Geometry All experiments are performed in a 55kip MTS Corporation tensile loading system using displacement control via the MTS Testar control software. Hydraulic, serrated friction grips are used to grip each end of the tensile specimen. Figure 20.13 shows the dog-bone-shaped FSW tensile specimen geometry. As in other welding methods, the FSW weld zone is generally separated into the Nugget region and the heat-affected zone (HAZ). The top surface of the FSW zone, where the rotating FSW pin is injected, is known as the crown and the bottom surface is designated the root. The two sides of

585

Part C 20.10

Constrained matching for 3-D displacement measurements has both advantages and disadvantages. Advantages of this approach include (a) direct measurement of object level parameters (e.g., orientation of surface normal) to provide cross-checking of the accuracy of measurements and (b) direct use of pinhole parameters developed during calibration. Disadvantages of the method include (a) the complexity of the resulting matching process, (b) potential bias in measurements due to the direct use of the perspective projection model in the presence of uncorrected distortions, and (c) additional complexity when attempting to modify the object shape from a locally planar surface. Due to the potential for bias when performing constrained matching, it is generally preferable to employ unconstrained matching and 3-D projection to obtain 3-D positions with a stereo-vision system. In general, the pan angle between two cameras (e.g., similar to the included angle between the optical axes in Fig. 20.10) should be near 90◦ to obtain equivalent accuracy in all three components of displacement. Since this situation is difficult to achieve in real applications due to geometric constraints on the experimental setup, it is always preferable to construct a two-camera stereo system with the included pan angle between optical axes in the range 40 degrees up to 90 degrees whenever possible for a given experiment.

Two-Dimensional Application

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Noncontact Methods

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Advancing side

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Part C 20.10

Base material

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Fig. 20.13 Specimen geometry for tensile experiments. Vertical transverse cross-section patterned for image correlation

measurements using white enamel coating and overspray of black enamel. Field of view is ≈ 45 mm × 7 mm with image centered in the nugget region. Scale factors Λx = Λ y ∼ = 31 pixels/mm

the weld have distinct designations. The advancing side has the rotating pin moving in the direction of pin translation, whereas the retreating side has the rotating pin moving in the opposite direction of pin translation. The region to be imaged is the through-thickness plane (also known as the transverse vertical crosssection) and is shown in Fig. 20.13. Note that, for these experiments, the thickness of the specimen is larger than the height of the vertical transverse cross-section being viewed during the experiment. A major advantage of viewing the edge of the specimen is that the stress–strain history of points from the crown side to the root side of the FSW zone can be estimated using the measured strain field and corresponding load levels by assuming a uniform stress state until localized necking occurs.

20.10.2 Single-Camera Imaging System In this study, a Q-Imaging QICAM FAST-1394 squarepixel camera with 8 bit intensity resolution and 1392 × 1040 spatial resolution is used to record all images. A 200 mm Nikon lens with a 2 × magnification lens is used to record all images. The camera was mounted to a Bogen tripod via a three-dimensional translation stage so that minor adjustments could be made in the position of the camera without moving the tripod.

The tripod–camera system is placed ≈ 2.5 m from the specimen in order to minimize the influence of outof-plane motion on deformation measurements. There is no need for an A/D board in this case since the Q-imaging camera has on-camera, 8 bit digitization. Since the experiment is quasistatic, each 1.4 MB image is transferred to a PC and stored in hard memory prior to acquiring the next deformed image of the specimen.

Correlate images for load Fi

Increment loading

Determine average stressstrain response at key points Determine displacement field (u (x, y), υ (x, y))i across weld zone via 2-D correlation Convert displacement field data to normal strain field data (εxx (x, y))i

Separate/collect stress-strain pairs at points of interest

Determine local stress-strain pairs (σxx (x, y), εxx (x, y))i

Fig. 20.14 Procedure used to determine stress-strain re-

sponse at local points in a heterogeneous material system. Strain response measured via 2D image correlation. Stresses estimated based on measured load and initial area

Digital Image Correlation for Shape and Deformation Measurements

20.10.3 Camera Calibration

20.10.4 Experimental Results The textured pattern was applied using spray paint. A thin layer of white paint is applied, followed by an overspray of black paint to obtain a random speckle pattern on the through-thickness cross-section. In order to improve bonding between the coating and the specimen, a Rustoleum Specialty High Heat white enamel spray paint and Rustoleum Stops Rust flat black enamel spray paint were applied. During the loading process, one hundred images of the specimen surface are acquired and stored for image analysis. The process for converting the images into the local stress strain field data is shown schematically in Fig. 20.14. In this study 25 × 25 pixel subsets and linear subset shape functions are used to perform the correlation process and obtain the center-point displacement field. A step size of two pixels is used during the correlation process, resulting in 500 × 600 pairs of displacement data at each load step. In this study, the commercial software VIC-2-D [20.43] was used to perform image correlation through subset-level pattern matching. Local least-squares smoothing using the ten nearest neighbors in all directions (local smoothing em-

587

ployed a quadratic fit using the 11 × 11 set of nearest neighbor displacement data) is performed to determine the strain field data at each loading step. Since the experiment is terminated just after localization/necking is observed in the undermatched weld region, a uniform stress field is assumed for all material points. Figure 20.15 shows a map of the yield stress field throughout the FSW zone for the fast, medium, and slow FSW processes. Also shown are three of the local stress strain curves used to extract the local yield stress maps.

20.10.5 Discussion To determine whether out-of-plane displacement w would affect the measurements, separate mechanical measurements were performed that showed wmax = 0.10 mm during the early stages of necking, predicting εrr ≈ w/Z < 40 × 10−6 [20.24], which is well below the range of interest in these experiments. Hence, out-ofplane motions have a negligible effect on deformation measurements in this application. To extract deformation data that reasonably reflects the expected spatial variations in properties for the FSW region shown in Fig. 20.13, it is necessary to minimize the subset size and thereby reduce local averaging of the material response. If the size of the object region to be imaged is fixed, then extra care must be taken when applying the textured pattern to ensure that the feature size is near the minimum resolvable in the imaging system. In this work, several patterns were applied and evaluated until the feature size was ≈ 4 × 4, an ideal average speckle size so that small subsets can be used to obtain local displacement measurements. All 2-D DIC image correlations were performed successfully using 25 × 25 subsets. By performing each 2-D image correlation with ΔX S = ΔYS = 2 pixels, and local least-squares smoothing using the ten nearest neighbors in each direction (11 × 11), the resulting strain field also had a spatial resolution of ≈ 0.80 mm. When applying a textured pattern using spray paint, it is noted that extended drying time may introduce debonding and excessive cracking in the applied pattern. For application of the high-temperature paint to aluminum, our experience has shown that performing the tensile experiment ≈ 1 h after patterning the specimen in a laboratory environment eliminates these effects.

Part C 20.10

In this study, distortion is not considered and the minimal calibration procedure outlined in Sect. 20.8 is used to determine the two scale factors. First, the specimen is patterned and placed in the friction clamping grips of the tensile test frame. Second, the camera is mounted to the translation stage, attached to the Bogen tripod, and the entire system moved into position for the experiment. Third, using a laboratory version of a carpenter’s square, the camera was aligned to be approximately perpendicular to the specimen surface. Fourth, a thin ruler with a minimum scale of 1 mm is placed vertically in the field of view on the front surface of the specimen; minor adjustments are made in the ruler orientation so that it is nearly parallel to the vertical YS -axis of the sensor plane. Then, images are acquired of the ruler. Fifth, the process is repeated by aligning the ruler with the horizontal X S -axis of the sensor plane and acquiring an additional set of images. Through pixel-level estimation of the locations for several 1 mm grid lines, an average value for Λx = Λ y ∼ = 31 ± 0.31 pixels/mm in this experiment.

Two-Dimensional Application

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Fig. 20.15a–c Yield stress maps and local stress-strain results for points near root of friction stir weld (FSW) joints in

AL2524-T351 specimens. All strain data obtained using 2D digital image correlation

20.11 Three-Dimensional Application: Tension Torsion Loading of Flawed Specimen Figure 20.16 shows the loading fixture and thin AL6061-T6 specimens used in the combined tension– torsion loading of flawed specimens. The steel fixture was machined and heat-treated to increase the yield strength above 900 MPa. All specimens were laser cut with an initial notch length of 28.6 mm and a total fatigue precrack length of ≈ 33.4 mm. The

specimen’s orientation is in the LT direction (i. e., the crack is aligned with the transverse direction, which is perpendicular to the sheet’s rolling direction). Prior to performing the experiment, the AL6061-T6 specimens were lightly coated with the same hightemperature paint discussed in Sect. 20.10.

Digital Image Correlation for Shape and Deformation Measurements

20.11.1 Experimental Setup with 3-D Imaging System

ϕ =30°

20.11.2 Experimental Procedures

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ing frame so that minor adjustments in the position of the system can be performed without dismounting the cameras. As noted earlier, to obtain displacement data with equivalent accuracy for the out-of-plane displacement (i. e., the displacement along a line that bisects the angle between the two optical axes), each camera would be placed at about 30−40◦ angle with the specimen normal (the angle between the optical axes of the two cameras should be above 30◦ ). However, due to the orientation of the specimen within the loading frame and the limited space available to arrange the stereo rig, the angle between the two cameras is on the order of 16◦ , nearly 5 × smaller than would be needed to ensure optimal accuracy in all the displacement components.

11.18

8 × Ø 6.35

In this example, loading is applied at ϕ = 30◦ , resulting in combined tension–torsion loading through the grips, using the same 55kip, MTS 810 tensile loading system used for the experiments in Sect. 20.10. Throughout the loading process, the displacement rate is held constant at 0.05 mm/s. During the experiment, rotation of the lower (moving) grip around the loading axis is fully constrained so that the boundary conditions can be more reliably modeled. To minimize image analysis errors, the cameras are synchronized with a maximum time difference of 10 μs. Images are taken in a time-controlled mode during continuous loading with the load level recorded as each image is acquired.

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Fig. 20.16 Tension torsion loading fixture and AL6061-T6 specimen dimensions (all dimensions are in mm). Thickness is 2.00 mm. Specimen is laser cut before fatigue pre-cracking

Moving grip

Fig. 20.17 Close-up of setup of 3D displacement measurement system and experimental setup for thin sheet, tension-torsion experiments

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Figure 20.17 shows a close-up of the experimental setup for the mixed-mode tension–torsion experiments. The digital image correlation system (3-D DIC) for threedimensional surface deformation measurement shown in Fig. 20.17 is composed of two Qimaging QICAM FAST-1394 cameras, with matching Nikon AF Nikkor 28 mm f/1:2.8D lenses. Two NA C-Mount adaptors were used to connect the cameras and the lenses. In order to minimize relative motion between the two cameras during the experiment, the two cameras are mounted to a small, rigid aluminum cross-member. The cross-member is then mounted to a translation stage, and the translation stage is mounted on the MTS load-

Three-Dimensional Application

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Camera 2

Part C 20.11 Fig. 20.18 Three pairs of images of a calibration grid at different orientations (the grid spacing is 4.00 mm). Images were used during calibration process

20.11.3 Calibration of Camera System for Tension–Torsion Experiments Prior to performing the tension–torsion experiment, the camera system is calibrated using a uniformly spaced dot pattern with 4 mm spacing. In this case, calibration is performed with the cameras mounted in position for the experiment. Due to a lack of space in the vicinity of the specimen, calibration required that the lower specimen grip be moved downward so that the grid could be place in the vicinity of the specimen for calibration.

Figure 20.18 shows three pairs of calibration images at different 3-D positions and orientations during calibration; 60 image pairs are acquired for calibration. Table 20.1 shows the camera parameters obtained using the calibration procedure described in Sect. 20.9.1. As noted in Table 20.1, the lens center is near the geometric center of the sensor plane (696.520), which is consistent with the use of high-quality lenses and cameras. The included pan angle, β, between the optical axes of the two cameras is 15.32◦ , which is smaller than one would like for a stereo-vision system; the smaller included angle is also evident in the small baseline distance between the two pinholes. The image scale factor for this setup is ≈ 25 pixels/mm. The distortion factor κ1 represents the contribution of third-order radial distortion. The remaining distortion factors, κ2 and κ3 , are the coefficients for the fifth- and seventh-order radial distortion terms and are set to zero in these analyses. Defining κ1normalized = κ1 · ( f λx )2 , corrections at the outer edge of the sensor plane are ≈ 10−4 in nondimensional form. When converted to pixel units, the true value for κ1 and κ2 are generally quite small, as shown below. Even with these small values, the corrections at the outer edge of the sensor plane are significant, on the order of 0.87 and 0.60 for camera 1 and camera 2, respectively. For the system used in this experiment, calibration results also show that the standard deviation of residuals in the pixel positions for all images is less than 0.03 pixels after calibration. Based on the above information, and assuming that the primary error source is in matching of the subsets, the estimated accuracy in the measured displacement components is 1.2 μm for the in-plane displacement measurement and 9.7 μm for the out-of-plane displacement component. Here, the larger error in the out-of-plane displacement component is consistent with the small angle between the stereo cameras.

Table 20.1 Calibration results for a stereo-rig system used for tension–torsion experiments

Center x (pixels) Center y (pixels)

Camera 1

Camera 2

680.092

687.472

α (deg)

339.191

β (deg)

487.869

f λx (pixels)

6385.54

6411.50

f λ y (pixels)

6385.54

6411.50

κ1normalized κ1

Camera 1 to 2 transformation

γ (deg)

− 0.104776

− 0.0761126

Tx (mm)

− 2.416 × 10−9

− 1.704 × 10−9

Ty (mm)

− 1.81489 − 15.3214 0.995723 75.2186 0.892859

κ2

0

0

Tz (mm)

21.5791

κ3

0

0

Baseline (mm)

78.2579

Digital Image Correlation for Shape and Deformation Measurements

20

0

W (mm)

10

–5 –10

0

0.006 0.004 0.002 0.000 –0.002 –0.004 –0.006 –0.008 –0.010 –0.012 –0.014 –0.017 –0.019 –0.021

Y (mm)

0 –10

10 20

X (mm)

–20 30

Y (mm) 10 eyy

5

0.00020 0.00012 0.00004 –0.00004 –0.00012 –0.00020

0 –5 –10 –15 –20 –25 –10

0

10

20

30

X (mm)

Fig. 20.19 Images of AL6061T6 specimen in undeformed state. Also shown is the initial specimen shape, confirming that flatness is within 30 μm prior to impact. Finally, the strain field shows the strain field, ε yy , obtained when comparing two sets of undeformed images

20.11.4 Experimental Results The commercial software VIC-3-D [20.44] was used to analyze a series of images acquired by the calibrated system. Figure 20.19 shows two reference stereo images and the corresponding initial surface profile of the specimen prior to loading. The profile is obtained by using (a) the calibration parameters in Table 20.1, (b) a dense set of matching subsets in camera 0 and camera 1, and (c) the procedure outlined in Sect. 20.9.3. To obtain the surface 3-D displacement field, (u, v, w), the procedure outlined in Sect. 20.9.3 for deformed images is employed. The surface strain field is obtained using the procedure described in Sect. 20.10 with ten nearest neighbors for each component of displacement. By performing this least-squares fitting process with differentiation for all components of dis-

placement, the displacement fields are converted into the full in-plane strain fields in the area of interest using the Lagrangian large-strain formulation in terms of the displacement gradients 1 εxx = ∂u/∂x + [(∂u/∂x)2 + (∂v/∂x)2 2 + (∂w/∂x)2 ] , 1 ε yy = ∂v/∂y + [(∂u/∂y)2 + (∂v/∂y)2 2 + (∂w/∂y)2 ] , 1 εxy = (∂u/∂y + ∂v/∂x) + [(∂u/∂x)(∂u/∂y) 2 + (∂v/∂x)(∂v/∂y) + (∂w/∂x)(∂w/∂y)] , (20.16)

where (u, v, w) are displacements relative to an (x, y, z) coordinate system, respectively, located at the initial

591

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W (mm)

Three-Dimensional Application

592

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Part C 20.11

W (mm)

20

0

W (mm)

10

–5 –10

0

2.0 1.5 1.0 0.5 0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 –3.5 –4.0 –4.5

Y (mm)

0 –10

10 20

X (mm)

–20

Y (mm) 10 eyy

5

0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 0 –0.010 –0.020

0 –5 –10 –15 –20 –25 –10

0

10

20

30

X (mm)

30

Fig. 20.20 Stereo image pair, measured deformed shape and Lagrangian large strain values for ε yy are shown. Quasistatic, tension-torsion loading at 30◦ is applied. Note that largest surface tensile strain is oriented more towards the direction of crack growth and asymmetry is reduced somewhat during continued crack extension

crack tip position. It is important to note that the strain tensor components defined by (20.16) are invariant with respect to rigid-body motion. By correlating two pairs of images at zero loading, and using the profile images as references, the initial strain field is determined and shown in Fig. 20.19. Since the measured strain field should be zero, the values in Fig. 20.19 are a measure of strain error. The data indicates that the errors are distributed somewhat randomly throughout the field, with a total range of ±200 με. The procedures outlined in Sect. 20.9.3 are used to determine the 3-D surface deformations. Figure 20.20 shows (a) stereo images of the deformed specimen during crack propagation, (b) the 3-D position of the specimen surface at this instant, and (c) the tensile surface strain field component, ε yy , as the specimen is being loaded and the crack is extending.

20.11.5 Discussion As shown in (20.16), 3-D surface measurement of the displacement fields for u, v, and w is necessary so that the complete surface strain field can be accurately measured, especially when large deformations (e.g., surface slopes, such as ∂w/∂x) are present. To maximize the range of motions that a specimen can be subject to and still remain in focus, the stereovision system should be positioned at one end of the depth of field so that the specimen moves into the focus field during the deformation process. This arrangement was essential in this experiment since the relatively high-magnification and short-focal-length lenses resulted in a relatively small depth of focus (≈ 30 mm). Since calibration was performed with the stereovision system mounted in their final position (Fig. 20.17)

Digital Image Correlation for Shape and Deformation Measurements

prior to installing the specimen-fixture combination in the load frame, the light sources had to be rearranged to ensure uniform high-intensity illumination of the grid during calibration. Once the calibration process was complete, and the specimen installed in the loading frame, the lighting was rearranged and adjusted for uniform illumination with maximize contrast without overdriving and saturating pixels in either camera. As shown in Table 20.1, the pan angle β is less than 20◦ due to physical space limitations. For appli-

Three-Dimensional Measurements

cations where β ≤ 20◦ and the field of view is on the order of 10 mm or less, it has been observed that (a) the center locations determined during the calibration process can have a wide range of values without significant effect on the accuracy of the solutions and (b) the outof-plane displacement will be less accurate than the in-plane values. For such high-magnification cases, it may be preferable to fix the center location at the midpoint of the sensor array, thereby eliminating potential convergence difficulties.

The overall specimen geometry and fatigue precrack length are the same in this case as used in the static experiments, as shown in Fig. 20.16. Prior to performing the experiment, the specimens are patterned in the manner described in Sect. 20.6.

(Fig. 20.16) through a pin connection. In this work, a weight of 2669 N struck the lower load transfer bar with an impact speed of 5.79 m/s.

20.12.1 Experimental Setup

Figure 20.21 also shows the stereo-vision setup, which must be located outside of the steel protection barrier surrounding the specimen and support structure. For this study, software and hardware were developed to acquire images with Phantom v7.1 cameras. To allow the cameras to be placed outside of the protection barrier, two Nikon AF Nikkor 200 mm f/1:4D lenses are used for the imaging process. In order to minimize the relative motion between the two cameras during the

Load cell

20.12.2 3-D High-Speed Imaging System

Load cell

Upper pin connection

Phantom cameras

Specimen at 30 degree Drop loading location

Drop loading location Lower pin connection

Fig. 20.21 Two images of experimental setup for impact measurements with stereovision systems for dynamic tension torsion loading. Left images show both impact rods and specimen. Right image shows stereo-rig setup using Phantom cameras

Part C 20.12

20.12 Three-Dimensional Measurements – Impact Tension Torsion Loading of Single-Edge-Cracked Specimen

Figure 20.21 shows the specimen and support structure for the dynamic experiments. All experiments are performed in a Dynatup drop-weight facility. The drop weight transfers load to the specimen through two vertical impact bars of the same length. The bars simultaneously impact a lower, hardened striker bar. The striker bar is attached to the tension torsion grip

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experiments, the two cameras are mounted to a rigid cross-member via a pair of translation stages that allow for minor adjustments without disassembly of the components. During the experiments, the cameras are triggered using the output signal of an accelerometer mounted to the lower load transfer bar. All images are acquired at 800 × 600 pixel resolution at a frame rate of 4800 fps (frames per second). The two cameras are synchronized to within ±1 μs to record surface deformations during dynamic crack propagation.

20.12.3 Camera Calibration Part C 20.12

Prior to performing the impact experiments, the highspeed stereo system is calibrated using the same grid and overall calibration procedure described in

Sects. 20.9.1 and 20.11.3. The calibration results for the Phantom camera arrangement are shown in Table 20.2. With the center location defined as shown, the calibration results indicate that the standard deviation of residuals for all images is ≈ 0.06 pixels, which is somewhat larger than that for the static experiments. The image scale factor for this setup is ≈ 13 pixel/mm. Based on this data, the estimated accuracy is 4 μm for each in-plane displacement and 20 μm for the out-ofplane displacement component. The parameter κ1normalized is defined in the same manner as described in Sect. 20.11.3. In normalized form, corrections at the outer edge of the sensor plane are ≈ 2 × 10−5 . When converted to pixel units, the corrections at the outer edge of the sensor plane are on the order of 0.71 mm and 0.62 mm for camera 1 and camera 2, respectively.

Y (mm) 10 W (mm)

0

W (mm)

–1

10 0

0 10

–10 20

X (mm)

30

–20

Y (mm)

0.002 0 –0.001 –0.003 –0.005 –0.007 –0.009 –0.011 –0.013 –0.015

5 eyy 0.00020 0.00012 0.00004 –0.00004 –0.00012 –0.00020

0 –5 –10 –15 –20 0

10

20

30

X (mm)

Fig. 20.22 Images of AL6061T6 specimen in undeformed state from high speed stereo system. Also shown is the initial

specimen shape, confirming the flatness of the specimen prior to impact. Finally, the strain field, ε yy , obtained when comparing two sets of undeformed images is shown

Digital Image Correlation for Shape and Deformation Measurements

Three-Dimensional Measurements

10 eyy

W (mm)

0

W (mm)

–1

10 0

0 10

0.047 0.010 –0.026 –0.063 –0.100 –0.136 –0.173 –0.210 –0.248 –0.283

Y (mm)

–10

5

0.080 0.074 0.068 0.062 0.055 0.049 0.043 0.037 0.031 0.025 0.018 0.012 0.08 0 –0.008

0 –5 –10 –15 –20 0

10

20

X (mm)

30

20

30

X (mm)

–20

Fig. 20.23 Images of AL6061T6 specimen from high speed stereo system, specimen shape and ε yy in crack tip region. Impact with loading angle of 30◦ . Data obtained at impact time of 208 μs and crack extension of 0.32 mm

20.12.4 Experimental Results Software VIC-3-D [20.44] is used to analyze a series of images acquired by the calibrated system during the test to obtain three-dimensional displacement fields. Figure 20.22 shows two reference stereo images and the corresponding initial surface profile of the specimen prior to loading. The profile is obtained by using (a) the calibration parameters in Table 20.2, (b) a dense set of matching subsets in camera 0 and camera 1, and (c) the procedure outlined in Sect. 20.9.3. To obtain the surface 3-D displacement field (u, v, w) the procedure outlined in Sect. 20.11.4 for deformed images is employed. By correlating two pairs of images at zero loading, and using the images in Fig. 20.22 as references, an estimate for the initial strain field is determined (also shown in Fig. 20.22). The data indicates that the errors are consistent with the

quasistatic experiments, falling within a total range of ±200 με and distributed somewhat randomly. The procedures outlined in Sect. 20.9.3 are used to determine the 3-D surface deformations. Figures 20.23 and 20.24 show (a) stereo images of the deformed specimen at two different times after impact, (b) 3-D position of the specimen surface at both times, and (c) the tensile surface strain field component, ε yy , during the crack extension process.

20.12.5 Discussion Synchronization of the image acquisition time for all cameras in a stereo-vision system is an essential part of the 3-D DIC measurement process; images within an image pair that are acquired at different times (but are assumed to be acquired at the same instant) will introduce potentially large errors in the measurements.

Part C 20.12

Y (mm)

595

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Part C 20.12

Y (mm) 10 eyy

W (mm) 0.299 0.118 –0.088 –0.251 –0.434 –0.618 –0.801 –0.984 –1.188 –1.351

0

W (mm)

–1

10 0

0 10

Y (mm)

5

0.180 0.165 0.150 0.135 0.119 0.104 0.089 0.074 0.059 0.044 0.029 0.013 –0.002 –0.017 –0.032

0 –5 –10 –15 –20

–10

0

10

20

X (mm)

30

20

30

X (mm)

–20

Fig. 20.24 Images of AL6061T6 specimen from high speed stereo system, specimen shape and ε yy in crack tip region. Impact with loading angle of 30◦ . Data obtained at impact time of 1042 μs and crack extension of 9.06 mm

Though generally not difficult to complete the synchronization process, care must be taken to ensure that the maximum difference in time between cameras for image acquisition is (a) within an acceptable range for the experiments being performed and (b) consistent throughout the image acquisition process.

The one difference between the calibration procedures used for the quasistatic and dynamic experiments is that the image center is constrained to a predefined location in the image plane. As noted previously, this modification is suggested by the relative insensitivity of the calibration parameter solution to the image cen-

Table 20.2 Calibration results of a setup of the camera system for impact experiments Center x (pixel) Center y (pixel) f λx (pixels) f λ y (pixels) κ1normalized κ1 κ2 κ3

Camera 1

Camera 2

Transformation

400 300 13 340.3 13 340.3 1.960 1.101 × 10−8 0 0

400 300 13 342.3 13 342.3 1.731 9.724 × 10−9 0 0

α (deg) β (deg) γ (deg) Tx (mm) Ty (mm) Tz (mm) Baseline (mm)

0.097603 22.0486 0.995723 − 0.102554 − 396.571 79.1882 404.411

Digital Image Correlation for Shape and Deformation Measurements

ter location for our specific camera arrangement. The insensitivity is introduced by the relatively small pan angle between the cameras and the high focal length of the lenses used to image the object. The high-speed imaging results shown above were obtained using two single-channel cameras with CMOS sensors. The author has also investigated multichannel, image-intensified cameras for ultrahigh-speed imaging (up to 200 million fps) in a stereo-vision arrangement. Results from this study indicate that (a) the two image-

20.14 Further Reading

intensified cameras can be synchronized to acquire simultaneous images within ±5 ns, (b) in-plane correlation can be performed with an accuracy of ±0.05 pixels, and (c) 3-D displacement fields can be converted to strain fields with point-to-point variability in strain ≈ 0.001. Thus, the results indicate that the imageintensification components in the ultrahigh-speed system do increase the variability in the measurements by a factor of five, for both strain and displacement measurements.

The data presented in Sects. 20.10-20.12 are a clear indication of the remarkable measurement capability of modern digital image correlation systems. Reliable fullfield 3-D displacements and full-field total surface strain data can be obtained even though the specimen undergoes

2. 3. 4. 5. 6.

• • • • •

The author firmly believes that the concepts developed and discussed herein provide an excellent foundation for understanding these recent advances. However, in each case there are specific aspects of the developments and applications that require detailed investigation and scientific study to identify key factors affecting the accuracy of the measurements (e.g., distortion, time variability, scan nonlinearities) so that confidence can be developed in the measurement technique. Since image-based measurement methods continue to grow and develop, with an ever-increasing range of applicability, it seems appropriate that these advances be highlighted in future publications, in both original and summary review articles.

•

Furthermore, it is important to note that a form of digital image correlation has been applied in a wide range of areas, many of which are either not discussed or are only briefly mentioned herein. These areas include 1. optical stereo microscopy [20.36]

scanning tunneling microscopy [20.37, 38, 45] atomic force microscopy [20.46] scanning electron microscopy [20.47, 48] computer-aided tomography [20.49, 50] magnetic resonance imaging [20.51]

20.14 Further Reading • • •

O. Faugeras: Three-Dimensional Computer Vision: A Geometric Viewpoint (MIT Press, Cambridge 1993) O. Faugeras, Q.T. Luong, T. Papadopoullos: The Geometry of Multiple Images (MIT Press, Cambridge 2001) R. Hartley, A. Zisserman: Multiple View Geometry in Computer Vision (Cambridge Press, Cambridge, UK 2003)

• • •

B. Jahne: Practical Handbook on Image Processing for Scientific Applications (CRC, Boca Raton 1997) B. Jahne: Digital Image Processing-Concepts, Algorithms and Scientific Applications (Springer, New York 1997) R.C. Gonzalez, RE Woods: Digital Image Processing, 2nd edn. (Addison Wesley, Boston 1987)

Part C 20.14

20.13 Closing Remarks

combined loads impact loading large in-plane rotations locally large strains large out-of-plane displacements and rotations (multiple-camera system) substantial warping during the deformation process (multiple-camera system)

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• • • • • •

J.D. Foley, A. van Dam, S.K. Feiner, J.F. Hughes: Computer Graphics: Principles and Practice, 2nd edn. (Addison–Wessley, New York 1996) M.V. Klein: Optics (Wiley, New York 1970) M. Born, E. Wolf: Principles of Optics, 3rd edn. (Pergamon, Cambridge, UK 1965) G. Strang: Linear Algebra and Its Applications 2nd edn. (Academic, New York 1980) G.W. Stewart: Introduction to Matrix Computations (Academic, New York 1973) J.F. Doyle: Modern Experimental Stress Analysis, 1st edn. (Wiley, New York 2004)

• • • • •

L.E. Malvern: Introduction to the Mechanics of a Continuous Medium (Prentice Hall, Upper Saddle River 1969) D.J. McGill. W.W. King: Engineering Mechanics: An Introduction to Statics and Dynamics (PWS Engineering, Boston 1985) D.E. Carlson: Continuum Mechanics, Theo and Appld Mech Dept, Univ. of Illinois, Notes 1981 D. Post, B.T. Han, P. Ifju: High Sensitivity Moiré (Springer, Berlin 1994) P.K. Rastogi: Photomechanics, Top. Appl. Phys 77, 323–372 (1999)

Part C 20

References 20.1

20.2

20.3

20.4

20.5

20.6

20.7

20.8

20.9

W.H. Peters, W.F. Ranson: Digital imaging techniques in experimental stress analysis, Opt. Eng. 21(3), 427–432 (1981) M.A. Sutton, W.J. Wolters, W.H. Peters, W.F. Ranson, S.R. McNeill: Determination of displacements using an improved digital correlation method, Image Vis. Comput. 1(3), 133–139 (1983) W.H. Peters, He Zheng-Hui, M.A. Sutton, W.F. Ranson: Two-dimensional fluid velocity measurements by use of digital speckle correlation techniques, Exp. Mech. 24(2), 117–121 (1984) J. Anderson, W.H. Peters, M.A. Sutton, W.F. Ranson, T.C. Chu: Application of digital correlation methods to rigid body mechanics, Opt. Eng. 22(6), 738–742 (1984) T.C. Chu, W.F. Ranson, M.A. Sutton, W.H. Peters: Applications of digital image correlation techniques to experimental mechanics, Exp. Mech. 25(3), 232–245 (1985) M.A. Sutton, S.R. McNeill, J. Jang, M. Babai: The effects of subpixel image restoration on digital correlation error estimates, Opt. Eng. 10, 870–877 (1988) M.A. Sutton, M. Cheng, S.R. McNeill, Y.J. Chao, W.H. Peters: Application of an optimized digital correlation method to planar deformation analysis, Image Vis. Comput. 4(3), 143–150 (1988) M.A. Sutton, H.A. Bruck, S.R. McNeill: Determination of deformations using digital correlation with the Newton–Raphson method for partial differential corrections, Exp. Mech. 29(3), 261–267 (1989) M.A. Sutton, H.A. Bruck, T.L. Chae, J.L. Turner: Development of a computer vision methodology for the analysis of surface deformations in magnified images. In: MICON-90: Advances in video technology for micro-structural evaluation of materials, ASTM STP-1094, ed. by G.F. Vander Voort (ASTM Int., West Conshohocken 1990) pp. 109–134

20.10

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20.12

20.13

20.14

20.15

20.16

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20.19

M.A. Sutton, J.L. Turner, T.L. Chae, H.A. Bruck: Full field representation of discretely sampled surface deformation for displacement and strain analysis, Exp. Mech. 31(2), 168–177 (1991) S.R. McNeill, W.H. Peters, M.A. Sutton, W.F. Ranson: A Least square estimation of stress intensity factor from video-computer displacement data, Proceeding of the 12th Southeastern Conference of Theoretical and Applied Mechanics (1984) pp. 188– 192 M.A. Sutton, H.A. Bruck, T.L. Chae, J.L. Turner: Experimental investigations of three-dimensional effects near a crack tip using computer vision, Int. J. Fract. 53, 201–228 (1991) G. Han, M.A. Sutton, Y.J. Chao: A Study of stable crack growth in thin SEC specimens of 304 stainless steel, Eng. Fract. Mech. 52(3), 525–555 (1995) G. Han, M.A. Sutton, Y.J. Chao: A Study of stationary crack tip deformation fields in thin sheets by computer vision, Exp. Mech. 34(4), 357–369 (1994) B.E. Amstutz, M.A. Sutton, D.S. Dawicke: Experimental study of mixed mode I/II stable crack growth in thin 2024-T3 aluminum, ASTM STP 1256 Fatigue Fract. 26, 256–273 (1995) J. Liu, M.A. Sutton, J.S. Lyons: Experimental characterization of crack tip deformations in alloy 718 at high temperatures, ASME J. Eng. Mater. Technol. 20(1), 71–78 (1998) M.A. Sutton, Y.J. Chao, J.S. Lyons: Computer vision methods for surface deformation measurements in fracture mechanics, ASME-AMD Novel Exp. Method. Fract. 176, 123–133 (1993) M.A. Sutton, Y.J. Chao: Experimental techniques in fracture. In: Computer Vision in Fracture Mechanics, ed. by J.S. Epstein (VCH, New York 1993) pp. 59–94 J.S. Lyons, J. Liu, M.A. Sutton: Deformation measurements at 650 ◦ C with computer vision, Exp. Mech. 36(1), 64–71 (1996)

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20.20

20.21

20.22

20.24

20.25

20.26

20.27

20.28

20.29

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20.31

20.32

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20.38

20.39

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non-linear minimization method, Int. Conference on Intelligent Robots and Systems (Grenoble, 1997) D. Garcia, J.J. Orteu, M. Devy: Accurate calibration of a stereovision sensor; Comparison of different approaches, 5-th Workshop on Vision, Modeling and Visualization (Saarbrucken, 2000) pp. 25–32 J.M. Lavest, M. Viala, M. Dhome: Do we really need an accurate calibration pattern to achieve a reliable camera calibration?, European Conference on Computer Vision (Frieburg, 1998) pp. 158–174 H.W. Schreier, D. Garcia, M.A. Sutton: Advances in stereo light microscopy, Exp. Mech. 44(3), 278–289 (2004) G. Vendroux, W.G. Knauss: Submicron deformation field measurements, Part II, Improved digital image correlation, Exp. Mech. 38(2), 86–92 (1998) G. Vendroux, N. Schmidt, W.G. Knauss: Submicron deformation field measurements, Part III, Demonstration of deformation determination, Exp. Mech. 38(3), 154–160 (1998) H.W. Schreier, J. Braasch, M.A. Sutton: On systematic errors in digital image correlation, Opt. Eng. 39(11), 2915–2921 (2000) H.W. Schreier, M.A. Sutton: Effect of higher order displacement fields on digital image correlation displacement component estimates, Int. J. Exp. Mech. 42(3), 303–311 (2002) C.R. Dohrman, H.R. Busby: Spline function smoothing and differentiation of noisy data on a rectangular grid, Proceedings of SEM Spring Conference (Albuquerque, 1990) pp. 76–83 Z. Feng, R.E. Rowlands: Continuous full-field representation and differentiation of threedimensional experimental vector data, Comput. Struct. 26(6), 979–990 (1987) VIC2-D Software, Correlated Solutions, Incorporated, 120 Kaminer Way, Parkway Suite A Columbia, SC 29210 www.correlatedsolutions.com VIC3-D Software, Correlated Solutions, Incorporated, 120 Kaminer Way, Parkway Suite A Columbia, SC 29210 www.correlatedsolutions.com G. Vendroux, W.G. Knauss: Submicron deformation field measurements, Part I, Developing a digital scanning tunneling microscope, Exp. Mech. 38(1), 18–23 (1998) S.W. Cho, I. Chasiotis, T.A. Friedmann, J.P. Sullivan: Young’s modulus, Poissons’ ratio and failure properties of tetrahedral amorphous diamond-like carbon for MEMS devices, J. Micromech. Microeng. 15(4), 728–735 (2005) M.A. Sutton, D. Garcia, N. Cornille, S.R. McNeill, J.J. Orteu: Initial experimental results using an ESEM for metrology, Proc. SEM X Int. Congr. Expo. on Exp. Appl. Mech. (Costa Mesa, 2004) M.A. Sutton, N. Li, D. Garcia, N. Cornille, J.J. Orteu, S.R. McNeill, H.W. Schreier, X. Li: Metrology in an SEM: Theoretical developments and experimen-

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20.23

M.A. Sutton, Y.J. Chao: Measurement of strains in a paper tensile specimen using computer vision and digital image correlation – Part 1: Data acquisition and image analysis system, Tappi J. 70(3), 153–155 (1988) M.A. Sutton, Y.J. Chao: Measurement of strains in a paper tensile specimen using computer vision and digital image correlation – Part 2: Tensile specimen test system, Tappi J. 71(3), 173–175 (1988) M.A. Sutton, S.R. McNeill, J.D. Helm, H.W. Schreier: Computer vision applied to shape and deformation measurement,. In: Trends in Optical Non-Destructive Testing and Inspection, ed. by P.K. Rastogi, D. Inaudi (Elsevier, Oxford 2000) pp. 571–591 M.A. Sutton, S.R. McNeill, J.D. Helm, Y.J. Chao: Advances in 2-D and 3-D computer vision for shape and deformation measurements. In: Photomechanics, Topics in Applied Physics, Vol. 77, ed. by P.K. Rastogi (Springer, Berlin 2000), 323–372 M. A. Sutton, Y. J. Yan, H. W. Schreier, J. J. Orteu: The effect of out of plane motion on 2D and 3D digital image correlation measurements, Opt. Eng, (in press) P.F. Luo, Y.J. Chao, M.A. Sutton: Application of stereo vision to 3-D deformation analysis in fracture mechanics, Opt. Eng. 33(3), 981–990 (1994) P.F. Luo, Y.J. Chao, M.A. Sutton: Accurate measurement of three-dimensional deformations in deformable and rigid bodies using computer vision, Exp. Mech. 33(3), 123–133 (1993) P.F. Luo, Y.J. Chao, M.A. Sutton: Computer vision methods for surface deformation measurements in fracture mechanics, ASME-AMD Novel Exp. Method. Fract. 176, 123–133 (1993) J.D. Helm, S.R. McNeill, M.A. Sutton: Improved 3-D image correlation for surface displacement measurement, Opt. Eng. 35(7), 1911–1920 (1996) J.D. Helm, M.A. Sutton, D.S. Dawicke, G. Hanna: Three-dimensional computer vision applications for aircraft fuselage materials and structures, 1st Joint DoD/FAA/NASA Conference on Aging Aircraft in (Ogden, 1997) pp. 1327–1341 J.D. Helm, M.A. Sutton, S.R. McNeill: Deformations in wide, center-notched, thin panels: Part I: Three dimensional shape and deformation measurements by computer vision, Opt. Eng. 42(5), 1293–1305 (2003) J.D. Helm, M.A. Sutton, S.R. McNeill: Deformations in wide, center-notched, thin panels: Part II: Finite element analysis and comparison to experimental measurements, Opt. Eng. 42(5), 1306–1320 (2003) O. Faugeras, F. Devernay: Computing differential properties of 3-D shapes from stereoscopic images without 3-D models, INRIA Report 2304 (1994) M. Devy, V. Garric, J.J. Orteu: Camera calibration from multiple views of a 2-D object using a global

References

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20.49

20.50

tal validation, Meas. Sci. Technol. 17(10), 2613–2622 (2006) B.K. Bay, T.S. Smith, D.P. Fyhrie, S. Malik: Digital volume correlation, three dimensional strain mapping using X-ray tomography, Exp. Mech. 39(3), 218–226 (1999) T.S. Smith, B.K. Bay, M.M. Rashid: Digital volume correlation including rotational degres of freedom

20.51

during minimization, Exp. Mech. 42(3), 272–278 (2002) C.P. Neu, M.L. Hull: Toward and MRI-based method to measure non-uniform cartilage deformation; An MRI cyclic loading apparatus system and steadystate cyclic displacement of articular cartilage under compressive loading, Trans. ASME 125, 180– 187 (2003)

Part C 20

601

Geometric Mo 21. Geometric Moiré

Bongtae Han, Daniel Post

21.1 Basic Features of Moiré ......................... 21.1.1 Gratings, Fringes, and Visibility...... 21.1.2 Intensity Distribution.................... 21.1.3 Multiplicative and Additive Intensities ................ 21.1.4 Moiré Fringes as Parametric Curves .

601 602 602 603 605

21.2 In-Plane Displacements ........................ 607 21.2.1 Fringe Formation by Pure Rotation and Extension ............................. 607

21.2.2 Physical Concepts: Absolute Displacements ................ 607 21.2.3 Experimental Demonstration: Relative Displacements ................. 609 21.2.4 Fringe Shifting ............................. 610 21.3 Out-Of-Plane Displacements: Shadow Moiré ...................................... 21.3.1 Shadow Moiré.............................. 21.3.2 Projection Moiré .......................... 21.3.3 Comparison ................................. 21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT)......................... 21.4.1 The Talbot Effect .......................... 21.4.2 Fringe Contrast Versus Gap ............ 21.4.3 Dynamic Range and Talbot Distance 21.4.4 Parameters for High-Sensitivity Measurements. 21.4.5 Implementation of SM-NT .............

611 611 615 617 617 617 618 619 620 621

21.5 Increased Sensitivity ............................. 623 21.5.1 Optical/Digital Fringe Multiplication (O/DFM) ....................................... 624 21.5.2 The Phase-Stepping (or Quasiheterodyne) Method ........ 624 References .................................................. 626

21.1 Basic Features of Moiré Moiré patterns are common. In Fig. 21.1a, two combs produce the moiré effect. In Fig. 21.1b, two wire mesh bowls used in food preparation produce moiré patterns. The broad dark and light bands are called moiré fringes. They are formed by the superposition of two amplitude gratings, each comprised of opaque bars and clear spaces. Whereas gratings with straight bars and spaces are typical, the bars and spaces can be curvilinear. The moiré fringes are contour maps of the difference between the gratings, and as such, they have been attractive in experimental solid

mechanics for the determination of surface deformations. The term moiré has been adopted in the optics literature to describe the visual effect produced by this interaction of two superimposed gratings. It stems from the French word for watered, relating moiré to watered silk, and earlier to watered mohair. These are fabrics that are processed to produce patterns with brighter and darker bands, both of the same color. Thus, the appearance of moiré fringe patterns resembles that of moiré fabrics.

Part C 21

Our goal in this chapter is to review the fundamental concepts of geometric moiré that are usually addressed in the study of optical methods of experimental mechanics and engineering practice. Moiré methods can measure the in-plane and out-of-plane components of displacement. Herein emphasis is placed on the measurement of deformations that occur when a body is subjected to applied forces or temperature changes. An additional goal is to build a foundation for easy assimilation of the concepts of high-sensitivity moiré interferometry presented in the next chapter. Much of this chapter is excerpted from [21.1, 2] with the kind permission of Springer Science and Business Media, and from [21.3].

602

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21.1.1 Gratings, Fringes, and Visibility

a)

The visibility of the moiré fringes is good in Fig. 21.1a, but poor in Fig. 21.1b. Moiré fringes of excellent visibility occur when the following conditions are present:

• •

b)

• •

Part C 21.1

Fig. 21.1a,b Moiré patterns

a)

b) g1

G

G

g θ I

g2 I

x

c)

I

x

G x

d)

I

x

Fig. 21.2a–d These moiré fringes have a triangular intensity distri-

bution when the emergent light is averaged over the pitch of the coarser grating. In (d), the rounding is caused by averaging over several pitches

the widths of the bars and spaces are equal, the two gratings are well defined, i. e., the distances between the bars and the curvatures of the bars in each grating are smoothly varying functions of the coordinates x and y, the angle of intersection of the two gratings is small, e.g., 3◦ or less, the ratio of the pitches of the two gratings is small, e.g., 1.05:1 or less.

A corollary to the fourth condition is that the distance between adjacent moiré fringes is large compared to the distance between adjacent bars of the gratings, e.g., in the ratio 20:1 or larger. Of course, excellent (and good) fringe visibility is a subjective assessment and less stringent conditions are frequently acceptable. The pitch of each grating is the distance between corresponding points in adjacent bars. The reciprocal of the pitch is the frequency, or the number of bars per unit length. The bars are often called lines, so the grating pitch, g, is the distance between adjacent lines and the grating frequency, f , is the number of lines per unit length; 10 lines/mm (250 lines/in) is a common example of a grating frequency for geometric moiré. For most applications, grating frequencies do not exceed 40 lines/mm (1000 lines/in). The pitch, G, of moiré fringes is the smallest distance between corresponding features on neighboring fringes. Their frequency, F, is the number of fringes per unit length, e.g., 2 fringes/cm or 5 fringes/in. Usually we are interested in the fringe frequencies along orthogonal x- and y-axes, that is, the number of fringes that cut lines that are parallel to the x- and y-axes, respectively, per unit length along these lines.

21.1.2 Intensity Distribution The observer sees the moiré fringes as a cyclic function of the intensity of light; the fringes appear as alternating bands of brightness and darkness. This is especially true in the usual case where the bars of the superimposed gratings are closely spaced and the observer cannot resolve them. Ideally, the observer perceives the average of the light energy that emerges from each small region, where the dimensions of the region are equal to

Geometric Moiré

a)

Observer or camera

Source Gratings Diffuser

b)

When several fringes cross x, the intensity distribution is depicted in Fig. 21.2c. Fringe spacings are typically one to two orders of magnitude larger than grating line spacings. Since the moiré fringes are much further apart, they can be resolved even when the grating lines cannot. The limited resolution of the observer (or camera) influences the perceived intensity distribution. With reduced resolution, the triangular intensity distributions become rounded near maxima and minima points. The rounding is the result of intensity averaging in regions larger than the grating pitch, for example, when averaging occurs over a few grating pitches. The effect is illustrated in Fig. 21.2d. Since the averaging distance is fixed for a given observation, the departure from a triangular distribution is more severe for fringes that are more closely spaced than for fringes that are further apart.

21.1.3 Multiplicative and Additive Intensities Multiplicative Intensities Figure 21.3 illustrates alternative means to superimpose gratings. In Fig. 21.3a the gratings are on transparent substrates and are in contact, or nearly in contact. They are illuminated by a broad diffuse source of light. In Fig. 21.3b, one grating is printed in black ink on a diffusely reflecting surface, (e.g., a white paper or a white matte painted surface) and the other grating is on glass. In both of these cases, light that would otherwise reach the observer is partly obstructed by the gratings. In the transmission system (Fig. 21.3a), the intensity of light that emerges from the first grating is the incident intensity I0 times the transmittance T1 (x, y) at each point of the first grating, or I0 T1 (x, y). The intensity of light that emerges from the second grating is

I = I0 T1 (x, y)T2 (x, y) .

(21.1)

For the bar and space gratings that are commonly used in geometric moiré, T ≈ 0 at the opaque bars and T ≈ 1 at the transparent spaces. Thus, there are two levels of intensity: zero in obstructed regions and I0 elsewhere. When the emergent intensities are averaged over the grating pitch, the conditions of Fig. 21.2 prevail. In the reflective system (Fig. 21.3b), light passes twice through the transmission grating and it is reflected once by the reflection grating. Accordingly, the intensity of the emergent light is I = I0 (T (x, y))2 R(x, y) ,

Fig. 21.3a,b Superposition of gratings in (a) a transmission system and (b) a reflection system

603

(21.2)

where R(x, y) is the reflectance at each x, y point of the reflective grating. At the reflective bars, R is some

Part C 21.1

the pitch of the coarser grating. This perceived intensity distribution is illustrated by the graphs in Fig. 21.2 for uniform linear gratings, each with bars and spaces of equal width. Pure rotation is depicted on the left, where equal gratings are superimposed with an angular displacement θ. Pure extension is illustrated on the right, where two gratings of slightly different pitches are superimposed. In both cases, some of the light that would otherwise emerge from the first grating is obstructed by the superimposed grating. At the centers of dark fringes, the bar of one grating covers the space of the other and no light comes through; the emergent intensity I is zero. Proceeding from there toward the next dark fringe, the amount of obstruction diminishes linearly and the amount of light (averaged across the grating pitch) increases linearly until the bar of one grating falls above the bar of the other. There, the maximum amount of light passes through the gratings. Then the obstruction increases again and the intensity (averaged) decreases again. This progression is indicated in the graphs, in which the coordinate x lies perpendicular to the moiré fringes. The triangular variation of intensity with x is obvious from the diamond-shaped spaces formed in the pure rotation case. For the pure extension case, the unobstructed spaces become wider in a linear progression, and then narrower, as seen in the figure. This linear progression matches the triangular distribution when the intensity is averaged over the pitch of the coarser grating.

21.1 Basic Features of Moiré

604

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Noncontact Methods

finite reflectance, perhaps 0.8 in many cases; at nonreflective spaces R ≈ 0. Again, T is approximately zero or one. Thus, the emergent intensity is zero in the obstructed regions and a constant finite value elsewhere. When averaged, the conditions of Fig. 21.2 prevail here, too. In view of (21.1) and (21.2), the patterns developed when two real gratings are in contact are called moiré patterns of multiplicative intensities; sometimes they are called multiplicative moiré patterns. They are characterized by moiré fringes of good visibility. What is meant by contact? The two gratings may be pressed tightly together, but that is not required. Essentially no deterioration of the moiré pattern will be seen when the gap between the two gratings is small compared to half the Talbot distance (or self-imaging distance defined as 2g2 /λ), e.g., 1% of 2g2 /λ, where λ is the wavelength of the light; see Sect. 21.5.1.

Part C 21.1

Additive Intensities A different optical system is illustrated in Fig. 21.4, where Fig. 21.4a represents the projection of a bar-andspace grating onto a diffuse screen. A lantern slide projector could be used. The observer sees a grating on the screen, comprised of dark and bright bars, or lines. For the sake of explanation, visualize two projected gratings superimposed on the screen, as illustrated schematically in Fig. 21.4b. Let the two projected gratings be slightly different from each other so that moiré fringes are formed on the screen. As in the prea)

vious case, the difference might be a small rotation, a small change of pitch, or both. Referring to Fig. 21.4a, with the single projector, let the observer have limited resolution. Let the smallest detail that can be resolved be equal in size to the pitch of the projected grating (or a small multiple of the pitch). Then the observer would perceive each and every pitch as having the average intensity of the dark bar and bright bar. The screen would appear to be illuminated by a uniform intensity. The second projector would produce the same effect; no light is obstructed. None of the dark and bright bars that actually exist on the screen would be perceived. If the observer cannot resolve the individual lines of the grating, only the average intensity is detected and no moiré fringes are seen. The effect of limited resolution is totally different from that found in Fig. 21.2 for multiplicative intensities. If the observer or camera can resolve the individual grating lines, a periodic structure is seen and perceived as moiré fringes. In regions where the bright bars of the two gratings fall directly on top of each other, bright and dark grating lines are visible. In regions where the bright bars of one grating fall midway between those of the other, a gray band is formed. The periodic structure of gray bands alternating with bands of grating lines comprises the moiré fringes. An example is shown a)

N=1

2

3

4

Observer

b) I Screen or film x

c)

Slide projector

b) d) I Screen or film x

Fig. 21.4 (a) Projection of a grating. (b) Two superimposed gratings by projection

Fig. 21.5 (a) Moiré of two projected gratings recorded with linear photographic sensitivity. (b) Averaged intensity for linear recording. (c) Moiré with nonlinear recording. (d) Averaged intensity for nonlinear recording

Geometric Moiré

21.1.4 Moiré Fringes as Parametric Curves A very nice treatment of the relationship between moiré fringes and the gratings that create them was presented by Durelli and Parks [21.4], and that approach will be introduced here. Moiré patterns feature three families of curves. The two superimposed gratings are two families of curves, which we will call families L and M. These families interact in such a way that they form a third family, the family of moiré fringes, which we will call family N. Figure 21.6 illustrates the three families, but for simplicity the grating lines and fringes are shown straight instead of curved. The centerlines of the bars of the L and M families are extended in the figure. Each member of each family is given an index number, representing the sequential order of the parametric family. A specific number applies to every point on one curve of the family. The figure shows that the centerlines of the bright moiré fringes pass through the intersections of the centerlines of the bars. Inspection shows that the centers of the moiré fringes comprise the locus of points where L − M is a constant integral number. These moiré fringes are a parametric system of curves defined by N = L − M, where N is the index number assigned to each point in the moiré pattern. Consistent with practice, N is called the moiré fringe order, and more specifically, it is called the subtractive moiré fringe order.

605

L = 3 4 5 6 7 8 9 10 11 12 13 14 15 8 9 6 7 5 3 4 N= 6 1 2 0 –1 (N + = 23) M = –3 –2 0.5 6.5

(N + = 22) 7

N= 6 8 7 9 8

9.5 10

9 9.5

10.5 11

10

Fig. 21.6 The L and M parametric families of grating lines

creates the N family of moiré fringes, where N = L − M at every point

The encircled region in Fig. 21.6 indicates that all points between curves of integral index numbers have intermediate index numbers. Every point in the field has index numbers L and M and fringe order N. We note that a fourth family of parametric curves appears in Fig. 21.6. It is the family of additive moiré defined by N + = L + M. The phenomenon is described and illustrated in [21.1]. Moiré as a Whole-Field Subtraction Process Moiré can be used to subtract data, and it can do the subtraction simultaneously everywhere in an extended field. Figure 21.7 shows an example. The specimen was cut from a plastic sheet, polymethylmethacrylate, of 3 mm (1/8 in) thickness. The objective was to map the changes of thickness near the notch caused by an applied tensile load; subwavelength sensitivity was required. Employing the experimental method of [21.5], Fizeau interferometry was used to record the contour maps of the specimen thickness before and after the tensile load was applied; they are shown in Fig. 21.7 as

Part C 21.1

in Fig. 21.5a, with the corresponding graph of averaged intensity shown in Fig. 21.5b. The visibility of fringes of additive intensity can be improved by nonlinear recording, e.g., as in Fig. 21.5c. A strong overexposure was used to saturate the photographic film. Thus, all the bright zones have the same brightness level; the nonlinear recording converted the gray levels of Fig. 21.5a to maximum brightness. When the photograph is viewed and the intensity is averaged across the grating pitch, the perceived intensity (idealized) is that graphed in Fig. 21.5d. Accordingly, the grating lines must be resolved on the screen or film, but then the moiré fringes can be discriminated by their variation of (averaged) intensity. When nonlinear recording is used, it is not necessary for the observer to resolve the individual lines to see the moiré, even though the grating lines must be resolved on the screen or film. Still, the contrast of fringes is low compared to moiré fringes of multiplicative intensities. Patterns formed by projected gratings are designated by the term moiré of additive intensities.

21.1 Basic Features of Moiré

606

Part C

Noncontact Methods

a)

b)

c)

–3

–7

–2 0 –1

d)

e)

g) Laser light

Specimen

Part C 21.1

127 mm 5 in.

Camera

t

Beam splitter

A B

f)

Fig. 21.7 (a) Contour maps of specimen thickness before and (b) after the tensile load was applied; the contour interval is λ/2n = 0.21 μm/fringe order. (c) Moiré pattern of a load-induced change of thickness. (d) The transparent specimen. (e) The optical system. (f) Reflected beams represented by rays A and B enter the camera and interfere to produce the maps of specimen thickness. (g) Photographic print of superimposed films of (a) and (b). Note: The patterns represent the region in the dashed box of (d)

(a) and (b), respectively. The initial (or zero-load) pattern is the M family of curves, in this case, and it was subtracted by superposition of (a) and (b). The resultant N family of parametric curves shown in (c) is the moiré pattern, where the fringes represent contours of constant change of thickness, specifically, contours of the loadinduced changes of thickness. The numbers attached to the moiré fringes are their fringe orders; when multiplied by the contour interval, 0.21 μm, the moiré pattern becomes a map of the thickness changes throughout the field. In practice, the photographic negatives of contour maps Fig. 21.7a and b are superimposed and inserted into a photographic enlarger to make a print. The result is shown in Fig. 21.7g, which is a negative image

of Fig. 21.7c. This format is sometimes preferred, since it is easier to sketch the centerlines of integral and half-order fringes directly on the print. The positive image (Fig. 21.7c) can be produced by using auto-positive printing paper in the photographic enlargement step; if the negatives are large enough, a xerographic copying machine or a computer scanner can be used instead. This concept of moiré as a whole-field subtraction process is powerful. In moiré interferometry, for example, it can be used to cancel distortions of the initial field; it can be used to determine deformations incurred between two different load levels in order to evaluate nonlinear effects; and it can be used to perform whole-field differentiation, to determine derivatives of displacements [21.1].

Geometric Moiré

21.2 In-Plane Displacements

607

21.2 In-Plane Displacements

21.2.1 Fringe Formation by Pure Rotation and Extension A beautiful feature of geometric moiré is that the relationships between the moiré fringes and grating lines can be determined simply by geometry. For the case of pure rotation, the shaded triangle in Fig. 21.2a gives, for small angles, g/2 = θG/2, where the grating pitch g and the fringe pitch G are defined in the figure. Thus, for small pure rotations, g (21.3) G= . θ

and in terms of frequency, the relationship is F = f1 − f2 ,

(21.6)

where f 1 and f 2 are the frequencies of the finer and coarser gratings, respectively. An example would be f 1 = 200 lines/cm, f 2 = 196 lines/cm, and F = 4 fringes/cm. The fringes of pure extension lie parallel to the grating lines.

21.2.2 Physical Concepts: Absolute Displacements Referring to Fig. 21.8a, let a point P on the surface of a solid body be displaced to point P  . Points P and P  represent the same physical point on the body; the displacement is caused by deformation of the body, or by rigid-body movement, or by a combination of both. The displacement of P is represented in Fig. 21.8b by the vector δ, and its components U, V, and W in the x-, y-, and z-directions, respectively. The displacement is usually described by the scalars U, V , W, which are the magnitudes of the corresponding vectors, with positive or negative signs for vector components that point in the positive or negative direction of the coordinate axes, respectively. U and V are called in-plane displacements since they lie in the original plane of the surface. W is perpendicular to the surface and is therefore called the out-of-plane displacement. Before discussing whole-field moiré measurements, consider the following concept. Let point P on the specimen act as a tiny light source that glows con-

Part C 21.2

Concepts relating in-plane displacements and moiré are developed in this section. Although strains can be extracted from in-plane displacement fields, the sensitivity of in-plane geometric moiré is not adequate in most cases for the determination of strain distributions. For in-plane displacement measurements, the technology has evolved from low-sensitivity geometric moiré to the powerful capabilities of moiré interferometry. Inplane moiré measurements are performed routinely in the interferometric domain with fringes representing subwavelength displacements per contour. Strain analysis will be treated in Chap. 22 Moiré Interferometry, in conjunction with the high-sensitivity measurement of in-plane displacements. However, it is useful to appreciate at the outset that the equations relating moiré fringes to in-plane displacements are identical for geometric moiré, moiré interferometry, and microscopic moiré interferometry.

In terms of frequency, the relationship is F = fθ .

(21.4)

a)

b) y

Note that the fringes of pure rotation lie perpendicular to the bisector of angle θ, or nearly perpendicular to the lines of the gratings. For pure extension (Fig. 21.2b) there is one more line in the finer grating than in the coarser one, for each moiré fringe. Thus, the pitch of the moiré fringes is given by G = ng2 = (n + 1)g1 ,

P'

P W

P

δ U

z x

where n is the number of pitches of the coarser grating that fall within G. By eliminating n this reduces to 1 1 1 − , = G g1 g2

V P'

(21.5)

Fig. 21.8 (a) Point P moves to P  when the body is deformed. The

x-component of the displacement can be measured by means of the grating. (b) A three-dimensional displacement is resolved into its U, V , and W components

608

Part C

Noncontact Methods

tinuously. Let a reference grating on a glass substrate be fixed in front of the specimen, as illustrated in Fig. 21.8a; let the reference grating pitch and frequency be g and f , respectively. Let the observer be far away, such that every point is viewed at (nearly) normal incidence. When the specimen is deformed or moved, P moves to P  while the reference grating remains stationary. The tiny light source appears to blink on and off as it is repeatedly obstructed by the opaque reference grating bars. In fact, the number of intensity cycles or blinks, N, seen by the observer is N=

U . g

a)

c)

P

Since the number of blinks depends upon the number of opaque bars crossed by the bright point, the number is independent of V and W; it depends upon U alone. Thus, the displacement U is the number of intensity cycles times the reference grating pitch, or U = gN =

1 N. f

(21.7)

U is the absolute displacement of point P, in the x-direction, since N is the total number of cycles, independent of whether the movement is caused by deformation or rigid-body motion. Next, let a grating of pitch g be printed on the specimen with its bars and spaces in registration with the

A'

Part C 21.2

y Reference grating x

Specimen grating

19 mm 0.75 in.

3

2 1

Ny

127 mm 5 in.

0

–3

b) Specimen

A

Reference

Source

d)

Video camera Diffuser f = 19.7 lines/mm 500 lines/in.

1

Nx

e)

y (mm)

0 1

40 20 –2

Ny

–1 1 –20 – 40

2

Fig. 21.9a–e In-plane displacements: (a) Specimen. (b) Optical arrangement. (c) N y or V displacement field. (d) N x or U field. (e) Fringe orders

along A A

Geometric Moiré

21.2.3 Experimental Demonstration: Relative Displacements The demonstration illustrated in Fig. 21.9 investigates in-plane deformation. In Fig. 21.9a, a deeply notched tensile specimen is imprinted with a uniform crossline grating on its surface. The specimen is transparent and illuminated from behind, as shown in Fig. 21.9b. A linear reference grating of the same frequency is superimposed in registration with the specimen grating; let the reference grating be oriented with its lines perpendicular to the y-axis, so that it interacts with the corresponding set of lines in the cross-line specimen grating. The observer sees a null field, a field devoid of moiré fringes, corresponding to the zero-load (zerodeformation) condition of the specimen. When tensile loads are applied, the specimen deforms, and the cross-line specimen grating deforms with it. Now, interaction of the specimen and reference grat-

ing produces the moiré pattern of Fig. 21.9c, with fringe orders N y . The moiré pattern of Fig. 21.9d is formed by rotating the linear reference grating 90◦ , to interact with the orthogonal set of specimen grating lines; the fringe orders are designated N x . As before, each pattern represents displacement components perpendicular to the lines of the reference grating. The displacements are 1 U(x, y) = gN x (x, y) = N x (x, y) f 1 (21.8) V (x, y) = gN y (x, y) = N y (x, y) , f where the fringe orders are taken at the corresponding x, y points and where g and f represent the reference grating. However, the fringe orders numbered in Fig. 21.9 are not absolute fringe orders. Instead, they were numbered by first assigning a zero-order fringe at a convenient location where the displacement was assumed to be zero. Thus, the moiré fringes in the figure represent displacements relative to an arbitrary zerodisplacement datum. The free choice of a zero datum is equivalent to adding a freely chosen rigid-body translation to the absolute displacements; this approach is permissible because rigid-body motions have no importance for deformation studies. Only relative displacements are needed. Accordingly, the normal procedure utilizes (21.8) together with an arbitrarily chosen zero-fringe-order location. The in-plane displacements of every x, y specimen point are determined from the fringe orders at the point, and the displacements are relative to the chosen zero-displacement datum. The remaining question is how the nonzero fringe orders are assigned. They must change systematically as a parametric family of curves. In many cases, the question of whether the order of an adjacent fringe changes by +1 or −1 (i. e., whether the fringe orders are increasing or decreasing) is answered by the physics of the problem. For example, the specimen here is stretched in the y-direction and therefore the relative displacements and fringe orders increase as y increases. In complex problems where the fringe ordering is not obvious, the question is answered by a simple experimental procedure described in [21.1]. Figure 21.9e shows a graph of fringe orders along the line A A , where the data points are taken at the centerlines of integral and half fringe orders. The graph emphasizes the fact that a fringe order exists at every point on the specimen. The intermediate fringe order of any x, y point can be estimated by visual interpola-

609

Part C 21.2

reference grating. Let light be emitted from every point in the spaces of the specimen grating. The two gratings are originally in registration, so none of the light emitted from the specimen grating is obstructed by the reference grating. The observer sees a uniform (averaged) bright field and N = 0 everywhere. If the reference grating is fixed, but the specimen experiences a rigid-body translation δ, the light emerging from the pair experiences N cycles of brightness/darkness. In this case, the instantaneous intensity is the same everywhere in the field; the number of wholefield intensity oscillations is given by (21.7). N is the moiré fringe order, and here it represents the absolute or total U displacement from the original position. If the specimen and specimen grating deform, the displacement δ will vary from point to point in the field and (unless cracks are produced) the U displacements will vary as a continuous function of the x, y coordinates of the body. Point P cited earlier represents any and all light-emitting points in the specimen and (21.7) applies for every point. Accordingly, N varies as a continuous function throughout the field. The absolute fringe order is determined by making the fringe count while the body is being deformed. Of course, it is usually impractical to count the fringes while the body is being loaded or deformed. Fortunately, only relative displacements are needed for deformation analyses, and these can be determined without recourse to counting fringes during the loading operation. The following example illustrates the experimental analysis.

21.2 In-Plane Displacements

610

Part C

Noncontact Methods

b)

c)

A'

3

2

d)

2.67

2.33 1

Ny

0.67

0.33

0

A

a)

e)

y (mm) 40

Part C 21.2

20 –2

Ny

–1

δ

1

2

–20 – 40

Fig. 21.10 (a) Fringe shifting is accomplished by translating the reference grating by δ. In (b)–(d) the shifts are δ = 0, g/3, and 2g/3, respectively. (e) Fringe order along the line A A

tion, or else it can be determined by interpolation from a graph such as that in Fig. 21.9e.

21.2.4 Fringe Shifting The fringes can be shifted and still represent the same displacement field. This is accomplished by shifting or moving the reference grating relative to the specimen grating. A shift of δ (perpendicular to the grating lines) causes every fringe to move; the change of fringe order at every point in the field is ΔN =

δ = fδ , g

(21.9)

where g and f are the pitch and frequency of the reference grating, respectively. Fringe shifting is demonstrated in Fig. 21.10, in which the mechanical shifts are δ = 0, g/3, and 2g/3, which produce fringe shifts of 0, 1/3, and 2/3 of a fringe order, respectively.

Fringe orders are marked on the patterns. In this case, fringe shifting provides three times the usual number of data points for plotting fringe orders along the line A A , or along any other line. The centers of dark fringes are marked by circles in the graph, but the centers of bright fringes can be marked, too, as in Fig. 21.9c. Again, the accuracy of each data point is not improved, but the increased number of data points provides a graph with statistically improved accuracy. Amplification by Moiré When two uniform gratings of slightly different pitches are superimposed, they form moiré fringes governed by (21.5) and (21.6). Two such gratings and their superposition are shown in Fig. 21.11. The moiré pattern in Fig. 21.11c represents the initial condition, before the grating of pitch g1 is moved horizontally by a distance δ. When it is moved by some integral multiple

Geometric Moiré

of δ, (δ = g1 , 2g1 , 3g1 , . . .), the resulting fringe pattern appears the same as in Fig. 21.11c. When δ = g1 /4, the moiré fringes move to new positions, displaced by Δ = G/4 from their original positions, as shown in Fig. 21.11d; in Fig. 21.11e, Δ = G/2. Generalizing   G (21.10) , Δ=δ g1

a) g2 Fixed

b)

g1 Moveable

c)

G δ=0

d)

Δ = G/4

e)

δ=

g1 4

δ=

g1 2

Δ = G/2

Fig. 21.11a–e Amplification: the fringe movement Δ is G/g1 times the grating movement δ. As an instructive exercise, the reader can make transparencies of (a) and (b) on a copying machine and manipulate the superimposed gratings

cordingly, a second photodiode is positioned at a point G/4 away from the first. A logic circuit interprets the two signals and outputs the correct position of the table.

21.3 Out-Of-Plane Displacements: Shadow Moiré Two techniques for measuring out-of-plane displacements, W, by geometric moiré are described. They are the shadow moiré and projection moiré methods. Both measure W uniquely. Although U and V may also be present, the methods sense and measure only W. Both methods determine the topography of the specimen surface, i. e., its deviation from a plane surface. In mechanics, they are used to measure the topography of initially flat surfaces to evaluate warpage caused by load, temperature, humidity, age, or other variables. In metrology, they are used to document the shape of three-dimensional bodies.

611

21.3.1 Shadow Moiré The technique called shadow moiré has evolved as the moiré method most widely chosen for out-of-plane measurements. Although its sensitivity is lower, engineering applications seldom require the subwavelength sensitivity of interferometric methods for out-of-plane measurements. Instead, the low sensitivity, plus the recently achieved intermediate sensitivity of shadow moiré, fit nearly all the practical applications. The most popular and practical shadow moiré arrangement is introduced here. Additional information

Part C 21.3

where G/g1 is an amplification factor. The fringe movement Δ amplifies the grating movement δ by this factor. When g1 < g2 , the fringes move in the same direction as the moveable grating. When g1 > g2 , the directions are opposite. This principle is widely used for the measurement and control of movements of machine tools. On a milling machine, for example, a moiré device can be used to measure (and control) the position of the workpiece relative to the cutting tool. A long grating, called a scale grating, is attached to the moveable table of the machine and a short reference grating is attached to the stationary base of the machine. A photodiode is used to sense the intensity of moiré fringes that pass a stationary point as the scale grating moves by. The photodiode produces oscillations of voltage in harmony with the oscillations of the moiré fringe intensity. The oscillations are counted electronically, and the fringe count, multiplied by the pitch of the scale grating (e.g., 1/50 mm), specifies the position of the table. A single sensor cannot discriminate the direction of movement, i. e., whether δ is positive or negative. Ac-

21.3 Out-Of-Plane Displacements: Shadow Moiré

612

Part C

Noncontact Methods

is given in [21.1, Chapter 3 and Appendix A], which provides a more general treatment of shadow moiré. The competing method of projection moiré is also addressed in this reference, although that technique is usually avoided because of its stringent requirements on optical resolution.

Part C 21.3

Basic Concept Figure 21.12 illustrates the principle of shadow moiré. The specimen surface is usually prepared by spraying it with a thin film of matte white paint. A linear reference grating of pitch g is fixed adjacent to the surface. This reference grating is comprised of black bars and clear spaces on a flat glass plate. A light source illuminates the grating and specimen, and the observer (or camera) receives the light that is scattered in its direction by the matte specimen. When the gap W is small, a well-defined shadow of the reference grating is cast upon the specimen surface. This shadow, consisting of dark bars and bright bars, is itself a grating. The observer sees the shadow grating and the superimposed reference grating, which interact to form a moiré pattern. In the cross-sectional view of Fig. 21.12, light that strikes points A, C, and E on the specimen passes through clear spaces in the reference grating to reach the observer. Light that strikes points B and D are obstructed by opaque bars of the reference grating. Thus, bright moiré fringes are formed in the regions near

points A, C, and E, while black fringes are formed near B and D. Of course, many more bars appear between A and B in an actual system, so the transition between bright and dark moiré fringes is gradual. With the oblique illumination, the positions of the shadow on the specimen depend upon the gap W, so the moiré pattern is a function of W. Let the fringe order at A be N z = 0, since the specimen and reference grating are in contact and W = 0. From the viewpoint of the observer, the number of shadow grating lines between A and C is compared to the number of reference grating lines between A and C’. The number of shadow grating lines is larger by one, and therefore the fringe order at C is N z = 1. Similarly at E, the number of shadow grating lines is larger by two and the moiré fringe order at E is N z = 2. a)

x

L

W Ψ

α

D

x Reference grating

b)

Observer

Video camera

Ψ

A

α

L

B C

E C'

Ψ W α

Nz g

D Nz g = W tan α +W tan Ψ g W= N tan α + tan Ψ z

E

D

g W

z Specimen surface

Source

Fig. 21.12 Principle of the shadow moiré method

Fig. 21.13a,b Shadow moiré arrangements in which tan α +

tan Ψ = constant

Geometric Moiré

a)

α

613

Fig. 21.14a,b Shadow moiré with collimated beams

b)

Ψ

21.3 Out-Of-Plane Displacements: Shadow Moiré

α

Fringe Visibility The moiré pattern formed by the reference grating and its shadow has the character illustrated in Fig. 21.2. The bright fringes are crossed with the lines of the reference grating. These extraneous lines can be made to disappear by a few different methods.

In an elegant method [21.6, 7], the reference grating is in motion during the time interval of the photographic exposure. The grating is translated in its plane, perpendicular to the direction of the grating lines. At points of destructive interference, as at point D of Fig. 21.12, either the incoming or outgoing ray is always obscured by the reference grating; destructive interference persists at D even as the grating translates. At points of constructive interference such as point E, light passes through the reference grating spaces during half of the exposure time and it is obstructed by the grating bars during the other half. Thus, translation of the reference grating has no influence on the positions of the moiré fringes, but the grating bars are smeared and become invisible because of their motion. Referring to Fig. 21.2b, the triangular intensity distribution is recorded even when the pattern is viewed with a high-resolution optical system. Averaging produced by reduced resolution is not needed. An equivalent scheme is to record the pattern with a double exposure, wherein the reference grating is translated by half its pitch (by g/2) between exposures. This method assumes the usual type of reference grating, with equal widths of bars and spaces. When a video camera is used, the reference grating lines disappear when the camera magnification is adjusted to provide a whole number of grating lines in the active width of the pixel. The intensity across each pixel is averaged, thus eliminating the grating lines. Of course, the averaging method already mentioned in connection with in-plane measurements is effective here, too. The camera aperture can be decreased to prevent resolution of the reference grating lines, while the coarser moiré fringes remain well resolved. A useful technique is to use a slit aperture in front of the camera lens (or inside it), with the length of the slit parallel

Part C 21.3

The relationship between the fringe order and the gap W is derived in the insert, with the result g (21.11) Nz . W= tan α + tan ψ Notice that the angles α and ψ are variables that change with the coordinate x. Therefore, the sensitivity of the measurement is not constant, i. e., the displacements are not directly proportional to fringe orders of the moiré pattern. However, if the light source and observer are at the same distance L from the specimen, and if D is the distance between the light source and observer, the variables reduce to D (21.12) =K, tan α + tan ψ = L where K is a constant, and g W = Nz . (21.13) K Thus, W is directly proportional to the moiré fringe order. Consistent with this, the configuration of Fig. 21.13b is the usual arrangement of choice. In addition to constant sensitivity across the field, it has the advantage that the specimen is viewed at normal incidence, thus avoiding geometrical distortions. When D = L, the contour interval of the moiré fringe pattern (i. e., displacement W per fringe order) is equal to the pitch of the reference grating. The collimated light arrangements of Fig. 21.14 also provide constant sensitivity. They are practical for small fields of view.

614

Part C

Noncontact Methods

to the reference grating lines. With a slit, more light is admitted into the camera, and the resolution in the direction parallel to the grating lines remains high.

Part C 21.3

Demonstration The post-buckling behavior of an I-shaped laminated graphite/epoxy column was investigated using shadow moiré [21.2]. The purpose of the experiment was to verify a theoretical model and to obtain experimental insight into the post-buckling behavior of such structures. The out-of-plane deflections of both the web and the flange were measured as a function of load, and the experimental data were used to guide and validate numerical modeling. Since the maximum out-of-plane displacement was expected to be quite high (approximately 6 mm), relatively low-frequency linear gratings called grilles were used. The reference grilles were made by printing an image created with drawing software onto acetate transparency material, using a 600 dpi (dots per inch) laser printer. The acetate material was bonded to a rigid transparent acrylic plate with acetone. Since it was desired to measure the out-of-plane deflections on both the web and the flange of the column simultaneously, two sets of reference grilles were produced. Deflections were monitored throughout the entire test, up to failure. Since the range of deflections was high and accuracy at each load was required, two grille frequencies were a)

used. A frequency of 3.94 lines/mm (100 lines/in) was used at the early stages of loading, and 1.97 lines/mm (50 lines/in) at later stages. The experimental setup is illustrated in Fig. 21.15a. The grilles were mounted on adjustable bases so that their positions relative to the specimen could be adjusted easily. Mirrors were used to direct the light to the reference grilles. This was necessary because the loading machine and fixtures would have blocked a large portion of the specimen if direct illumination had been used. Figure 21.15a shows the mirror–grille assemblies separated from the specimen for illustration purposes. During testing, both grilles were moved to nearly contact the specimen. Two clear glass incandescent light bulbs, mounted on tripods, were used for the light sources. The angle of illumination was 45◦ , and the viewing angle was normal to the specimen surface. Both the light source and camera position were located at the same perpendicular distance from the specimen surface, making the optical setup equivalent to that in Fig. 21.13. The specimen was sprayed with flat white paint to provide a matte surface. Images were recorded by two 35 mm cameras using technical pan film. Figure 21.15b illustrates examples of the shadow moiré fringe patterns on the web and flange of the specimen. Nearly the entire web and flange were observed. From these patterns, the out-of-plane deflections were determined as a function of load. b)

Load

Moiré grille

Moiré grille Light source

Light source 45° 45°

To camera

To camera

W displacement fields Contour interval 0.254 mm (0.01 in.) 1 cm

I-shaped column Adjustable base

Adjustable base

Web

Flange

Fig. 21.15 (a) Experimental arrangement for post-buckling deformation of column and (b) shadow moiré patterns for web and flange. (Courtesy of Prof. Peter G. Ifju)

Geometric Moiré

21.3 Out-Of-Plane Displacements: Shadow Moiré

21.3.2 Projection Moiré

W=

gN z , tan α

where N z is the fringe order at B. The relationship is the same as that for shadow moiré. In fact, when the optical configuration is generalized to that of Fig. 21.17, the patterns of projection moiré and shadow moiré are identical. The gratings projected onto the original and deformed specimen surfaces are the same as the reference grating and its shadow in the corresponding shadow moiré system. The equations are identical; (21.11) and (21.13) apply for projection moiré when collimated light is used. The projector of Fig. 21.17 uses a fine grating to project a comparatively coarse grating. An alternative system for creating the projected grating is sketched in Fig. 21.17b. The input laser beam is divided into two beams which appear to diverge from two adjacent points in space. Their separation is adjusted by the inclination of the mirrors. After passing the lens, two collimated beams intersect at an angle 2θ to produce a projected grating of pitch g . The frequency F of this projected grating is determined by F=

2 sin θ . λ

g

B

α

tan α = C

W

b gNz

gNz W

a

c

B

b

A

α

a W

Deformed specimen

Camera g'

g' = g cos α

Original position

Fig. 21.16 Schematic of double-exposure projection moiré for deformation measurements

In practice, θ is very small. The assembly of Fig. 21.17b and its use for creating two intersecting beams is well known; the arrangement is sometimes called a Michelson interferometer. In principle, projected gratings of excellent quality can be produced, but in practice speckle noise associated with coherent illumination of a matte specimen surface causes some deterioration. a) W

Ψ α

Camera

Projected grating

Projector

Original surface Deformed surface

b)

2θ Two divergent beams

Beam splitter and mirrors Laser beam and spatial filter

Fig. 21.17 (a) Projection moiré using a slide projector. (b) Alternative means to produce the projected grating by interference of light; this assembly is used instead of the projector

Part C 21.3

One Projector: Deformations Variations of projection moiré will be considered. The first is illustrated in Fig. 21.16, where a specimen (with a matte white surface) is illuminated by an array of uniformly spaced walls of light. Let the shaded bars represent the presence of light and the blank spaces represent the absence of light. The camera sees bright bars on the surface of the specimen, separated by dark spaces from which no light is emitted. The illuminated specimen is photographed first in its original condition; later it is photographed on another film in its deformed condition, after loads are applied. After development, the two films are bar-and-space gratings, and they create a moiré pattern when they are superimposed. By inspection of Fig. 21.16, it is seen that the band of light that intercepted point b when the specimen was in its original condition is three pitches removed from the light that strikes B, which is the same point after deformation. Thus, the moiré fringe order at B is 3. Similarly, it is 2.5 at A and 4 at C. From the triangle a b B, we find that the out-of-plane displacement at B is

z

615

616

Part C

Noncontact Methods

Ground-glass screen

Ψ

Camera

α Camera lens A

Grating

Projector

Part C 21.3

Fig. 21.18 Projection moiré for surface topography

One Projector: Topography When three-dimensional shapes must be documented, rather than changes of shape, the advantage of beforeand-after grating images is lost. It is not possible to superimpose images that are recorded before and after the changes. Then, the scheme of Fig. 21.18 can be applied. A real bar-and-space grating is laminated onto the ground-glass screen in the observation leg. Cama)

b)

ace bdf

g α α

A

c)

a

g

c A

α

A=

g/2 tan α

Fig. 21.19 (a) Two-projector method. (b) The geometry of intersecting walls of light. (c) The geometry within one pitch of the projected specimen grating

era lens A is focused to image the specimen surface on the ground-glass screen. Thus, an image of the distorted grating from the curved surface is superimposed upon the real grating to form a moiré. The pitch and alignment of the real grating corresponds to the image that would be formed by placing a flat reference plane in the specimen space. Therefore, the moiré pattern is a contour map of the separation W between the specimen surface and the flat reference plane. The moiré pattern is recorded by a camera located behind the screen. The result is multiplicative moiré, generally characterized by fringes of good visibility. However, the distorted grating lines on the specimen must be resolved sharply by the camera lens A, and this means that very fine gratings cannot be used. When deeply warped surfaces are observed, the frequency of moiré fringes can become similar to the frequency of the grating lines, thus reducing the fringe visibility. As with shadow moiré, an elegant technique to wash out the grating lines uses moving gratings [21.8]. The technique can produce moiré fringes of excellent visibility. Reference [21.8] also cites numerous publications on projection moiré. Two-Projector Moiré Another variation is illustrated in Fig. 21.19. In this case two sets of parallel walls of light are incident from symmetrical angles and cross in space as depicted in Fig. 21.19b. The specimen intercepts the light, which casts two sets of bright bars on the matte specimen surface. They interact to produce a moiré pattern of additive intensity. Let the shaded walls represent the presence of light. The patterns of light received locally on the deformed specimen surface depends upon where the array of Fig. 21.19b is cut by the specimen surface. If some region of the specimen surface is represented by line a, that region is illuminated uniformly; in regions where the specimen surface coincides with lines b, d, and f, the surface is illuminated by alternating bright bars and dark spaces. The pattern on the specimen is characterized by moiré fringes of additive intensities, as depicted in Fig. 21.5a. To record the moiré fringes in the camera, the individual grating lines must be coarse enough to be individually resolved. The out-of-plane displacement of points on one moiré fringe relative to the neighboring fringe is A, where A is the distance between planes a and c in Fig. 21.19b. From the triangle sketched in Fig. 21.19c, A = g/(2 tan α). The out-of-plane displace-

Geometric Moiré

ment of any point is W = AN z and therefore g W(x, y) = N z (x, y) , 2 tan α

(21.14)

where W is relative to an arbitrary datum where the moiré fringe order is assigned N z = 0. Note that the relationship is the same as that for shadow moiré when |Ψ | = α. This method has the advantage that the specimen can always be viewed at normal incidence. It has the disadvantages of additive intensity moiré, which requires resolution of the individual grating lines. Reference [21.8] provides means to transform the pattern to multiplicative moiré fringes

21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT)

617

and to wash out the grating lines; however, resolution of the individual grating lines is still required in the observation leg.

21.3.3 Comparison Compared to shadow moiré, projection moiré has the advantage that no element of the apparatus is required to be close to the specimen. It has the disadvantage that the optical systems are more complicated, and the individual lines of the projected gratings must be resolved in the observation leg. With shadow moiré, only the moiré fringes must be resolved.

21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT) 21.4.1 The Talbot Effect

α

To camera

2g2 (21.15) cos3 α λ when the grating is illuminated at incident angle α by a monochromatic collimated beam of wavelength λ. In the conventional practice of shadow moiré using coarse gratings, DTα is relatively large and the specimen can be positioned within a small fraction of DTα /2 from the reference grating. Within that region, the assumption of rectilinear propagation of light is reasonable, and the virtual gratings and the corresponding shadow moiré fringes have sufficient contrast. In nonconventional practice, shadow moiré has recently been adopted in the electronics industry to measure coplanarity, i. e., the warpage of microelectronic devices [21.11, 12], where measurement resolution as small as 5 μm can be required. Although the desired resolution is possible in theory by employing a coarse reference grating and the phase-shifting technique, the measurement accuracy is affected by the nonsinusoidal DTα =

Reference grating D αT /2

α x g

z

D αT /2 D αT /2

δx = z tan α Virtual gratings

Fig. 21.20 Illustration of virtual gratings of optimum contrast formed at preferred distances; DTα is the Talbot distance. Note: an alternative definition of the Talbot effect was used in [21.1] whereby the distance between successive virtual gratings is DT instead of DT /2

Part C 21.4

Equation (21.11) implies that shadow moiré fringes are formed for any value of z. In fact, the contrast of the moiré fringe pattern varies with z, such that the pattern disappears and reappears cyclically as z increases. This is caused by the phenomenon known as the Talbot effect, also known as the grating self-imaging effect [21.9]. Light diffracted in multiple orders by the real reference grating recombines to form a series of vir-

tual images of the grating in space, as illustrated in Fig. 21.20. These virtual grating images have optimum contrast at preferred planes which lie at successive distances m DTα /2 (m = 0, 1, 2, 3, . . .) from the reference grating. Virtual gratings of lower contrast are visible near these preferred planes, with their contrast gradually diminishing to zero midway between the preferred planes. To obtain a clear image of the grating on the specimen surface, the gap between the grating and the specimen must be restricted. The specimen must lie near a preferred plane. The Talbot distance for oblique illumination, DTα , is [21.10]

618

Part C

Noncontact Methods

intensity distribution of the shadow moiré fringes and the desired displacement resolution will be difficult to achieve in practice [21.3]. Therefore, the basic measurement sensitivity must be increased to attain the desired resolution. For configurations of relatively high basic measurement sensitivity where the fine gratings are used, DTα /2 is much smaller. Then, it can be impractical to position the specimen within a small fraction of DTα /2 from the reference grating. This condition is further exacerbated when the method is employed to document thermally or mechanically induced warpage. For such applications, it can be most practical to position the specimen at multiples of half the Talbot distance. The following sections treat the conditions encountered when shadow moiré is used with relatively fine gratings. They summarize the results of a comprehensive theoretical and experimental study reported in [21.3, 13–15].

Part C 21.4

21.4.2 Fringe Contrast Versus Gap

a) Contrast

The moiré image seen in the camera (Fig. 21.13b) depends critically upon two factors; one is called the aperture effect, illustrated in Fig. 21.21, which shows that the lateral shift of the shadow grating is half the pitch of the reference grating, which is supposed to produce a dark fringe. Light scattered by the specimen is completely blocked by the reference grating for a pinhole aperture in the camera (Fig. 21.21a), but only partially blocked for a large aperture (Fig. 21.21b). Thus, the contrast of moiré fringes diminishes as the a)

b)

d

diameter of aperture d increases and as z increases. The effective aperture, de = d/L, and the grating pitch g are the pertinent parameters that control fringe contrast versus gap z. The relationship for circular apertures is graphed in Fig. 21.22 for two values of de . The effect is a linear decrease of contrast as z increases, with a rapid loss for relatively large apertures. The second factor is called the Talbot effect, or the diffraction effect. The incident beam in Fig. 21.20 is diffracted by the reference grating into numerous subsidiary beams, each propagating in slightly different directions. Mathematical summation of the electric field strengths of the subsidiary beams as they propagate in space predicts the Talbot effect. Specifically, it predicts the existence of virtual gratings in the space beyond the real grating. Experimental corroboration is seen in Fig. 21.23 [21.13]. The specimen was a 12 mm square flat glass plate coated with white paint, mounted on a microm-

zw

1 Experiment Talbot effect Aperture effect Combined effect

0.8 0.6 0.4 0.2 0

0

0.25

0.5

0.75

1 z/D αT

b) Contrast

d

1

Aperture stop

Talbot effect Aperture effect Combined effect

0.8 0.6 L

α

α

0.4 0.2

Reference grating

δx

g

0 z

Shadow grating

Fig. 21.21a,b Illustration of the effect of aperture on the contrast of shadow moiré fringes: (a) pinhole aperture and (b) finite aperture

0

0.25

0.5

0.75

1 z/D αT

Fig. 21.22a,b Fringe contrast showing the added influence of the aperture effect. The optical parameters are α = 63.4◦ , λ = 661 nm, g = 0.2 mm, 100 μm/fringe, and (a) de = 0.0245; (b) de = 0.005

Geometric Moiré

a)

2L Aperture stop

d

Light source

L Reference grating g = 0.2 mm

63°

z

21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT)

619

The moiré fringes are formed by the interaction of two gratings: the virtual grating on the specimen surface and the reference grating. The cyclic variation of contrast of the virtual grating is inferred by the cyclic variation of contrast of the moiré fringes. With contrast at each point defined by CT =

I max − I min , I max + I min

(21.16)

1°

Specimen surface Micrometer stage

c)

d)

e)

f)

g)

Fig. 21.23 (a) Experimental setup to measure the contrast of shadow moiré fringes, and fringe patterns obtained at z = (b) 0, (c) 1/8DTα , (d) 1/4DTα , (e) 3/8DTα , (f) 1/2DTα , and (g) 5/8DTα . de = 0.0245; λ = 661 nm; contour interval = 100 μm/fringe

eter stage. The specimen was tilted by approximately 1◦ , which produced two fringes on the specimen. Numerous fringe patterns were recorded, each for slightly different values of z. Representative fringe patterns obtained at z = 0, 1/8DTα , 1/4DTα , 3/8DTα , 1/2DTα , and 5/8DTα are shown in Fig. 21.23, where the white curves represent the intensity distribution averaged along lines perpendicular to the fringes. The figure shows the influence of both the Talbot effect and the aperture effect. The light source had essentially zero width. Since a diverging beam was used in the setup, the illumination angle varied across the specimen. However, with the source to specimen distance of 570 mm, the illumination angle varied only 0.3◦ across the one-fringe interval, which had a negligible effect on the Talbot distance calculation.

21.4.3 Dynamic Range and Talbot Distance The dynamic range of an instrument is its useful range of measurement, in this case, the range of out-of-plane measurements, W. For high-sensitivity shadow moiré where the Talbot distance is small, the contrast of fringes diminishes rapidly and thus the dynamic range of measurement is limited. Figure 21.24 shows an example [21.14]. This optical setup for high-sensitivity shadow moiré provides a contour interval of 50 μm/fringe; here g = 0.1 mm, α = 63◦ , and de = 0.01. The Talbot distance is 3.25 mm. An optical flat was tilted to produce a linearly varying displacement field. The combined contrast function is plotted in (Fig. 21.24b) and the fringe pattern obtained from the setup is shown in (Fig. 21.24c). Due to the Talbot effect, the fringe contrast reduces abruptly and becomes zero at quarter the Talbot distance. The fringe contrast increases again as z reaches half the Talbot distance.

Part C 21.4

b)

(where the intensities, I , refer to the maximum and minimum intensities of the moiré fringes) the theoretical and experimental results could be compared. The dotted curves in Fig. 21.22 show the theoretical results for the Talbot effect alone, i. e., the diffraction effect, assuming a pinhole camera aperture. It predicts maximum fringe contrast at multiples of half the Talbot distance, and zero contrast at intermediate quarter distances. The solid curves show theoretical results for the combined Talbot effect and aperture effect. The contrasts calculated from the experimental data are plotted in Fig. 21.22 together with the theoretical predictions; the solid circles represent the contrasts obtained from the fringe patterns shown in Fig. 21.23, and the open circles are from additional fringe patterns recorded in the same experiment. The experimental results corroborate the validity of the theoretical analysis. Obviously greater fringe contrast will be achieved with smaller effective apertures. Note, too, that the maximum local fringe contrast occurs when the gap is slightly less than DTα /2.

620

Part C

Noncontact Methods

Note that the use of a broad spectrum (e.g., white light) has a beneficial effect [21.14, 15]. Compared to Fig. 21.22, the contrast function becomes relatively smooth. Criteria are required to generalize the parameters for a practical dynamic range. The criterion accepted here for contrast requires that the dynamic range cannot exceed one-quarter of the Talbot distance; this is marked in Fig. 21.24b as DRmax . Furthermore, a dynamic range of ten contour intervals (ten fringes) is considered ideal, so ΔN = 10 is accepted as an additional criterion; this is marked as DR on the graph. Although the contrast, or visibility, at which the fringes cease to be useful is subjective, these are practical criteria for usual applications. a)

White light source

Aperture stop

d

Part C 21.4

d /L = 0.01 63°

Reference grating g = 0.1 mm Glass, painted white D Rmax

w = 1 mm

b)

Γ=

0.8 0.6 0.4

Zero Talbot area

0.2 0

0

DTα =

Half Talbot area 0.5

1

1.5

2

2.5 z (mm)

c) Zero Talbot area

Γ2 sin 2α sin α . λ

(21.17)

Here, Γ is the contour interval (displacement per fringe), extracted from (21.11) for normal viewing, i. e.,

DR

Contrast 1

21.4.4 Parameters for High-Sensitivity Measurements For high-sensitivity measurements, the example of Fig. 21.24 illustrates that a larger Talbot distance is the key to achieving a larger dynamic range. An informative relationship is derived by combining (21.15) and (21.11), which gives

L

z

In practice, a small gap between the specimen and reference grating should be provided to prevent accidental contact. A minimum gap, w, of 1 mm is assumed as a further criterion. Again, the specimen is usually viewed at normal incidence to avoid distortion of its image; observation at normal incidence is an additional criterion. Implicit, too, is that the reference grating is comprised of bars and spaces of equal widths, and the camera aperture is circular. For the high-sensitivity arrangement of Fig. 21.24, when subject to these conditions, the specimen cannot lie in the zero Talbot area. However, effective measurements can be made in the half Talbot area, where both the graph and the fringe pattern show that the contrast and dynamic range are fully adequate.

Half Talbot area

Fig. 21.24a–c Demonstration of the dynamic range in half the Talbot distance; (a) shadow moiré setup, (b) theoretical fringe contrast, and (c) the fringe pattern representing linearly varying displacements

g . tan α

(21.18)

Note that the contour interval is the inverse of the sensitivity (fringes/unit displacement). Results are graphed in Fig. 21.25 for contour intervals of 100 μm/fringe and 50 μm/fringe. The analysis proves that the maximum Talbot distance is always achieved when the angle of incidence is α = 54.7◦ , and that the maximum Talbot distance is determined by substituting this angle into (21.17). When designing an experiment to achieve a desired sensitivity, the pitch of the reference grating and the angle of illumination can be chosen to provide a large Talbot distance. The curves of Fig. 21.25 raise an important question. For a desired sensitivity, the same Talbot distance can be achieved with two different experimental arrangements. As an example, α = 45◦ with g = 0.05 mm and α = 63.4◦ with g = 0.10 mm both provide (essentially) the same Talbot distance and the same contour interval. Which choice is superior? The answer is that

Geometric Moiré

a) Talbot distance (mm)

Pitch of grating (mm) 0.6

6

Talbot distance Maximum Pitch of grating

5

0.5

4

0.4

3

0.3

2

0.2

1

0.1

0

0

10

20

30

40

50

b) Talbot distance (mm) 20

0 70 80 90 Alpha (degree) Pitch of grating (mm) 1

60

Talbot distance Maximum Pitch of grating

16

0.8 0.6

8

0.4

4

0.2

0

0

10

20

30

40

50

60

0 70 80 90 Alpha (degree)

Fig. 21.25a,b Maximum Talbot distance and grating pitch for contour intervals of (a) 100 μm/fringe and (b) 50 μm/fringe. λ = 661 nm

contrast is achieved with the larger reference grating pitch. Figure 21.26 illustrates the theoretical results in a form that is especially useful to guide the design of high-sensitivity measurements. The dashed curve represents (21.17) (with α = 54.7), establishing the maximum Talbot distance for contour interval, i. e., for any sensitivity. The dot-dashed curve represents the Talbot distance that allows the specimen to be located in the zero Talbot area; the experimental conditions assume the aforementioned criteria plus one more: that the sum of w plus 10Γ is less than one-eighth of the Talbot distance. (As an example, in Fig. 21.24b the one-eighth Talbot distance is at z = 0.41 mm.) The solid curve represents the Talbot distance that is required to satisfy the stated criteria when the specimen is located in the half Talbot distance area. These curves are drawn for a wavelength of λ = 661 nm. The criteria are satisfied for contour intervals where the required Talbot distance is less than the maximum Talbot distance. Thus, the conventional shadow moiré arrangement, with the specimen located in the zero Talbot area, can be used for contour intervals exceeding 110 μm/fringe. The nonzero arrangement, with the specimen in the half Talbot area, can be used for contour intervals exceeding 43 μm/fringe. Smaller contour intervals cannot satisfy these practical criteria. Additional provisions to guide the optimization of shadow moiré measurements can be found in [21.14].

21.4.5 Implementation of SM-NT the choice with the larger grating pitch is superior. The reason is higher fringe contrast, related to the aperture effect (see Fig. 21.22). For any value of z, the contrast varies with the grating pitch as CT = 1 − k/g, where k is a constant (k = 3de z/3π) [21.3]. Thus, greater

In this example, the implementation of SM-NT to measure the warpage of an electronic package is described. The specimen is a four flip-chip plastic ball grid array (FC-PBGA) package (Fig. 21.27). The size of the

Talbot distance (mm) 60 50 40

Non-practical area

Shadow Moiré using non-zero Talbot area

Conventional shadow Moiré

30

Maximum Talbot distance

Required Talbot distance when zero Talbot area is utilized

20 Required Talbot distance when half Talbot area is utilized

10 0

0

50

100 150 200 Contour interval (μm/fringe)

Fig. 21.26 Graphical representation of the sensitivity of shadow moiré, utilizing the nonzero Talbot distance. λ = 661 nm

621

Part C 21.4

12

21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT)

622

Part C

Noncontact Methods

a)

b)

60 mm

54 mm

Fig. 21.27a,b Four-chip FC-PBGA package: (a) top view and (b) bottom view

a)

b)

Part C 21.4 Fig. 21.28a,b Shadow fringes of the back surface of a four-chip FC-PBGA package documented (a) at the zero Talbot area and (b) the half Talbot area; g = 0.1 mm and α = 63◦ , de = 0.01, Γ = 50 μm/fringe. (Courtesy of Dr. C.-W. Han)

a)

b) 0.7 mm

Chip

37 mm

Substrate

Fig. 21.29a,b Single-chip FC-PBGA package: (a) top view and (b) side view

package is 54 mm × 60 mm. The warpage is attributed to a large mismatch of the coefficients of thermal expansion (CTE) of the chip and substrate. In the sub-

sequent assembly process, electrical and mechanical connections are made by solder balls between the substrate and a printed circuit board. If the bottom side of the substrate warps significantly at the solder reflow temperature, it yields an uneven height of solder interconnections, which could cause premature failure of the assembly. Detailed knowledge of the out-of-plane deformation is essential to optimize the design and process parameters for reliable assemblies. The experimental arrangement of Fig. 21.24a, using a white-light source, was employed to measure the warpage of the back surface of the package. The Talbot distance was approximately 3.25 mm. The warpage of the specimen was observed at the zero Talbot area as well as the half Talbot area. The fringe pattern obtained at the zero Talbot area is shown in Fig. 21.28a. Note that the fringes near the quarter Talbot distance wash out completely. The corresponding results obtained at the half Talbot area are shown in Fig. 21.28b. The maximum deflection between the center and the corners of the package is approximately 0.4 mm, which produces eight fringes. The SM-NT pattern documents the deformations faithfully, as evidenced by the excellent visibility of individual fringes. It is to be noted that the small circles on the images are not caused by optical noise; instead, they represent circular copper pads on the substrate where solder connections will be made in subsequent operations. The pads are about 25 μm thick, which is recognized by local changes of about half a fringe order. Warpage Measurement for Non-coplanar Surfaces Another FC-PBGA package is shown in Fig. 21.29. In the package, a single chip is attached to an organic substrate. The size of the package is 37 mm × 37 mm and the height of the chip is 0.7 mm. For high-performance FC-PBGA packages, a heat sink is usually required to dissipate the excess heat. Consequently, the warpage of the chip and substrate becomes an important design parameter for an optimum thermal solution, especially when interstitial materials between the heat sink and the silicon are employed [21.12]. Figure 21.30a illustrates that the deformation of both surfaces cannot be revealed in the zero Talbot area when the contour interval is 50 μm/fringe. The optical arrangement was the same as that in Fig. 21.24a. Figure 21.30a also shows the combined contrast function for the zero Talbot area. Superimposed in the figure is a cross-sectional view of part of the specimen, showing

Geometric Moiré

Fig. 21.30a–c Fringes and contrast function of shadow moiré (a) using the zero Talbot area for the setup of white light; g = 0.1 mm, tan α = 2, de = 0.01, Γ = 50 μm/fringe; (b) using both the zero Talbot area and the half Talbot area for the setup of white light; g = 0.1 mm, tan α = 2.5, de = 0.01, Γ = 40 μm/fringe; and (c) using the half Talbot area for the setup of white light; g = 0.1 mm, tan α = 2, de = 0.01, Γ = 50 μm/fringe. (Courtesy of Dr. C.-W. Han)

a)

its thickness and its position in the zero Talbot area; the large white circle represents a solder ball in the crosssectional view. The solid circles show the positions of the specimen surfaces relative to the contrast function. The substrate surface was near the zero contrast level, so fringes appear only on the chip surface. The demonstration violated the 1.0 mm safety gap between the specimen and reference grating. Of course, any finite gap is acceptable, but 1.0 mm is recommended for routine practice. Figure 21.30b demonstrates another option. The angle of incidence was adjusted until the substrate surface can be positioned near half the Talbot distance. Considering the physical gap of 0.7 mm, the incident angle was set at 68.2◦ (tan α = 2.5) and its effect on the Talbot distance is also illustrated in Fig. 21.30b. Note that the contour interval of the new configuration is 40 μm/fringe. It is important to note that the fringes of the chip surface were obtained at the zero Talbot area while the fringes of the substrate surface were obtained at the half Talbot area. Although it was implemented successfully for the specimen, this approach has a critical limitation. The optical setup will be extremely case sensitive; the angle of incidence and the grating pitch will have to be optimized and selected based on the gap between the two surfaces and, for some cases, a practical configuration may not be available.

b)

21.5 Increased Sensitivity

Contrast 1 0.8 0.6 0.4 0.2 0

0

0.5

1

1.5

2

2.5 3 z (mm)

1

1.5

2

2.5 3 z (mm)

1

1.5

2

2.5 3 z (mm)

Contrast 1 0.8 0.6 0.4 0.2 0

0.5

Contrast 1 0.8 0.6 0.4 0.2 0

0

0.5

A more attractive solution is to utilize the half Talbot area by taking advantage of the large dynamic range. This approach is illustrated in Fig. 21.30c, where the actual fringe pattern obtained from the optimized SM-NT setup is shown together with the combined contrast function. Good fringe contrast of the fringes on both surfaces is achieved.

tary image analysis techniques are used to provide additional data for determination of the deformation. In the following sections, two image processing techniques that are used to enhance sensitivity are presented. Shadow moiré experiments are used to illustrate the methods.

Part C 21.5

c)

0

21.5 Increased Sensitivity Geometric moiré applications shown in other sections of the Handbook illustrate examples in a wide range of out-of-plane displacements. In many present-day applications, especially shadow moiré analyses of small objects, the deformation is small and only a few moiré fringes are present. Consequently, supplemen-

623

624

Part C

Noncontact Methods

21.5.1 Optical/Digital Fringe Multiplication (O/DFM) The O/DFM method utilizes phase stepping of β equal steps to achieve fringe multiplication by a factor of β. In the case of shadow moiré, the reference grating is moved in the z-direction by small increments g/β K between recordings. According to (21.13), g/K is the displacement per fringe order. The center of every moiré fringe shifts by 1/β of a fringe order, and the phase φ shifts by 1/β of 2π at every point in the field. The charge-coupled device (CCD) camera records the intensity at every pixel in the field for the phase-shifted images. a)

I

I 0π = I0 shifted π

I0

Part C 21.5

0

b)

1

2

N

1

2

N

Ir π

I0 –I 0 0

c) |Ir |

π

|I0 – I 0 |

The fringe multiplication factor β is an even integer. For each of the β patterns, there is a complementary pattern, i. e., another pattern with its phase shifted by π at every point in the field. The O/DFM algorithm is depicted in Fig. 21.31 for one pair of complementary patterns. Note that the dark fringe contours in Fig. 21.31d are produced at points where the complementary fringe patterns have equal intensities, independent of the slopes at crossing points in Fig. 21.31a. Two fringe contours are produced for every fringe in the original moiré pattern. Full-field patterns corresponding to Fig. 21.31a, c, and d are shown in Fig. 21.32. If additional moiré patterns are recorded with phase steps of π/4 and 3π/4, they can be processed as a complementary pair to provide intermediate fringe contours. The result is portrayed in Fig. 21.32d, where the fringe contours are combined in a single computer output map. Four fringe contours are produced for every moiré fringe in the original pattern, providing fringe multiplication by a factor of 4. Higher multiplication factors can be produced by processing β phase-stepped moiré patterns, e.g., β = 6, 8, . . . . The algorithm was developed originally for microscopic moiré interferometry that produces fringes of sinusoidal intensity distribution. However, O/DFM is not based upon fringes of sinusoidal intensity distribution. Instead, a broad class of periodic functions is applicable, including the case of geometric moiré. The mathematical analysis and a discussion of its implications are given in [21.1, 16]. O/DFM applied to moiré interferometry is demonstrated in this Handbook in the subsequent chapter entitled Chap. 22 Moiré Interferometry.

Truncation 0

d) |Ir |b

1

2

N

i=0

0

e) |Ir |b

1

2

N

2

N

i = 0, 1

0

1

Fig. 21.31a–e Steps in the O/DFM algorithm, applied with

the quasitriangular intensity distribution of shadow moiré

21.5.2 The Phase-Stepping (or Quasiheterodyne) Method In the technique that is becoming better known as phase stepping [21.17], but previously called the quasiheterodyne method [21.18], three or more phase-shifted images of the moiré pattern are recorded by a CCD camera. In practice, a small camera lens aperture is used to suppress the resolution of the shadow moiré fringe pattern so that the intensity distribution approaches a sinusoidal function. Then, the intensities at any single point in the moiré patterns are approximated by I (x, y) = Im (x, y) + Ia (x, y) cos[φ(x, y)] ,

(21.19)

where Im (x, y) is the mean intensity (the constant DC component of the intensity at the point), Ia (x, y) is the

Geometric Moiré

Fig. 21.32 (a) The initial shadow moiré pattern representing the warpage of an electronic package. (b) The complementary pattern. (c) Patterns corresponding to steps (c) and (d) in the O/DFM algorithm in Fig. 21.31. (d) Fringe contours for multiplication by β = 4

intensity modulation (half the amplitude of the maximum change when φ is varied), φ(x, y) is the angular phase information of the fringe pattern, and (x, y) represents all the points in the x–y plane of the specimen; φ represents the fringe order N at each point of the pattern by φ(x, y) = 2π N(x, y), where N is fringe order. The most widely used phase-shifting algorithm uses four phase-shifted images. The set of four images can be expressed as Ii (x, y) = Im (x, y)

a)

b)

c)

d)

21.5 Increased Sensitivity



Then, the unknown phase at the point is determined from the four intensities measured by the CCD camera and the data is sent to a personal computer to evaluate the fringe order at every pixel by using   I4 (x, y) − I2 (x, y) (21.21) . φ(x, y) = arctan I1 (x, y) − I3 (x, y) Like O/DFM, any effects of nonuniform background intensity and also any optical noise (if it is constant, i. e., if it does not change with phase stepping) is incorporated in Im and does not degrade the result. Since (21.21) yields the phase in the restricted range 0 to 2π, an unwrapping algorithm is used to establish the integral part M of the fringe order N at each pixel by

Part C 21.5

 π(i − 1) + Ia (x, y) cos φ(x, y) + 2 (21.20) i = 1, 2, 3, 4 .

φ +M 2π M = ±0, 1, 2, 3 . . . ,

pseudocolor map, or three-dimensional (3-D) graph. Figure 21.23 shows the results from the phase-shifting a)

b)

N=

(21.22)

where M is determined from point-to-point changes of phase throughout the field. Accordingly the fringe order is determined at each pixel to within a small fraction of a fringe order, and by (21.7), the displacement field is determined with enhanced sensitivity. As in the O/DFM method, the displacement field can then be displayed as a contour map,

c)

–3.17 ×101 –1.08 ×101 1.01×101 3.10 ×101

z

5.19 ×101 7.28 ×101 9.37 ×101

Fig. 21.33a–c Results of the phase-stepping algorithm using data from the fringe patterns of Fig. 21.32; (a) wrapped phase map, (b) unwrapped color phase map, and (c) 3-D

map of displacement

625

1.15 ×102 1.35 ×102 1.56 ×102

x

1.77 ×102

626

Part C

Noncontact Methods

technique, where the data from the four phase-stepped patterns employed for Fig. 21.22d were used for this technique. It shows the wrapped phase map, the unwrapped phase map, and the 3-D map of the W displacement field. Whereas the sensitivity of geometric moiré can be increased by the phase-stepping technique, the degree of enhancement is limited by the nonsinusoidal intensity distribution. A theoretical analysis was conducted to quantify the error produced by the nonsinusoidal intensity distribution of shadow moiré fringes [21.15]. The analysis revealed that the maximum error was approximately 0.017 fringes (1.7% of the contour inter-

val); thus the displacement measurement error increases as measurement sensitivity decreases (or the contour interval increases). In fact, considering other sources of errors, the maximum uncertainty in displacement resolution can be higher for both the phase-stepping and O/DFM techniques. Thus, in the design of the experiment, the pitch of the reference grating cannot be arbitrarily large, e.g. 1 mm, with the thought that phase-shifting routines can provide ever-greater precision of measurement. Instead, higher basic measurement sensitivity, providing a greater number of fringes, is required for precision shadow moiré measurements.

References 21.1

Part C 21

21.2

21.3

21.4 21.5

21.6 21.7

21.8

21.9

D. Post, B. Han, P. Ifju: High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer, New York 1994) D. Post, B. Han, P. Ifju: Moiré methods for engineering and science – moiré interferometry and shadow moiré. In: Photomechanics, ed. by P.K. Rastogi (Springer, New York 2000) C.W. Han: Shadow Moire Using Non-zero Talbot Distance and Application of Diffraction Theory to Moire Interferometry. Ph.D. Thesis (University of Maryland, Maryland 2005) A.J. Durelli, V.J. Parks: Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs 1970) G. Mesmer: The interference screen method for isopachic patterns, Proc. Soc. Exp. Stress Anal. XIII(2), 21–26 (1956) H. Takasaki: Moiré topography, Appl. Opt. 9, 1467– 1472 (1970) J.B. Allen, D.M. Meadows: Removal of unwanted patterns from moiré contour maps by grid translation techniques, Appl. Opt. 10, 210–212 (1971) M. Halioua, R.S. Krishnamurthy, H. Liu, F.P. Chiang: projection moiré with moving gratings for automated 3-D topography, Appl. Opt. 22, 850–855 (1983) K. Patorski: The self-imaging phenomenon and its applications. In: Progress in Optics, Vol. XXVII, ed. by E. Wolf (North-Holland, Amsterdam 1989)

21.10

21.11

21.12

21.13

21.14

21.15

21.16

21.17

21.18

M. Testorf, J. Jahns, N. Khilo, A. Goncharenko: Talbot effect for oblique angle of light propagation, Opt. Commun. 129, 167–172 (1996) K. Verma, D. Columbus, B. Han: Development of real time/variable sensitivity warpage measurement technique and its application to plastic ball grid array package, IEEE Trans. Electron. Packag. Manuf. 22, 63–70 (1999) B. Han: Thermal stresses in microelectronics subassemblies: Quantitative characterization using photomechanics methods, J. Therm. Stress. 26, 583–613 (2003) C.-W. Han, B. Han: Contrast of shadow moiré at high order Talbot distances, Opt. Eng. 44, 1–6 (2005) C. Han, B. Han: High sensitivity shadow moiré using nonzero-order Talbot distance, Exp. Mech. 46, 543–554 (2006) C.-W. Han, B. Han: Error analysis of phase shifting technique when applied to shadow moiré, Appl. Opt. 45, 1124–1133 (2006) B. Han: Interferometric methods with enhanced sensitivity by optical/digital fringe multiplication, Appl. Opt. 32(25), 4713–4718 (1993) J. Huntley: Automated fringe pattern analysis in experimental mechanics: A review, J. Strain Anal. 33(2), 105–125 (1998) P. Hariharan: Quasi-heterodyne hologram interferometry, Opt. Eng. 24(4), 632–638 (1985)

627

Moiré Interfe 22. Moiré Interferometry

Daniel Post, Bongtae Han

The basic concepts and practice of moiré interferometry are reviewed. Moiré interferometry provides contour maps of in-plane displacement fields with high sensitivity and high spatial resolution. It has matured into an invaluable tool for engineering analyses, proved by many industrial and scientific applications. With the typical reference grating frequency of 2400 lines/mm, the contour interval is 0.417 µm displacement per fringe order. For microscopic moiré interferometry, sensitivity in the nanometer range has been achieved. Reliable normal strains and shear strains are extracted from the displacement data for bodies under mechanical, thermal, and environmental loading.

22.2.4 Insensitivity to Out-of-Plane Deformation ........ 22.2.5 Accidental Rigid-Body Rotation ..... 22.2.6 Carrier Fringes ............................. 22.2.7 Loading: Mechanical, Thermal, etc. ............. 22.2.8 Bithermal Loading........................ 22.2.9 Curved Surfaces............................ 22.2.10 Data Enhancement/Phase Stepping 22.2.11 Microscopic Moiré Interferometry .............................

635 637 637 638 638 640 640 643

22.3 Challenges ............................................ 644 22.3.1 Strain Analysis ............................. 644 22.3.2 Replication of Deformed Gratings ... 644 22.4 Characterization of Moiré Interferometry . 645

630 630 632 632 633 634

22.2 Important Concepts ............................... 22.2.1 Physical Description...................... 22.2.2 Theoretical Limit .......................... 22.2.3 Black Holes..................................

634 634 635 635

This section introduces basic concepts of moiré interferometry and their evolution, leading to the current practice. Moiré interferometry has become an important tool in industrial, research, and academic organizations. The tool is used to measure tiny deformations of solid bodies, caused by mechanical forces, temperature changes, or other environmental changes. It is the in-plane deformations that are measured, namely the U and V components of displacement that are parallel to the surface of the body. These are the displacements from which the induced strains and stresses are determined.

22.5 Moiré Interferometry in the Microelectronics Industry .............. 22.5.1 Temperature-Dependent Deformation ................................ 22.5.2 Hygroscopic Deformation .............. 22.5.3 Standard Qualification Test of Optoelectronics Package ............ 22.5.4 Micromechanics Studies by Microscopic Moiré Interferometry

646 646 648 650 651

References .................................................. 652

Moiré interferometry has been applied for studies of composite materials, polycrystalline materials, layered materials, piezoelectric materials, fracture mechanics, biomechanics, structural elements, and structural joints. It is practised extensively in the microelectronics industry to measure thermally induced deformation of electronic packages. For in-plane displacement measurements, the technology has evolved beautifully from low-sensitivity geometric moiré to the powerful capabilities of moiré interferometry. Moiré measurements are performed routinely in the interferometric domain with fringes

Part C 22

22.1 Current Practice ..................................... 22.1.1 Specimen Gratings ....................... 22.1.2 Optical Systems ............................ 22.1.3 The Equations .............................. 22.1.4 Fringe Counting ........................... 22.1.5 Strain Analysis .............................

628

Part C

Noncontact Methods

49

Ny = 117

Specimen grating

17 5

10

76 51 mm

–118

–50

Fig. 22.1 Demonstration of moiré interferometry at 97.6% of the theoretical limit of sensitivity. f = 4000 lines/mm. The patterns are from the shaded regions. Excellent fringe resolution was evident in the full 76 × 51 mm specimen grating area

Part C 22

representing subwavelength displacements per contour. Since moiré responds only to geometric changes, it is equally effective for elastic, viscoelastic, and plastic deformations, for isotropic, orthotropic, and anisotropic materials, and for mechanical, thermal, and dynamic loadings. The history of moiré interferometry is described in [22.1], in which Walker recounts the pioneering developments in Japan, which predated accomplish-

ments in the Western world by several years, but were not widely known in the West. He outlined work by Sciamarella and by Post and their students in the USA, and by several investigators in Europe – especially the outstanding work at Strathclyde University by McKelvie, Walker, and their colleagues. Moiré interferometry has matured through worldwide efforts. x

Parabolic mirror

y

z

ω2

–α

Specimen

Laser

f x

Specimen grating

y

Specimen grating

–α α

Mirror

fs Camera α

ω1

Camera

Fig. 22.3 The deformed specimen grating interacts with Fig. 22.2 Optical system for Fig. 22.1: Lloyd’s mirror arrangement

the virtual reference grating to form the moiré pattern. The phenomenon is analogous to geometric moiré

Moiré Interferometry

a)

Diffraction B

ω

ω α

B

ω1

b) Interference

ω

ω1 –α α ω ω1

Fig. 22.4a,b Virtual gratings formed (a) by a real grating, and (b) by two coherent beams

a virtual reference grating and it can replace the real reference grating. Thus, in Fig. 22.2 the direct and the reflected beams, intersecting at angle 2α, provided the virtual reference grating. (An important equivalency principle is acting here: interference of two beams creates a virtual grating of frequency f ; a real grating of frequency f creates the same two beams [22.4]. In fact, this equivalency is also the basis of holography.) It became clear that phase gratings (ridges and furrows) could be used as well as the amplitude gratings (bars and spaces) of geometric moiré. Several innovations evolved to link the techniques of the 1970s to the present. A significant factor was the practical availability of the laser, which provided very high monochromatic purity and allowed high-contrast twobeam optical interference even when the beams traveled substantially unequal path lengths. The laser led to a practical technique to produce high-frequency phasetype specimen gratings. For this process, a setup with a beam-splitter and mirrors formed the two mutually coherent beams of Fig. 22.4b. A plate with a photosensitive coating – pho-

Part C 22

Work at Virginia Tech began in 1979 and was responsible for numerous advances. The power of the method was demonstrated in 1981 when Weissman [22.2] produced the fringe pattern shown in Fig. 22.1, which depicts the vertical displacement field surrounding a hole in a plate loaded in tension, demonstrating the moiré effect at 4000 lines/mm, 97.6% of its theoretical limit of sensitivity. This demonstrated the V deformation field with a huge number of fringes, or displacement contours, and the fringes had superb visibility. The instrumentation was simple, as illustrated in Fig. 22.2. A phase grating of 2000 lines/mm was formed on the specimen, and it deformed together with the specimen as the specimen was loaded. A Lloyd’s mirror arrangement created a 4000 lines/mm interference fringe pattern in the same space, and that pattern acted as the reference grating. This is called a virtual reference grating. (It was not a real reference grating of the sort used in geometric moiré, but instead was an interference pattern formed in space that functioned as a reference grating.) The specimen grating and the reference grating interacted to form the moiré fringes shown in Fig. 22.1. The scheme is illustrated in Fig. 22.3, where the two coherent beams from the Lloyd’s mirror arrangement interfere to form the virtual reference grating of frequency f , which interacts with the specimen grating of frequency f s to create the moiré pattern. Stepping back, the transition from geometric moiré to moiré interferometry evolved largely in the 1970s. Whereas only low-frequency gratings could be applied to specimens, high-frequency reference gratings could be used, together with an optical filtering arrangement, to produce moiré fringe multiplication. The fringe multiplication factor, β, is the ratio of grating frequencies, and the resulting sensitivity is determined by the reference grating frequency, f ; the sensitivity is then β times that of standard geometric moiré, where specimen and reference gratings of equal frequencies are used. Multiplication by β = 60 was documented [22.3], but at a substantial sacrifice of the efficiency of light utilization. Then it was realized that light diffracted by a real reference grating (BB) creates a virtual reference grating, as illustrated in Fig. 22.4a. It is the virtual grating that interacts with the specimen grating to create the moiré pattern. Consequently, it was realized that the interference pattern produced by the intersection of any two mutually coherent beams (Fig. 22.4b) can act as

629

630

Part C

Noncontact Methods

Part C 22.1

tographic emulsion or photoresist – was inserted to record the virtual grating. When developed, the plate exhibited the ridges and furrows of a phase grating, in exact registration with the bands of constructive and destructive interference of the virtual grating. Subsequently, the phase grating was replicated on the specimen by known techniques, using silicone rubber, epoxy, or various plastics to reproduce the array of ridges and furrows on the specimen. Specimen gratings of very high frequency could be made, so the need for high fringe multiplication factors vanished. The moiré interferometry technique described here is actually moiré fringe multiplication with a multiplication factor of two. It utilizes the firstorder diffractions from the specimen grating instead of high diffraction orders. This choice circumvents the inefficient light utilization and increased optical noise associated with higher orders, while maintaining the advantage of viewing the specimen at normal incidence. Numerous refinements were introduced in the 1980s. Cross-line specimen gratings, together with four-beam moiré interferometers became standard practice; this eliminated the classical uncertainty of shear measurements and enabled accurate determination of both shear and normal strains. The use of carrier fringes became common. Practical techniques for measurement of thermal strains were developed. Achromatic moiré

interferometers were developed. Microscopic moiré interferometry was introduced. In the 1990s, charge-coupled device (CCD) cameras became popular for recording the fringe patterns. This led to phase stepping and automated full-field strain maps. Phase stepping proved to be an important asset for cases where fringes were sparse, because it increases the quantity of data for analysis. (For typical applications, where the fringe data is abundant, phase stepping is superfluous and sometimes counterproductive.) A very important milestone occurred in the 1990s. IBM Corporation discovered the value of moiré interferometry for experimental analysis of thermal deformation of microelectronic devices. It was used to measure – sometimes to discover – the deformation behavior of their small complex structures, and to guide and verify their numerical analyses. Since then, moiré interferometry has propagated extensively in the electronic packaging industry, to become a standard tool for experimental analysis. The reference [22.4] provides comprehensive coverage of the theory and practice, and diverse applications of moiré interferometry in one volume. The present chapter is intended to review the basic concepts and qualities in lesser detail, but sufficient for understanding the practice and for recognizing the power and place of moiré interferometry as a tool of experimental solid mechanics.

22.1 Current Practice Moiré interferometry was developed to measure the in-plane displacements of essentially flat surfaces. If demanded, it can cope with curved surfaces, but with greater effort; Fig. 22.15, to be addressed later, shows an example of this. Two displacement fields fully define the state of engineering strain along the surface. Normally two orthogonal displacement fields are recorded: the U and V fields. They represent x- and y-components of displacement at every point in the field. The data are received as moiré fringe patterns, where the fringes are contours of constant U or V displacements. Fringe orders are given the symbols N x and N y , where N x = fU and N y = fV . The fringe orders are proportional to the displacements, and the constant of proportionality is the frequency, f , of the (virtual) reference grating. Of course, we can write the relationships in terms of the pitch g of the reference grating, since f = 1/g.

22.1.1 Specimen Gratings Typically, the virtual reference grating frequency is f = 2400 lines/mm (60 960 lines/in.); the moiré interferometer projects this grating onto the specimen. The specimen grating is a cross-line phase grating with 1200 lines/mm in both the x- and y-directions. It is usually formed on the specimen by replication from a mold, which is itself a cross-line grating. A greatly enlarged view is shown in Fig. 22.5a, illustrating the orthogonal array of hills that comprise the cross-line grating. The replication method of Fig. 22.5b is used for diverse applications that involve larger regions of interest, e.g., larger than 1 cm2 . The mold is prepared with a reflective metallic film, usually evaporated aluminum. A liquid adhesive, usually an epoxy, is squeezed into a thin layer between the specimen and mold. They are separated when the adhesive solidifies. The weakest

Moiré Interferometry

22.1 Current Practice

631

a)

b)

c) Optical tissue

Specimen Epoxy Uncured adhesive

Metallic film

Mold Cross-line grating mold (with metallic film)

Part C 22.1

Uncured adhesive Cotton swab Specimen

Alignment bar

Specimen and grating Cured adhesive Metallic film Mold

Fig. 22.5 (a) Cross-line 1200 lines/mm specimen grating. (b) Replication procedure for typical specimen gratings. (c) Procedure for small specimens of complex geometry

interface is between the metallic film and the underlying mold, so the reflective metallic film is transferred to the specimen together with the hills and valleys of the grating. The total grating thickness is usually about 25 μm.

The technique of Fig. 22.5c is used for small specimens of complex geometry, such as those encountered in electronic packaging tests. In this case, the specimen is prepared with a smooth, flat surface. A very thin layer of low-viscosity adhesive is spread on the mold by the

632

Part C

Noncontact Methods

y x z

Specimen

from the edges of the specimen; therefore, no cleaning operation is required to remove the excess adhesive. With this procedure, the grating thickness is usually about 2 μm. Again, the result is a thin, compliant, reflective cross-line grating, which deforms together with the underlying specimen.

22.1.2 Optical Systems

B3 x α

The generic optical system is illustrated in Fig. 22.6. As in Fig. 22.3, two coherent, collimated beams marked B1 B2 and B2 create a virtual reference grating with its lines α perpendicular to the x-direction. They interact with the Nx or Ny fringes corresponding array of lines on the specimen grating to create the N x fringe pattern, depicting the x-component Camera of displacement. The fringe pattern is recorded by the lens camera, which is focused on the specimen. Typically, B1 it is a digital CCD camera. Then beams B3 and B4 are used to record the N y pattern. The virtual reference grating frequency is usually 2400 lines/mm. For light in the B4 visible wavelength range, α is near 45◦ . The contour inFig. 22.6 Schematic diagram for four-beam moiré interferoterval for the fringe pattern is 1/ f , i. e., a displacement metry of 0.417 μm per fringe order. Numerous different optical systems can be designed drag method and the specimen is pressed into the ad- to provide the essential elements of Fig. 22.6. A few syshesive. Surface tension draws the thin adhesive away tems are illustrated in [22.4]. Two-beam systems have been used for special applications. For strain and stress analysis, the four-beam optical system offers a vital advantage: shear strains can be extracted with the same Ny accuracy as normal strains. Achromatic optical systems have also been devised, and Fig. 22.7 is an example [22.4]. A temporally inco20 herent light source can be used. With this arrangement, 20 the angle of diffraction, α, at the upper grating changes in exact harmony with the wavelength λ to maintain 15 15 a fixed frequency f of the virtual reference grating. The upper grating is called a compensator grating, since it 11 10 compensates for variations of the wavelength. Whereas 2 monochromatic purity is not needed, spatial coherence 0 5 remains a requirement. Grating

Part C 22.1

0

22.1.3 The Equations

y

The pertinent equations are

• z

2 mm

Fig. 22.7 Fringe counting. The fringe pattern shows

weld defects in a stainless-steel tension specimen. f = 2400 lines/mm. (Courtesy of S. A. Chavez)

For optical interference of two beams (Fig. 22.4b) 2 (22.1) sin θ, λ where F is the fringe frequency (fringes/mm) in the region of intersection, λ is the wavelength, and θ is the half-angle of intersection. F=

Moiré Interferometry

•

•

•

•

Figure 22.8 shows the deformation of a tensile coupon cut from a stainless-steel plate in the region of a weld. Cracks appear in the weld, but along the left edge of the specimen, the material is continuous. Since the strain is tensile, or positive, the fringe orders along the left edge increase monotonically in the +y-direction, as shown. Then the fringe orders can be assigned at every point in the field by following continuous fringes across the field. Fringe orders along the right edges were assigned accordingly. Where a crack is present, the V displacements are different along the upper and lower lips of the crack. This accounts for the crack opening that results from the tensile load. For example, at the right edge the fringe order changes from N y = 3 to N y = 10, indicating a crack-opening displacement of seven fringe orders, or 2.9 μm (115 μin). Clues derived from known loading conditions and known specimen geometry are often sufficient to establish the sign of the fringe gradient at every point, i. e., to establish whether the fringe order is increasing or decreasing as the count progresses from one fringe to the neighboring fringe. Occasionally the clues might not be sufficient, but there is always a simple experimental way to determine the sign of the fringe gradient. If, during the experiment, the specimen is moved gently in the +x-direction, the fringe order N x at every point increases. This means that the fringes all move toward the direction of lower-order fringes. f/2

22.1.4 Fringe Counting The rules of topography of continuous surfaces govern the order of fringes [22.5]. Fringes of unequal orders cannot intersect. The fringe order at any point is unique, independent of the path of the fringe count used to reach the point. The location of a zero-order fringe in a moiré pattern can be selected arbitrarily. Any fringe – black, white or gray – can be assigned as the zero-order fringe. This is because rigid-body translations are not important in deformation analysis. Absolute displacement information is not required and relative displacements can be determined using an arbitrary datum. In Fig. 22.8, the zero fringe is assigned in the lower left region of the specimen. Of course, at every point along this continuous fringe, N y = 0.

From source

Camera lens

f/2

Fig. 22.8 Achromatic moiré interferometer

Specimen

633

Part C 22.1

•

For the moiré interferometer λf sin α = (22.2) , f = 2 fs , 2 where f and f s are the frequency of the virtual reference grating and the specimen grating, respectively. For the displacements at each point in the field 1 1 (22.3) U = Nx , V = N y , f f where U and V are displacements in the x- and ydirections, respectively. For the strains at each point   ∂U 1 ∂N x εx = = , ∂x f ∂x   ∂V 1 ∂N y (22.4) , = εy = ∂y f ∂y   ∂U ∂V 1 ∂N x ∂N y γxy = + = + , (22.5) ∂y ∂x f ∂y ∂x where ε and γ are normal and shear strains, respectively. Thus, the strains are determined by the rate of change of fringe orders in the patterns, or the fringe gradient surrounding each point. For fringe gradients ∂N x ΔN x (22.6) ≈ ; etc. ∂x Δx The derivatives are usually approximated by their finite increments, i. e., the change of fringe order that occurs in a finite distance Δx. For the stresses Stresses are determined from the strains, using the stress–strain relationships (or the constitutive equations) for the specimen material.

22.1 Current Practice

634

Part C

Noncontact Methods

Thus, if the N x fringes move in the negative xdirection, the gradient ∂N x /∂x is positive. The argument is the same for the y-direction. A convenient alternative is available if the moiré interferometer is equipped for phase stepping (fringe shifting). The investigator can watch the pattern while the fringes are shifted. Again, the fringes move toward lower fringe orders when the phase is increased. Thus, the sign of the fringe gradients is readily determined at any point.

22.1.5 Strain Analysis Strains are determined from the two displacement fields by the relationships for engineering strain, (22.4) and (22.5). In principle, the exact differential can be

extracted at any point by plotting a curve of fringe orders along a line through the point and measuring the slope of the curve at the point. Often, however, the finite increment approximation is sufficient, whereby (as an example) ∂N x /∂x is taken to be equal to ΔN x /Δx. In that case, strain is determined by measuring Δx, the distance between neighboring fringes, in the immediate neighborhood of the point of interest. Shear strains are determined as readily as normal strains. Numerous examples of fringe counting and strain analysis are given in [22.4]. In nearly all cases of strain analysis, the strains are sought at specific points (e.g., where the fringes are most closely spaced, indicating strain maxima), or along specific lines. Manual and computer-assisted methods are most practical for such cases.

22.2 Important Concepts 22.2.1 Physical Description

Part C 22.2

The description offered in Fig. 22.3 is a casual explanation based upon the analogy to geometric moiré. Nevertheless, it is effective. The rigorous analysis of moiré interferometry [22.4], based on diffraction and optical interference, shows that the fringe order–displacement relationship is identical for moiré interferometry and Diffraction orders 3 2 1

fs = f /2 Beam 1 x Beam 1

w1

z

0 –α

–1

k

P w'1

w2''

α

1 0

βm –1

–3 Specimen

S

w'2 w1''

Camera Beam 2

w2

–2 Beam 2

Fig. 22.9 Diffraction by the specimen grating produces beams with

plane wavefronts for the unloaded condition. Warped wavefronts result from inhomogeneous deformation of the specimen

geometric moiré. The simple, intuitive procedures to extract data from geometric moiré patterns apply also to moiré interferometry patterns. This is important for our colleagues who might practise moiré interferometry, but specialize in diverse aspects of engineering and materials science, as distinct from specialists in optical techniques. For our colleagues in photomechanics, the physical description illustrated in Fig. 22.9 is more satisfying. Before loads are applied to the specimen, the grating on its surface has a uniform frequency of f s . The incident beam 1 is diffracted such that its first order emerges normal to the specimen, with wavefront w1 . This requires that sin α = λ f s . Similarly, beam 2 is diffracted to form emergent wave front w2 . For these initial conditions, the emergent beams have parallel wavefronts and they interfere to create a null field, i. e., a uniform intensity or the infinite fringe. Incidentally, we note from (22.1) that, when two beams intersect with this half-angle α, they generate interference fringes of frequency f , where f ≡ 2 f s . Thus, the frequency of the virtual reference grating of Fig. 22.3 is twice the initial frequency of the specimen grating. This is the condition that provides normal viewing, without distortion of the image. For a complex specimen, the specimen grating deforms nonuniformly when loads are applied. The grating pitch and orientation vary as continuous functions along the specimen. Consequently the directions of diffracted rays change continuously, and the two beams emerge as warped wavefronts w1 and w2 . They

Moiré Interferometry

interfere to generate contours of their separation S, and this interference pattern is recorded by the camera. With the camera focused on the plane of the specimen, the image is the moiré pattern – the map that depicts the inplane deformation of the specimen. Summing up, moiré interferometry is a case of two-beam interferometry. It is important to focus the camera on the plane of the specimen. Since wavefront warpage changes as the wave travels away from the specimen, this is especially important for cases of strong strain gradients and the concomitant strong warpage of the emerging wavefronts. This issue is addressed in [22.4].

22.2.2 Theoretical Limit

The camera lens can be translated laterally to capture the light of the moiré pattern that is otherwise lost. In doing so, it is possible that other regions will become black. An alternative is to replace the camera lens with a lens of larger numerical aperture, although this might necessitate viewing the specimen at higher magnification.

22.2.4 Insensitivity to Out-of-Plane Deformation This is largely a pedagogical issue, inasmuch as the literature may not be sufficiently clear. Various publications show sensitivity to out-of-plane rotations. However, the rigorous mathematical analysis of [22.4] proves that moiré interferometry is insensitive to all out-of-plane motions, including rotations. Where do the publications deviate? Figure 22.10 should help clarify the issue. It represents the planes, or walls of interference of the virtual reference grating, and it also represents the plane of the undeformed specimen grating as plane abcd. Let P be a point fixed on the specimen grating, and let P move to another point in space, P , when the specimen is deformed. The deformed surface surrounding P might have any slope, as indicated by the dashed box. As the point moves in the x-direction, it cuts through walls of interference and therefore causes a change of fringe order in the moiré pattern. Expressed in other

Beam 2

22.2.3 Black Holes Referring to Fig. 22.9, we can visualize that the angles of diffraction from any region of the specimen stems from two effects: (1) the load-induced strain and (2) the load-induced surface slope. If the angles of diffraction become large enough, light from that region can miss the lens and never enter the camera. The region appears black in the image – it is called a black hole. In such cases, the main cause is usually the surface slope. It is rare that a homogeneous body would exhibit large slopes. A composite body, fabricated by joining two dissimilar materials, can exhibit large slopes near the interface as a result of a Poisson’s ratio mismatch. Experience indicates that the slopes are seldom severe enough to cause black regions until very high strain levels are reached.

635

a P'

c

W

Beam 1

V b

U

P d

Fig. 22.10 Walls of interference of the virtual reference grating. Only the U component of displacement crosses the walls of the reference grating

Part C 22.2

From (22.2), we see that the theoretical upper limit for the reference grating frequency is approached as α (Fig. 22.6) approaches 90◦ . The theoretical limit is f = 2/λ. The corresponding theoretical upper limit of sensitivity is 2/λ fringes per unit displacement, which corresponds to a contour interval of λ/2 displacement per fringe order. In the experiment illustrated in Fig. 22.1, α was 77.4◦ and λ was 488.0 nm. This produced a virtual reference grating of 4000 lines/mm (101 600 lines/in). For this wavelength, the theoretical limit is f = 4098 lines/mm, which means that the experiment was conducted at 97.6% of the theoretical limit of sensitivity. The theoretical limit pertains to virtual reference gratings formed in air. In the experimental arrangement described later for microscopic moiré interferometry, the grating is formed in a refractive medium.

22.2 Important Concepts

636

Part C

Noncontact Methods

words, the point crosses lines of the (virtual) reference grating. For displacement components V and W, the point does not cross any wall, and therefore these coma)

ponents have no effect on the fringe order. The moiré pattern remains insensitive to V and W regardless of the surface slope. This corroborates the proof that moiré

b)

15 10 10

–5

0

7 6

20

–4

5 6

–3

4

21

5 3.5 6

3 2

7

–2

4

5

4

25

3 0

8 0

7

–1

23

–4

Part C 22.2

5 Nx or U filed

c)

Ny or V filed

d)

y, V Graphite/epoxy 90° fiber direction f = 2400 lines/mm x, U

Fig. 22.11a–d

Clamping device

Load transfer fixture

Specimen 19 mm Load transfer fixture

Clamping device Ny or V filed with rotation

Displacement fields for a compact shear specimen. Carrier fringes of rotation transform the V field. Local irregularities in the fringes are caused by local variations in the composite material. (Courtesy of P. G. Ifju)

Moiré Interferometry

22.2.5 Accidental Rigid-Body Rotation The previous paragraph refers to out of-plane deformations induced by the loads, not rigid-body motions. In practice, the specimen is observed in its unloaded condition and the moiré interferometer is adjusted to produce a null field, i. e., a uniform fringe order throughout the field. Then the load is applied and the displacement fields are recorded. Sometimes, an out-of-plane rigid-body rotation is introduced accidentally in the loading process. Rotation about an axis that is perpendicular to the lines of the reference grating has no effect on the fringe pattern. Rotation by Ψ about an axis parallel to the grating lines is seen as an apparent foreshortening of the specimen grating. It introduces an extraneous fringe gradient that corresponds to an apparent compressive strain εapp ,

637

where Ψ2 (22.7) , 2 provided the rotation Ψ is not large. The superscript signifies an extraneous strain, or error. The extraneous fringe gradient is a second-order effect and can be neglected in the usual case where it is small compared to the strain-induced fringe gradient. Otherwise, a correction can be applied [22.4]. εapp = −

22.2.6 Carrier Fringes An array of uniformly spaced fringes can be produced by adjustment of the moiré interferometer; these are called carrier fringes. With the specimen in the unloaded condition, a carrier of extension is produced by changing the angle α (Fig. 22.6). These fringes are parallel to the lines of the reference grating, just like the fringes of a pure tensile (or compression) specimen. Carrier fringes of rotation are produced by a rigid-body rotation of the specimen or the interferometer, and these are perpendicular to the lines of the reference grating. It is frequently valuable to modify the load-induced fringe patterns with carrier fringes [22.6]. Figure 22.11 shows an example [22.7], where the load-induced fringes are shown in Fig. 22.11a and Fig. 22.11b. The specimen is cut from a graphite–epoxy composite material and is loaded in a compact shear fixture (Fig. 22.11c). In Fig. 22.11d, the fringes in the test section between the notches are subtracted off by carrier fringes of rotation. The carrier fringes are vertical and uniformly spaced, and their gradient is opposite to that of the shear-induced fringes. What is the benefit? This shows that the normal strain ε y is zero along the vertical center line of the specimen, that ε y is extremely small throughout the test zone, and that the shear deformation is nearly uniform over a wide test region. Carrier fringes can be introduced easily, typically by turning an adjusting screw in the moiré interferometer. They are used, as needed, for a variety of purposes, especially to remove ambiguities and enhance the accuracy of data reduction. For in-plane rigid-body rotation ∂N y ∂N x ≡− . ∂y ∂x

(22.8)

Substitution into (22.5) shows that shear strains calculated from the moiré fringes are not affected by carrier fringes of rotation (when a four-beam moiré interferometer is used). Also, the calculated shear strains are

Part C 22.2

interferometry senses only the component of displacement perpendicular to the walls of the virtual reference grating. Engineering strain is the ratio of the local change of length to the original length, when both are projected onto the original specimen surface abcd. Thus, engineering strains are correctly determined by (22.4) and (22.5), regardless of surface slopes. Note that outof-plane displacements are referenced to the original surface, too. For exceptional studies in which the surface slope is large enough, the investigator might be interested in these lengths measured on the specimen surface instead of the plane abcd. This is where the publications deviate. In such cases, a mathematical transformation is required which involves additional variables, specifically the components of the surface slopes. These slopes must be measured (or otherwise determined) to augment the moiré data at each point where the strain is to be calculated. Thus, if the investigator feels the outof-plane slopes are significant enough to degrade the accuracy that is sought for a specific analysis, then a Lagrangian strain or other three-dimensional strain should be calculated. Otherwise, the investigator accepts that the influence of the slope is negligible for calculating the strain. Note, too, that special apparatus and procedures may be required to record an image of the moiré fringes when surface slopes become large. Otherwise, the moiré pattern may display black holes in regions of large slopes. In general, it is the engineering strain that is sought by moiré interferometry, not a transformed strain, and (22.4) and (22.5) are fully effective.

22.2 Important Concepts

638

Part C

Noncontact Methods

Portable interferometer

Specimen

Convection oven

Fig. 22.12 Arrangement for real-time thermal deformation

tests

not affected by carrier fringes of extension. The calculated normal strains are (essentially) unaffected by carrier fringes of rotation. These conditions allow beneficial changes in the fringe patterns without altering the reliability of strain calculations.

Part C 22.2

moisture ingress and egress, during chemical activity, etc. Hence, it enables real-time measurements. Figure 22.13 illustrates an example of real-time thermomechanical analysis of an electronic package during a thermal cycle with temperature extremes from −20 ◦ C to 125 ◦ C [22.8, 9]. The device is an assembly of diverse materials with different coefficients of thermal expansion (CTE), so thermal stresses, strains, and deformation result from any change of temperature. The fringes are remarkably clear at this magnification, except for those in the printed circuit board (PCB) at 125 ◦ C; the PCB is a heterogeneous material – a composite of woven glass fibers in an epoxy matrix – and because of the weave its displacement contours are very complicated. Higher magnification is indicated. Visual inspection of the V field shows opposite directions of curvature in the chip region before and after the 125 ◦ C temperature; this behavior results from creep of the solder and molding compound. Whereas extensive computational analysis is undertaken for the mechanical design of electronic packages, the complexities of geometry and materials necessitate experimental guidance and verification. Additional examples of moiré interferometry applied to electronic and photonic packages are discussed in Sect. 22.5.

22.2.7 Loading: Mechanical, Thermal, etc. 22.2.8 Bithermal Loading Experimental arrangements for mechanical loading are very diverse; they depend upon the size and nature of the specimen and the magnitude of the loads. In some cases it is necessary to mount the moiré interferometer on the structure of the testing machine, or to attach it to the assembly being investigated. In other cases, the specimen can be loaded in a fixture that rests on an optical table together with the moiré interferometer. For real-time thermal loading, or for deformation caused by changes of moisture content, chemical activity or radioactivity, the environmental chamber must be positioned adjacent to the moiré interferometer. Figure 22.12 illustrates a scheme used for thermal deformation measurements in the electronic packaging industry. An important factor is that the specimen is connected rigidly to the interferometer. The connecting rods do not contact the chamber (only loose insulation or compliant baffles fill the air gap at the chamber wall), so vibrations in the forced air convection chamber are not transmitted to the specimen. This arrangement has proved to be very effective. It enables the recording of fringe patterns during the transition between temperature changes, during

Thermal deformations can also be analyzed by roomtemperature observations. In this technique, the specimen grating is applied at an elevated temperature, and then the specimen is allowed to cool to room temperature before it is observed in the moiré interferometer. Thus, the deformation incurred by the temperature increment is locked into the grating and recorded at room temperature. An adhesive that cures slowly at elevated temperature is used, usually an epoxy. The specimen and mold are preheated to the application temperature; then the adhesive is applied, the mold is installed, the adhesive is allowed to cure, and the mold is removed – all at the elevated temperature. The mold is a grating on a zero-expansion substrate, so its frequency is the same at elevated and room temperatures. Otherwise, a correction is required for the thermal expansion of the mold. These measurements can also be performed at cryogenic temperatures. In one test, the specimen grating was applied at −40 ◦ C using an adhesive that cured in ultraviolet light.

Moiré Interferometry

Molding compound

22.2 Important Concepts

639

Chip y,V

PCB substrate Solder ball PCB

x,U

80 °C

125 °C

80 °C

–20 °C

U field

Part C 22.2

20 mm

80 °C

125 °C

80 °C

–20 °C

V field

Fig. 22.13 U and V displacement fields for an electronic package subjected to a thermal cycle in an environmental

chamber. f = 2400 lines/mm

640

Part C

Noncontact Methods

Fig. 22.14 Thermal deformation y

y'

Steel x

z

for a bimaterial joint. The N y pattern, with carrier fringes of extension, shows the abrupt transition near the interface. ΔT = 133 ◦ C; a = 55.9 mm; f = 2400 lines/mm. (Courtesy of J. D. Wood)

Brass α/2 2α 0

Part C 22.2

The example shown in Fig. 22.14 is taken from an investigation of thermal stresses in a bimaterial joint [22.10]. The specimen grating was applied and cured at 157 ◦ C and subsequently observed at 24 ◦ C (room temperature). The two materials restrained the natural contraction of each other, causing large thermal stresses and strains. The V displacement field for the 133 ◦ C temperature increment is shown here; carrier fringes of extension were applied to portray the abrupt transition near the interface. Clearly, the fringe gradient ΔN y /Δy is negative in the brass and positive in the steel. Strains were extracted, and since the elastic constants were known, stresses were calculated. A remarkable condition was found near the interface, namely a peak compressive stress in the brass and a peak tensile stress in the steel. Subsequent analysis by microscopic moiré interferometry documented a severe stress gradient in the 50 μm zone surrounding the interface. The peak stresses were elastic and the peaks were connected by the severe gradient in the transition zone.

22.2.9 Curved Surfaces Moiré interferometry has been developed for routine analysis of flat surfaces. However, certain accommodations can be made for curved surfaces when the problem is important enough to invest extra effort. The most straightforward approach is to treat any local region as a flat surface. A small grating mold (cut from a larger mold) could be used to replicate the grating on the curved surface of the specimen. Other innovative techniques can be employed for cylindrical surfaces, or other developable surfaces. Results shown in Fig. 22.15 are from an important investigation. It is from a study of the ply-by-ply

20

deformation at a central hole in multiply composite plates [22.11]. The specimens were thick laminated composite plates, each with a 25.4 mm-diameter central hole. The grating was applied to the cylindrical surface of the hole and successive replicas of the deformed grating were made with the specimen at different tensile load levels. First, a cross-line grating was formed on the cylindrical surface of a disk and was used as a mold to apply the grating to the specimen. Then, with the specimen under load, the deformed grating was replicated (or copied) on another disk. The replica was inserted into a moiré interferometer and deformation data were recorded by a moiré interferometer as a series of narrow strips, each approximating a flat surface. Figure 22.15 shows a mosaic of such strips for a 90◦ portion of the hole in a cross-ply [04 /904 ]3S laminate of IM7/5250-4 (graphite fibers in a bismalimide-cyanate ester matrix). An enlarged view of the Nθ fringe pattern at the θ = 75◦ location is shown in Fig. 22.15d, together with a graph of the ply-byply distribution of shear strains. This is the location of greatest fringe density, and yet the data are revealed with excellent fidelity, enabling dependable determination of strain distributions. The interlaminar shear strains at the 0/90 interfaces are about five times greater than the tensile strain at θ = 90◦ . A primary purpose of the work was to provide experimental data for the evaluation of computational techniques for composite structures.

22.2.10 Data Enhancement/Phase Stepping An abundance of load-induced fringes is typical for most applications, sufficient for reliable analyses. For small specimens, however, or within a small field of

Moiré Interferometry

a) Recorded fringe patterns

z θ

x

641

Fig. 22.15a–d

Tension

y

22.2 Important Concepts

z θ

b) 90 87 84 81 78 75 72 69 66 63 60 57 54 51 48 45 42 39 36 33 30 27 24 21 18 15 12 9 6 3 0

Ply-by-ply displacements at the cylindrical surface of a hole in a cross-ply composite plate. (d) shows an enlarged view of the Uθ pattern at the 75◦ location, and the shear strains at that location. (Courtesy of D. H. Mollenhauer)

Uθ fringe patterns

c) 90 87 84 81 78 75 72 69 66 63 60 57 54 51 48 45 42 39 36 33 30 27 24 21 18 15 12 9 6 3 0

d)

γθz ×106 20 000

0

90

0

90

0

90

0

90

0

90

0

90

15 000 10 000 5000 0 –5000 –10 000 –15 000 –20 000 –25 000

Moiré strain Actual ply boundaries 0

1

2 3 4 5 6 Distance from upper laminate face (mm)

Part C 22.2

Uz fringe patterns

642

Part C

Noncontact Methods

a) Moiré fringes I Microscope

Nonuniform illumination with noise

I π (x)

I (x)

A Prism B

A', B' Mirrorized

N=0

1

3

1

/2

5

2

/2

/2

x

Immersion fluid

α

Specimen grating

b) Subtract shifted intensity I I r (x) = I (x) – I π (x)

(+)

Result: influence of noise is canceled where I (x) = I π (x)

0

x

Immersion interferometer

Specimen Virtual reference grating Microscope objective lens

(–)

c) Take absolute values I |Ir|

Part C 22.2

Truncation

x

d) Binarize near |Ir| = 0 I

Mirrorized surface

Result: 2 data points per fringe of I (x)

Immersion interferometer

Fig. 22.17 Optical paths in an immersion interferometer

and arrangement for U and V fields N = 1/4

3

/4

5

/4

7

/4

9

/4

x

Fig. 22.16a–d Stepsin the optical/digital fringe multiplica-

tion (O/DFM) algorithm

view for larger specimens, the displacement accumulated across the field will be small (even if the strains are not small); then, the number of fringes is not sufficient and additional data are desired. Phase stepping, also called fringe shifting, provides additional data. Referring to Fig. 22.9, we recall that the fringe pattern is a contour map of the separation S of wavefronts that emerge from the specimen. In phase stepping, the separation S is increased (or decreased) by a fraction of a wavelength, uniformly across the field. The result is a uniform change of fringe order throughout the field of view; the intensity at every pixel, as

measured by the digital camera, is changed. Usually, the phase change is accomplished by translating a mirror, or other optical element in the moiré interferometer, using a piezoelectric actuator to displace the element by a tiny increment. For an achromatic system like that of Fig. 22.7, the phase change can be executed by in-plane translation of the compensator grating. A common procedure for full-field data enhancement is the quasiheterodyne method [22.12, 13]. In its simplest implementation, three shifted images of the fringe pattern are recorded, with phase steps of 0, 2π/3, and 4π/3 (fringe shifts of 0, 1/3, and 2/3 of a fringe order). The CCD camera, frame grabber, and computer gather the data from the three images, recording the intensity at each pixel as I1 , I2 , I3 . With these data, the

Moiré Interferometry

0

500 μm

quasiheterodyne algorithm calculates the fractional part of the fringe order N  at each pixel by √ 3 (I2 − I3 ) 1 N = . arctan 2π 2I1 − I2 − I3 

U field 0

(22.9)

V field 500 μm

y z

Fig. 22.19 Microscopic moiré fringe contours for a solder ball interconnection in an electronic package. ΔT = 60 ◦ C; f = 4800 lines/mm; β = 4; 52 nm/contour

Then the fringe order of each pixel is determined by an unwrapping algorithm, whereby neighboring pixels are compared to determine whether an integral fringe order should be added (or subtracted) to the fractional part. Thus, fringe orders are established throughout the field of view. Various other implementations of the quasiheterodyne method provide flexibility and redundancy by using more than three phase steps. A lesser known algorithm, called the optical/digital fringe multiplication (O/DFM) method, has also been used for moiré interferometry [22.4]. It shares numerous features with the quasiheterodyne method, including insensitivity to nonuniform illumination and optical noise. It offers a unique feature, however, because the data that is used comes exclusively from the points in the fringe pattern where the intensity is changing most rapidly. Compared to data taken at (or near) the points of maximum and minimum intensity in the fringe pattern, the phase increment per grey level of the CCD camera is many times smaller (approximately 1 : 50), so the data of greatest accuracy is used exclusively. The algorithm is portrayed in Fig. 22.16 for multiplication by 2. Two fringe patterns are used: patterns stepped by 0 and by π (i. e., two complementary patterns). Their intensities at each pixel are subtracted, truncated, and binarized. The result is a contour map of the fringe orders, with two contours per fringe. Multiplication by 4 is obtained using data stepped by 0, π/2, π, and 3π/2. Multiplication by β requires β stepped fringe patterns. Multiplication by β = 12 has been demonstrated.

22.2.11 Microscopic Moiré Interferometry Within a tiny field of view, the relative displacements are small and few fringes appear across the field. Increased sensitivity is desired. Two techniques have been implemented to progressively increase the number of load-induced fringes. The basic sensitivity is increased by a factor of 2 by means of an immersion interferometer, and then phase stepping and the O/DFM algorithm are used to produce a suitably dense fringe pattern. Figure 22.17 illustrates an immersion interferometer [22.4]; many alternative designs are possible. This one was implemented, with λ = 514 nm (in air), α = 54.3◦ , and n = 1.52 (where n is the index of refraction of the interferometer block). Within the refractive medium the wavelength was reduced to 338 nm, providing a virtual reference grating frequency of 4800 lines/mm. This is twice the frequency used (without immersion) for most macroscopic applications, and

643

Part C 22.2

Fig. 22.18 Boron/aluminum composite subjected to a change of temperature of 110 ◦ C. Microscopic moiré interferometry was used with f = 4800 lines/mm and β = 6; 35 nm/contour

22.2 Important Concepts

644

Part C

Noncontact Methods

it exceeded the theoretical limit for virtual reference gratings produced in air. By (22.3), twice as many fringes are obtained for a given displacement, compared to designs without immersion. For phase stepping, the element labeled ‘immersion interferometer’ is moved laterally by a piezoelectric actuator. Movement of 1/ f causes a phase change of 2π, so β steps of 1/β f each are used to implement the fringe multiplication. Figure 22.18 shows an example. The specimen was a unidirectional boron/aluminum composite subjected to thermal loading. The bithermal method was used, where ΔT was 110 ◦ C. Here, f = 4800 lines/mm,

β = 6, and the contour interval was 35 nm per contour. The pattern shows the combined effects of thermal strains plus free thermal expansion (αΔT ). When the uniform αΔT is subtracted off for each material, the substantial range of stress-induced strains in the aluminum matrix becomes evident. Figure 22.19 is an example from electronic packaging, showing the displacement fields for a single solder interconnect. Here, ΔT = 60 ◦ C, f = 4800 lines/mm, β = 4, and the contour interval is 52 nm/contour. Details of the small deformation are clearly documented.

22.3 Challenges The authors visualize two developments that would be important contributions to the experimental mechanics community. We propose that members of the community, worldwide, start modest endeavors to develop these capabilities. The first would enhance the ease and accuracy of strain determinations. The second would further enhance the versatility of moiré interferometry.

Part C 22.3

22.3.1 Strain Analysis For most macromechanics analyses, moiré interferometry provides a great number of stress-induced fringes, sufficient for a detailed analysis. Yet, in current practice, there is a tendency to use phase-stepping schemes to reduce these data to graphs of displacements and strain distributions. These phase-stepping algorithms enjoy popularity mostly because they enable automatic data reduction. However, there are many pitfalls. Automated analyses do not recognize extraneous input such as those from scratches or other imperfections of the specimen grating. Automated analyses do not cope well with rapidly changing displacement fields, for example, those that can be encountered in composite structures; in particular, automated analyses must not be allowed to extend across regions of dissimilar materials. Thus, for the majority of applications, interactive schemes should be developed in which the investigator controls the process and makes the decisions. The moiré pattern is itself a map of deformation. The investigator should choose the regions of interest – usually the regions where the fringes are closest together and strains are highest. Techniques should be developed to extract these strains with the highest accuracy. The investigator should deal with possible imperfections in the fringe

patterns. He/she should decide whether carrier fringes would aid the analysis. The authors have proposed ideas on methods to explore and perfect. The challenge is published by the Society for Experimental Mechanics in Experimental Techniques [22.14], where the worldwide community is invited to participate. The objective is to devise and share a superior technique for extracting strains, and in an on-going evolution to improve and refine the technique into an easy, accurate, and efficient algorithm. The Society for Experimental Mechanics will maintain an open website to share the contributions and make them available to everyone.

22.3.2 Replication of Deformed Gratings In a classical paper by McKelvie and Walker [22.15], replication was advocated for remote sites and harsh environments, followed by analysis of the replicas in the laboratory. Now, replication is envisioned as a routine practice. In this practice, a grating is applied to the specimen or workpiece in the usual way. Then, the workpiece is subjected to its working loads, for example in a mechanical testing machine, which deforms the specimen and the specimen grating. Replicas of the deformed grating are made at desired intervals in the loading process, and subsequently the replicas are analyzed in a moiré interferometer. Replication provides numerous advantages for typical applications. No limits are applied to the size of the workpiece. Familiar equipment can be used for loading, including large or special-purpose machines. Vibrations and air currents become inconsequential. The technique can be applied in the field, far from the laboratory,

Moiré Interferometry

and in difficult environments. In many applications, replicas can be made for cases where the loading is not mechanical, but stems from changes of temperature, humidity, chemical or radiation environments, etc. The replicas can be made on transparent substrates, enabling use of transmission systems of moiré interferometry instead of the current reflection systems. When transmission systems are used, the camera lens can be located very close to the replica and, therefore, high magnifications and high spatial resolution become convenient. Additionally, the replicas become permanent records of the deformation, so there is easy recourse

22.4 Characterization of Moiré Interferometry

645

to checking and extending an analysis after an initial investigation. Thus, replication is extremely attractive for routine use as well as special applications. What developments are required? Although the replication method can be practised with current technology (Fig. 22.15 is a compelling example) it would be advantageous to find materials and techniques for quick and routine replication of deformed gratings. We believe these techniques will be optimized and broadly implemented; the experimental mechanics community is invited and encouraged to participate in their development.

22.4 Characterization of Moiré Interferometry

• • • • • •

full-field technique – quantitative measurements can be made throughout the field, high sensitivity to in-plane displacements U and V – typically 0.417 μm per fringe order, but extended into the nanometer range by microscopic moiré interferometry, insensitive to out-of-plane displacements W – outof-plane deformation does not affect the accuracy of in-plane displacement measurements, high spatial resolution – measurements can be made in tiny zones, high contrast – the fringe patterns have excellent visibility, large dynamic range – the method is compatible with large and small displacements, large and small

• •

strains, and large and small strain gradients; there are no correlation requirements, shear strains are determined as accurately as normal strains, real-time technique – the displacement fields can be viewed as the loads are applied.

Moiré interferometry differs from classical interferometry and holographic interferometry, which are most effective for measuring out-of-plane displacements. It is distinguished from the various methods of speckle interferometry for measuring in-plane displacements, which cannot exhibit the fringe visibility and extensive dynamic range of moiré interferometry. Moiré interferometry has a proven record of applications in engineering and science.

•

• •

Bibliography For a comprehensive treatment of moiré interferometry for strain and stress analysis; theory, practice and applications see [22.4]. Note: excerpts are included in this chapter with the kind permission of Springer-Verlag, Berlin, New York. For background literature on moiré phenomena and mathematical treatments, not directed to strain analysis, see [22.16–18] For a review article and historic references until the 1980s, see [22.19].

Part C 22.4

The preceding description and examples lead to these conclusions. Moiré interferometry combines the simplicity of geometrical moiré with the high sensitivity of optical interferometry. It measures in-plane displacements with very high sensitivity. Because of the great abundance of displacement data, reliable strain distributions – normal and shear strains – can be extracted from the patterns. With knowledge of the material properties, local and global stresses can be determined. Moiré interferometry is characterized by a list of excellent qualities, including

646

Part C

Noncontact Methods

22.5 Moiré Interferometry in the Microelectronics Industry

Part C 22.5

Diverse applications from the microelectronics industry are exhibited in this section. However, the studies are not presented in detail; the cited references provide the details. Instead, the raw data are emphasized, that is, the moiré fringe patterns that provide the wealth of information to be extracted from each experiment. The main objective of this section is to illustrate the high quality of the fringe patterns. Note that the patterns can be enlarged – e.g., 200–500% – to evaluate their quality more fully. Of course, when the raw data are clear and dependable, the experimental results are reliable. These fringe patterns are contour maps of the deformation caused by changes of temperature and/or moisture content. They are fully quantitative; fringe orders are readily assigned to every fringe (every contour) and the U or V displacement at each point along that contour is simply proportional to the fringe order. Moiré interferometry provides submicron sensitivity; microscopic moiré interferometry provides sensitivity in the nanometer range. Strains can be extracted, too, inasmuch as strains are determined by the rate of a change of fringe order, e.g., ΔN x /Δx; thus, strains are highest where the fringes are closest together. Moiré interferometry has become extremely important in the electronics industry for electronic packaging studies. The applications – mostly to evaluate thermal strains, but also for hygroscopic deformation, mechanical loading, and material characterization – are introduced at the design and development, evaluation, and process control stages. The following experiments illustrate the applications.

22.5.1 Temperature-Dependent Deformation Figure 22.13 illustrates data from a wire-bond plastic ball grid array (WB-PBGA) package subjected to a thermal cycle; maximum and minimum temperatures were 125 ◦ C and −40 ◦ C. The thermal deformation is complicated, exacerbated by creep and relaxation of the solder and molding compound. Please refer to [22.8] for analysis and discussion. Ceramic BGA Package Assembly The objective of this application was to study the temperature and time-dependent thermomechanical behavior of a ceramic ball grid array (CBGA) package assembly subjected to an accelerated thermal cycling

condition. The assembly was a 25 mm CBGA package with 361 input/outputs (I/Os) (19 × 19 solder interconnection array) on an FR-4 PCB. A specimen with a strip array configuration was prepared from the assembly, containing five central rows of solder interconnections. The solder interconnection of the package assembly consisted of a high-melting-point solder ball (90%Pb/10%Sn) and a lower-melting-point eutectic solder fillet (63%Pb/37%Sn), as illustrated by the insert in Fig. 22.20. The high-melting-point solder ball does not reflow during the assembly process, so it provides a consistent and reproducible standoff between the ceramic package and the PCB. Figure 22.20 shows the temperature profile used in the thermal cycle. To ensure a uniform temperature distribution, the specimen was kept at each target temperature for 5 min before measurement. The U and V fringe patterns with fringe orders N x and N y , respectively, were recorded at each of the lettered temperature levels. Representative patterns at 55 ◦ C (point B) are shown in Fig. 22.20. The dominant mode of deformation of the solder interconnection is shear, which is caused by the mismatch of the CTE of the ceramic module and the PCB. Consequently, the shear strains at the interconnection increase as the distance from the neutral point (DNP) increases, so the last solder ball experiences the highest shear; the second insert in Fig. 22.20 shows the fringes in a magnified view of this interconnection. Note that the fringe orders are marked on the fringe patterns. At the top of the rightmost interconnection, N x = 2.5 is counted along the upper section, starting from the N x = 0 fringe; similarly N x = 9.5 is counted along the lower section of the assembly. In the same way, the fringe orders at the top and bottom of the interconnection can be determined from patterns recorded at each temperature level. This solder interconnection was analyzed to investigate the effect of the two solder materials. The development of inelastic strains in the solder joints during the thermal cycle is seen in Fig. 22.21, in which the horizontal displacements along the vertical centerline are plotted for various stages in the thermal cycle. It is important to note that the reflow process produced a thin eutectic solder bench between the solder ball and the copper pad, while there was essentially no gap between the solder ball and the ceramic module. Consequently, the shear deformation of the top eutectic fillet was constrained by the solder ball, but the bottom

Moiré Interferometry

eutectic fillet was free to deform in shear. This sequence of fringe patterns shows that plastic deformation occurred at the higher temperatures. It was concentrated in the eutectic (lower-melting-point) solder, and this irreversible strain was present in all the subsequent patterns. Although the fringe patterns from the full specimen were recorded at each temperature level, the region isolated in this series was less than 1 mm2 . Informa-

a)

120 D

C E

5 min

60 B

40

I

F G

0 –20 –40

H

0

10

20

30

40

50

60

70

80 90 100 Time (min)

b)

i)

4

8 12

c)

2.5

1 0.8 0.6 0.4 0.2 0 –4 0

90Pb –10Sn ball Bottom eutectic fillet Copper pad 9.5

PCB

1 0.8 0.6 0.4 0.2 0 –4 0

Nx = 0

4

8 12

0

c)

5

10

Ny = 0

5

1 0.8 0.6 0.4 0.2 0 –4 0

4

8 12

e)

55°C (B: Heating)

Fig. 22.20a–c Temperature profile used in the thermal cycle(a); views of the rightmost solder interconnection (b); and horizontal and (c) vertical displacement fields of the

assembly at 55 ◦ C (point B), f = 2400 lines/mm

1 0.8 0.6 0.4 0.2 0 –4 0

4

8 12

4

8 12

4

8 12

4

8 12

h)

1 0.8 0.6 0.4 0.2 0 –4 0

g)

Cooling

d)

1 0.8 0.6 0.4 0.2 0 –4 0

Part C 22.5

Ceramic substrate

Eutectic fillet

b)

Room temperature

20 A

Center line

Heating

80

Solder height (normalized) 1 0.8 0.6 0.4 0.2 0 –4 0 4 8 12 Displacement (μm)

Heating

C'

647

tion from the large specimen was needed to establish fringe orders in the tiny interconnection. It is a remarkable feature of moiré interferometry that such detailed local results could be discerned even when the image magnification was low enough to view the full specimen. Please refer to [22.9] for analysis of the data and discussion of the results.

a) Temperature (°C) 100

22.5 Moiré Interferometry in the Microelectronics Industry

1 0.8 0.6 0.4 0.2 0 –4 0

f)

4

8 12

1 0.8 0.6 0.4 0.2 0 –4 0

Fig. 22.21a–f U fieldfringe patterns of the rightmost solder interconnection and the corresponding horizontal displacements determined along the vertical centerline

648

Part C

Noncontact Methods

22.5.2 Hygroscopic Deformation Plastic encapsulated microcircuits are used extensively. In spite of many advantages over hermetic packages in terms of size, weight, performance, and cost, an important disadvantage is that the polymeric mold compound absorbs moisture when exposed to a humid environment. Hygroscopic stresses arise when the mold compound and other polymeric materials swell upon absorbing moisture while the adjacent nonpolymeric materials, such as the lead frame, die paddle, and silicon chip, do not experience swelling. This differential swelling leads to hygroscopic mismatch stresses in the package. Moiré interferometry is used effectively to analyze both hygroscopic and thermal deformations.

Part C 22.5

Coefficient of Hygroscopic Swelling Figure 22.22 is taken from a series of tests to determine the coefficient of hygroscopic swelling (CHS) of a specific mold compound [22.20]. CHS measurements are made at an elevated temperature of 85 ◦ C. Two equal specimens were used in the test; one was a reference specimen, kept at 85 ◦ C and zero moisture content. The other begins the test at 85 ◦ C and moisture saturation; fringe patterns were recorded as moisture was desorbed as a function of time. Long time intervals were employed, as noted in Fig. 22.22, where the data extends from 0 to 400 h. Moisture content was determined at every interval by weighing the specimen. Figure 22.23 shows the results for three nominally equal samples, where a linear relationship is documented for longitudinal strain versus moisture content. Hygroscopic Stress in a Plastic Quad Flat Package (PQFP) Hygroscopic mismatch stresses are documented in [22.21]. The package and the specimen are illustrated in Fig. 22.24. The specimen was conditioned to zero moisture content and the bithermal loading method was used to record the patterns of Fig. 22.25. Figure 22.25a shows the starting condition – the null field representing the specimen at 85 ◦ C and zero moisture. Figure 22.25b shows the patterns for 25 ◦ C and zero moisture, thus depicting the deformation incurred by thermal stress alone. Subsequently, the specimen was brought to its moisture saturation level. Figure 22.26 shows the deformation at 85 ◦ C for moisture saturation and for moisture desorption after 8 h. These represent deformation from moisture change alone.

a)

b)

20 mm

13 mm

c)

d)

Fig. 22.22a–d V field moiré patterns obtained from a mold compound at 85 ◦ C. (a) Null field for the reference sample; (b) fringe patterns of the test sample at time intervals of zero, (c) 16 h, and (d) 400 h, f = 2400 lines/mm

εh (10 – 4 ) 16 14 12 10

y, V

8 x, U

6

Sample 1 -V Sample 2 -V Sample 3 -V

4 2 0

0

0.15

0.3

0.45 0.6 Moisture content (%)

Fig. 22.23 Hygroscopic strain versus moisture content ob-

tained from the moiré fringes

Moiré Interferometry

Specimen strip

x, U

Collimating lens

28 mm

Focusing lens

3.36 mm

A

A'

649

Reliability analyses must include predictive capabilities of hygroscopic swelling as well as thermal deformation. The analysis and discussion in [22.21]

y, V

z, W

22.5 Moiré Interferometry in the Microelectronics Industry

Package body (Kovar)

PQFP package

Fig. 22.24 PQFP package for moiré experiments

Chip carrier

Isolator

Chip

Fiber

Pb –Sn solder

Ceramic plate Package-base (CuW)

a)

Chip

Die paddle

Metal column

U field

U

b)

Focusing lens

TEC assembly

V Chip

Die paddle

U

V field

Part C 22.5

V

Fig. 22.25 (a) Null field patterns recorded at 85 ◦ C before moisture absorption, and (b) fringe patterns after cooling

the package to 25 ◦ C (ΔT = 60 ◦ C, zero moisture), f = 2400 lines/mm

a)

Chip

Die paddle

U V

b)

Chip

Direction of increasing fringe order

Solder mask

U

Metal via

V

Photosensitive material

8 h at 85 ◦ C, f = 2400 lines/mm

5 Scale (mm)

Fig. 22.27 Schematic illustration of a generic optoelectronics package, and the U and V patterns caused by the high-temperaturestorage (HTS) test. Note: the fringes are multiplied by O/DFM by a factor of β = 4; f = 2400 lines/mm; the contour interval is 0.104 μm/fringe

Die paddle

Fig. 22.26a,b Fringe patterns obtained at 85 ◦ C during the desorption process (a) at moisture saturation and (b) after

0

Plated-through hole Build-up layer Substrate core-glass/ epoxy panel

Power line

Fig. 22.28 Schematic diagram of a high-density organic substrate with built-up structures

650

Part C

Noncontact Methods

a) z

Fig. 22.29a,b Schematic diagram of the flipchip assembly on a high-density substrate (a) before and (b) after specimen preparation. SLC (surface laminar circuit); SLL (surface laminar layer)

b)

y

x

0.7 mm

10 mm

Chip

Sectioned plane Underfill

Solder mask

C4

1.25 mm

SLC substrate

tSLL = 190 μm

SLL

Metal via

a)

22.5.3 Standard Qualification Test of Optoelectronics Package

Chip Underfill

y Substrate DNP

x

Chip Underfill

Bump Metal via

Built-uplayer Substrate core Substrate

Part C 22.5

b)

Nx* = 0

5

c)

Flip-chip assembly

10

–5

5 0 5

U field

0

200

V field

U field Nx* = 0

–5

–10

–10

Ny* = 0

Ny* = 0 10

10

20

V field

Scale (μm)

Fig. 22.30 (a) Micrographs of the region of interest. Microscopic U and V displacement fields of (b) bare substrate and (c) flip-chip assembly, ΔT = −70 ◦ C; f = 4800 lines/mm; β = 4; the contour interval is 52 nm/fringe

shows that hygroscopic strains can exceed thermal strains, thus reflecting the importance of both.

Long-haul telecommunication systems are powered by high-performance semiconductor lasers. The light, generated by a laser chip, is coupled into an optical fiber using a system of lenses. The laser temperature is controlled by a thermoelectric cooler (TEC) in order to produce a constant wavelength. Figure 22.27 shows a generic package. The package body and the base are joined by brazing. The lenses and fiber are fixed in place by various means, such as solder, epoxy, and laser welding. Thus, the assembly is a complex system that consists of many materials with different thermal coefficients of expansion, joined by a hierarchy of attachment methods. The spot size of the light focused at the fiber is approximately equal to the diameter of the fiber core, typically 8 μm. Tiny displacements of any optical component are sufficient to shift the focused light spot away from the fiber core and induce a serious loss of coupling efficiency. Accordingly, characterization of the structural behavior is crucial to ensure a stable optical power output. An experimental investigation of the global deformation of the package–TEC assembly was conducted using moiré interferometry. The package–TEC assembly was cut to expose its cross-section, yet preserve the three-dimensional integrity of the assembly. The cross-section was ground flat and a specimen grating was replicated at room temperature. The interferometer was adjusted with the specimen to provide null fringe patterns in the U and V fields. This adjustment was preserved while the package assembly was subjected to the standard high-temperature storage (HTS) test condition (85 ◦ C for 100 h). Then, the specimen was cooled to

Moiré Interferometry

room temperature and the U and V fields were recorded. The resultant fringe patterns after 100 h of HTS are shown in Fig. 22.27, where the fringes were multiplied by the O/DFM method. The fringe multiplication factor was 4, and the resultant contour interval was 0.104 μm per fringe. The fringes document the deformation of the assembly after the HTS. The results show bending in the base and the attached TEC. These two dissimilar materials were joined by soldering at elevated temperature; when they cooled to room temperature, thermal stresses developed in the solder. Subsequently, the thermal stresses relaxed during the HTS test, resulting in the bending seen in the figure. These observation provides important guidelines for the placement of the laser chip and collimating lens mounted on the TEC. It is clear that they should be located toward the front of the TEC or closer to the focusing lens. Otherwise, the change in deformation would lead to a change in angle of the light beam, causing a loss of optical power coupled into the fiber [22.22].

22.5.4 Micromechanics Studies by Microscopic Moiré Interferometry

651

y

x

A

U field

PTH

B

0

300

V field

Scale (μm)

Fig. 22.31 Cross-sectional view of substrate with plated-throughholes; contour maps of the Uand V displacement fields for ΔT = −80 ◦ C, f = 4800 lines/mm; β = 2; the contour interval is 104 nm/fringe (Courtesy S. Cho, Intel)

detailed microstructures of interest. An epoxy specimen grating was applied at an elevated temperature of 92 ◦ C in a small region containing the microstructures. The fringes were recorded at a room temperature of 22 ◦ C, capturing the thermal deformation incurred by ΔT = −70 ◦ C. The critical region studied here is marked by a dashed box in Fig. 22.30a; it is approximately 500 μm by 375 μm. The resultant fringe patterns are shown in Fig. 22.30 for the bare substrate and the flip-chip assembly. The moiré interferometer produced a virtual reference grating of 4800 lines/mm; the O/DFM method was used to provide a fringe multiplication factor of β = 4. Thus, the contour interval in Fig. 22.30 is 52 nm/fringe order. Fringe orders are marked in the figure. These contour maps of the U and V displacement fields show a remarkable wealth of detail. The contours are unambiguous, clearly defined, and readily analyzed. Again, the reader should increase the magnification of the image to gain further appreciation of its clarity. Please see [22.23] for analysis and discussion.

Part C 22.5

Microvia in Built-Up Structure One of several purposes of a chip carrier, or substrate, is to provide conducting paths between the extremely compact circuits on the chip and the more widely spaced terminals on the PCB. Recent microvia technology enables the industry to produce laminate substrates with imbedded conductors of high density and fine pitch, as required for advanced assemblies. A cross-sectional view of a high-density organic substrate is illustrated in Fig. 22.28. To produce the substrate, photosensitive dielectric layers (insulators) are built up sequentially, with each layer containing a patterned film of copper plating, typically 25 μm thick, to form the imbedded conductors. In subsequent operations, solder bumps connect the active chip to the substrate, and an underfill adhesive is used to fill the air space between the chip and substrate. Microscopic moiré interferometry was employed to quantify the effect of the underfill on the deformation of the microstructures within the built-up layers. Two specimen configurations were analyzed: a bare substrate and a flip-chip package. The flip-chip assembly is illustrated schematically in Fig. 22.29a with its relevant dimensions. The specimens were cut and ground to expose the desired microstructures as illustrated schematically in Fig. 22.29b, where the insert depicts the

22.5 Moiré Interferometry in the Microelectronics Industry

652

Part C

Noncontact Methods

Local CTE Variations in High-Performance Substrates A high-performance substrate accommodates various electrical components such as central processing units (CPU), capacitors, and resisters. The substrate is a complex composite system, which includes layers of epoxy, woven fiberglass, and copper planes. For reliability assessment of first-level interconnects, the global CTE mismatch between the silicon chip (≈ 3 ppm/◦ C) and the substrate (≈ 17 ppm/◦ C), and also the local CTE variations, must be considered. Microscopic moiré interferometry was employed to determine the local CTE variations around two plated-through-holes (PTH). Figure 22.31 shows the microscopic U and V displacement fields of the PTH

area, induced by ΔT of −80 ◦ C, where the contour interval is 104 nm/fringe. The fringe patterns clearly show the homogenous nature of the plug material inside the PTH, indicated by the uniformly spaced U and V fringes, and the heterogeneous nature of the fiber–resin laminated areas located between the PTHs. For the regions marked ‘A’ and ‘B’ in Fig. 22.31, the effective CTE on the upper surface differ by 8 times in the U field (33.7 and 4.0 ppm/◦ C). This resulted from variations in the materials and features comprising the substrate. The local fluctuations of CTE around these features are the critical information needed to identify areas of failure risk within the substrate. Again, please view the patterns at increased magnification, and refer to [22.22] for additional information.

References 22.1 22.2

22.3

Part C 22

22.4

22.5 22.6

22.7

22.8

22.9

22.10

22.11

C.A. Walker: A historical review of moiré interferometry, Exp. Mech. 34(4), 281–299 (1994) E.M. Weissman, D. Post: Moiré interferometry near the theoretical limit, Appl. Opt. 21(9), 1621–1623 (1982) D. Post: Moiré fringe multiplication with a non-symmetrical doubly-blazed reference grating, Appl. Opt. 10(4), 901–907 (1971) D. Post, B. Han, P.G. Ifju: High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer, New York 1994) A.J. Durelli, V.J. Parks: Moiré Analysis of Strain (Prentice Hall, Englewood Cliffs 1970) Y. Guo, D. Post, R. Czarnek: The magic of carrier fringes in moiré interferometry, Exp. Mech. 29(2), 169–173 (1989) P.G. Ifju, D. Post: A compact double notched specimen for in-plane shear testing, Proc. SEM Conf. Experimental Mechanics (Society for Experimental Mechanics, Bethel 1989) pp. 337–342 S.-M. Cho, S.-Y. Cho, B. Han: Observing real-time thermal deformations in electronic packaging, Exp. Tech. 26(3), 25 (2002) S.-M. Cho, B. Han, J. Joo: Temperature dependent deformation analysis of ball grid array package assembly under accelerated thermal cycling condition, J. Electron. Packag. Trans. ASME 126, 41–47 (2004) D. Post, J.D. Wood, B. Han, V.J. Parks, F.P. Gerstle Jr.: Thermal stresses in a bimaterial joint: an experimental analysis, ASME J. Appl. Mech. 61(1), 192–198 (1994) D.H. Mollenhauer, K.L. Reifsnider: Interlaminar deformation along the cylindrical surface of a hole in laminated composites – experimental analysis

22.12 22.13

22.14 22.15

22.16 22.17 22.18

22.19 22.20

22.21

22.22

by moiré interferometry, J. Compos. Technol. Res. 23(3), 177–188 (2001) K. Creath: Phase measurement interferometry techniques, Prog. Opt. 26, 349–393 (1988) K. Creath: Temporal phase measurement methods. In: Interferogram Analysis, ed. by D.W. Robinson, G.T. Reid (Institute of Physics, London 1993) pp. 94– 140 D. Post: Rapid analysis of moiré fringes, Exp. Techn. 29(5), 12–14 (2005) J. McKelvie, C.A. Walker: A practical multiplied moiré-fringe technique, Exp. Mech. 18(8), 316–320 (1978) K. Patorski: Handbook of the Moiré Fringe Technique (Elsevier, New York 1993) O. Kafri, I. Glatt: The Physics of Moiré Metrology (Wiley, New York 1990) J. Guild: The Interference Systems of Crossed Diffraction Gratings, Theory of Moiré Fringes (Oxford Univ. Press, New York 1956) C.A. Sciammarella: The moiré method, A review, Exp. Mech. 22(11), 418–433 (1982) E. Stellrecht, B. Han, M. Pecht: Measurement of the hygroscopic swelling coefficient in mold compounds using moiré interferometry, Exp. Techn. 27(4), 40–44 (2003) E. Stellrecht, B. Han, M. Pecht: Characterization of hygroscopic swelling behavior of mold compounds and plastic packages, IEEE Trans. Compon. Packag. Technol. 27(3), 499–506 (2004) B. Han: Characterization of stresses and strains in microelectronic and photonic devices using photomechanics methods. In: Micro- and Optoelectronic Materials and Structures, ed. by Y.C. Lee, A. Suhir (Springer, New York 2007)

Moiré Interferometry

22.23

B. Han, P. Kunthong: Micro-mechanical deformation analysis of surface laminar circuit in organic

References

653

flip-chip package: An experimental study, J. Electron. Packag. Trans. ASME 122(3), 294–300 (2000)

Part C 22

655

Speckle Meth 23. Speckle Methods

Yimin Gan, Wolfgang Steinchen (deceased)

Speckle methods, which are based on the wellknown speckle effect, play an important role in experimental mechanics. The formation and some important properties of speckle pattern are introduced firstly. Dealing with changes in surface intensity and positions of speckle patterns, speckle methods can be grouped into speckle interferometry and speckle correlation. The basic concepts and evaluations of speckle interferometry and speckle correlation are reviewed. Shearography, a special class of speckle interferometry suitable for measurement in the field (e.g., industrial applications), is introduced and used as a representative method for strain measurement and nondestructive testing (NDT).

23.2 Speckle Metrology ................................ 23.2.1 Speckle Correlation ....................... 23.2.2 Speckle Interferometry.................. 23.2.3 Shearography .............................. 23.2.4 Quantitative Evaluation (SI) ...........

658 658 660 662 665

23.3 Applications ......................................... 668 23.3.1 NDT/NDE ...................................... 668 23.3.2 Strain Measurement ..................... 670 23.4 Bibliography ........................................ 672 References .................................................. 672

lished by von Laue [23.3], who also worked out the first- and second-order speckle probability density functions. A noisy, random granular speckle pattern is also a carrier of important information. The first metrological application using a speckle pattern was published by Groh [23.4] in 1970. Today, speckle metrology is an important and rapidly growing area of optical metrology in experimental mechanics.

23.1 Laser Speckle 23.1.1 Laser Speckle Phenomenon When a temporally and spatially coherent laser beam is incident on a surface that is optically rough (with a surface roughness Rz that is greater than the laser wavelength λ), the microscopic small waves of the scattered light obtain a different phase relative to the phase of the incident wave. The dephased, but still coher-

ent wavelets interfere constructively (in antiphase) or destructively (in equiphase) in space and generate a statistically distributed, granular spatial speckle pattern. Such granular patterns can also be observed for optically rough surfaces illuminated by much less coherent light, e.g., mercury lamp and laser diode. According to the experimental setup, speckles can be divided into two categories: objective and subjec-

Part C 23

Speckle pattern is an interference phenomenon which has been investigated since the time of Newton. The earliest observation of the phenomenon of speckling appears to have been by Exner [23.1], nearly a century ago, in connection with a study of the Fraunhofer rings formed when a beam of coherent light is diffracted by a number of particles of the same size distributed at random [23.2]. A photograph of the Fraunhofer rings covered with a speckle pattern was first pub-

23.1 Laser Speckle........................................ 655 23.1.1 Laser Speckle Phenomenon ........... 655 23.1.2 Some Properties of Speckles .......... 656

656

Part C

Noncontact Methods

Illuminating laser wave

Coherent light

Observation plane

k dz Speckle in space D

ds

Speckles on CCD chip Lens Rough object surface

Image z Surface roughness

z

Fig. 23.2 Formation of subjective speckle patterns

Fig. 23.1 Formation of objective speckles in free space

Part C 23.1

tive speckles. Laser speckles existing in free space are known as objective or far-field speckles (Fig. 23.1). When an imaging system is used to observe speckle, the recorded information is known as a subjective speckle pattern (Fig. 23.2), in which case the properties depend on the imaging system. Subjective speckles are formed by the superposition of complex amplitudes of scattered wavelets in the image plane. Each point of the image plane registers only light beams which are reflected by the illuminated part of the object surface. Therefore they depend on the scattered light collected by the image aperture, and the speckle size is determined by the spatial frequencies passed through the lens system [23.5]. An observer who looks at the object surface perceives the subjective speckle effect because the human eye has a similar aperture and lens optics. Usually the speckle pattern in speckle metrology is observed and recorded by an optical system similar to that shown in Fig. 23.2 and the subjective or image speckles are simply called speckles.

23.1.2 Some Properties of Speckles Statistical Intensity Distribution Stochastic spatially distributed speckles are best described quantitatively by probability and statistics [23.6]. A laser emits linearly polarized light, but the direction of preference becomes lost when the rays are reflected by an optically rough surface. When the intensity field incident at the point P(x, y, z) on the object surface is perfectly monochromatic and perfectly polarized, the field is then represented by the following signal:

U(x, y, z, t) = A(x, y, z) exp[i(2π ft)] ,

(23.1)

where f is the optical frequency and A(x, y, z) is the complex phasor amplitude. The directly observable magnitude is the intensity at P(x, y, z), which is given by 1 I (x, y, z) = lim T f →∞ T f

+T  f/2

|U(x, y, z; t)|2 dt −T f/2

= |a(x, y, z)|2 ,

(23.2)

where T f is the exposure time. The complex phasor amplitude at point P(x, y, z) of the surface can be described mathematically as A(x, y, z) =

N  |ak | ei Φk ,

(23.3)

k=1

where ak is the amplitude and Φk is the phase of the k-th scattered elementary wave. In a fully developed p (I ) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

⎛ 1 ⎛ 1 exp ⎜ ⎜ I≥0 I ⎝ I ⎝

0

1

2

3

4

5 Intensity I / I

Fig. 23.3 Probability density function based on numerous measurements taken from a speckle pattern

Speckle Methods

speckle pattern ak and Φk are statistically independent of each other and the phase Φk is uniformly distributed between −π and +π. The complex amplitude of the speckle pattern obeys circular Gaussian statistics in the complex plane. The resulting intensity of the speckle pattern has a negative exponential probability distribution given by p(I ) =

  −I 1 exp I  I 

I ≥0,

ds ≈ 2.44

λf dAp

⎛2J (x) ⎛ 2 y= ⎜ 1 ⎜ ⎝ x ⎝

x=

2πa sin α λ

Fig. 23.4 Intensity distribution varying with the diffraction order of a plane wave through a circular aperture

diffraction angle Φ ≈ 1.22λ/2r within the boundary of resolution is given by ds ≈ 2.44

λf . dAp

(23.5)

In most imaging systems, the F-number F = f/dAp and the laser wavelength λ are known quantities. The speckle size increases with increasing F-number and laser wavelength. This characteristic of the subjective speckles is useful because the speckle size can be adjusted by the aperture lens for different measuring problem requirements [23.7–10]. Speckle size can also be extracted using an autocorrelation function, as described by Goodman [23.5]. The results of both methods are identical. Definiteness of Laser Speckles The speckles, which are randomly distributed in space, are irregularly shaped. The speckle pattern, however, is temporally constant and spatially determined. Each surface structure generates its own speckle pattern or wavefront in space. Thus, the speckle pattern provides a fingerprint of the illuminated area for the microstructure. When the microstructure of the object surface is changed due to movement or deformation, the observed speckle pattern changes accordingly. Because of this property, the speckle pattern can be used as a tool for metrological applications.

Part C 23.1

Speckle Size The scattered wavefront from the neighboring points P1 and P2 of the object surface are phase shifted as a result of the chain of hills and valleys created by its roughness, and the conjugated image points in the observation plane are P1 and P2 . Each image point yields an intensity field, as shown in Fig. 23.4, for a diffraction restricted by the point-spread function (PSF). The Fraunhofer diffraction function I = I0 {2J1 [2πa sin(a)/λ][2πa sin(α)λ]}2 yields the intensity of the Airy disk for perpendicular incidence, where a = dAp /2, α is the angle of the diffracted light deviating from the normal direction, and J1 is the Bessel function of first order. The phase shift of the intensity distributions is caused by a difference in the light waves; when the neighboring point-spread functions are overlapping, constructive or destructive interference is formed (i. e., microscopic interference). The overlapping region caused by the diffraction of the aperture plate is determined by the diameter of the aperture dAp , which is on the order of the size of Airy disk and corresponds to the diameter of the speckle ds (23.5) where f is the distance from the lens to the image plane [23.7]. The diameter of the first dark ring observed at the

657

I I0

(23.4)

where I  is the mean value [23.6]. The negative exponential distribution p(I ) (Fig. 23.3) establishes that the intensity I of the point P(x, y, z) lies between I and I + dI . The most probable intensity of the speckle distribution in the image plane is zero. The contrast γ of the speckle pattern is defined as γ = σ/I . Because the standard deviation σ and the mean value I  are identical, a speckle pattern which shows a good contrast. The speckle intensity and speckle phase are the quantities in speckle interferometry from which displacement, strain, surface roughness etc. are derived by means of first-order statistics. The second-order statistics of the scattered field contains the description of the spatial structure of the speckle field.

23.1 Laser Speckle

658

Part C

Noncontact Methods

23.2 Speckle Metrology A single speckle pattern contains information about the surface microstructure. However, it is not possible to deduce the phase information from this individual interferogram. At least two speckle patterns belonging to the same surface structure are required to compare separate states of deformation. A comparison of the two speckle patterns obtained from the object surface provides two sets of information: a change in the surface geometry and a change in the surface intensity distribution. A speckle pattern experiences both changes simultaneously. However, for many applications these changes can be studied separately. Measuring techniques dealing with changes in surface intensity are grouped under speckle interferometry (SI) and those dealing with geometric changes are grouped under speckle correlation (SC).

23.2.1 Speckle Correlation

Part C 23.2

When positional shifting between two speckle patterns in the image plane is studied, the method is known as speckle photography (SP) or speckle correlation (SC). These technique were developed around 1968 by Burch and Tokarski [23.11]. Archbold and Ennos [23.12] recorded two states (before and after loading) of an object surface illuminated with an expanded laser beam on a single photographic plate. When the specklegram developed by this process is illuminated by a narrow laser beam, Young’s fringes result in the far field; their frequency carries information about the displacement between the two exposures. Recently, speckle patterns have been recorded digitally by charge-coupled device (CCD) sensors and stored in separate frames. Figure 23.5 shows the experimental setup. The displacement field can be determined numerically using a two-dimensional cross-correlation

algorithm. Digital speckle correlation (DSC), also known as digital speckle photography (DSP), is much faster and easier to use than conventional SC because there is no need for film development or additional processing for reconstruction. Principles of Digital Speckle Correlation (DSC) In DSC the two speckle patterns I and I  acquired before and after deformation are usually divided into a set of subimages. The cross-correlation algorithm does not track the position change of each speckle, but rather the movement of a number of speckles acting together as a subimage. The correlation function is calculated for each pair of corresponding subimages and the respective displacement is derived from the position of the maximum. Figure 23.6 schematically illustrates the principle of calculating the displacement field. A subimage A in image I is allowed to sweep over the image I  . When an area is found where the statistical agreement (crosscorrelation) is highest, this area is labeled as subimage A and is considered to correspond to subimage A. The discrete cross-correlation between A and A can be calculated as

R A A (dx , d y ) =

N−1 M−1 1  A(i, j) NM i=0 j=0



A (i + dx , j + d y ) ;

the discrete normalized cross-covariance is defined as r A A (dx , d y ) =

R A A − A¯ A¯  , σ A σ A

Beam exclusion x y

z

A¯ =

N M 1  A(i, j) NM

and

i=0 j=0

θ Camera

Imaging lens Aperture

(23.7)

where dx and d y are displacements in the x- and ydirections, respectively.

Laser

Specimen

(23.6)

To image processing system

Fig. 23.5 The experimental setup for digital speckle correlation

A¯  =

N M 1   A (i + dx , j + d y ) NM i=0 j=0

are the mean values of subimages A and A .

Speckle Methods

23.2 Speckle Metrology

659

Fig. 23.6 Principle of displacement field measurement [23.13]

1 0.5 0 –16

–8

0

8

–16

–8

0

8

16

σ A and σ A are the variances of A and A and can be calculated as σ A2 =

N M 1  2 A (i, j) − A¯ 2 NM

and

i=0 j=0

σ A2  =

N M 1   2 A (i, j) − A¯ 2 . NM i=0 j=0

R A A (dx , d y ) = FT −1 [FT (A)∗ FT (A )] ,

(23.8)

White-Light Speckle Correlation The deformation or movement of an object can be detected by calculating the positional changes of two speckle patterns according to two object states using

Part C 23.2

The movement over the surface from position A to A is found by the position of the correlation peak and is given by the displacement vector of the midpoint of subimage A. These calculations are done for all subimages of I until a displacement field of the whole surface is obtained, meaning that the movement is determined in two directions, x and y. The height of the correlation peak indicates how similar the cross-correlated subimages are and hence yields a value of the accuracy of the measurement. Usually, the complete cross-correlation function is required and the direct calculation by (23.7) is rather time consuming. It is much more efficient to perform the cross-correlation in the Fourier domain by using a fast Fourier transform (FFT) algorithm. In virtue of the Wiener–Khintchine theorem [23.13], the crosscorrelation is the Fourier transform of the cross-power spectrum which can be calculated as

where FT and FT −1 are the discrete Fourier transform and its inverse, respectively. The asterisk denotes the complex conjugate. The procedure is illustrated in Fig. 23.7. The size limitation of the object being studied depends mainly on the magnification of the imaging system. The accuracy of the correlation technique is about 1% of the pixel pitch (i. e., it depends on the size of the studied area, the magnification, and the number of pixels of the detector). If the object being studied is small, smaller displacements can be determined than if the same detector were used to study a larger object. A drawback with this method is that the images are divided into subimages, thereby lowering the spatial resolution. The larger the the size of subimages, the lower the resolution. Using the conventional imaging geometry of SC the three-dimensional deformation can be obtained from a pair of images by the tilt of an object γ against the starting position, i. e., the gradient of the outof-plane deformation determined from corresponding subimages with a subsequent integration over the whole image (Fig. 23.2). The tilt can be calculated from the frequency dependence of the two correlating speckle pattern [23.13].

660

Part C

Noncontact Methods

Fig. 23.7 Flow chart of the cor-

relation algorithm using Fourier transformation [23.13] FT

1

Δx

x

0.5

FT –1 1 –8

–4

0

4

8

I (pixel)

Part C 23.2

DSC. Actually, speckle patterns need not be generated by laser; they can also be formed artificially. In practical applications the object surface is first painted matt white and then sprayed with black speckles. When the object surface is illuminated by white light, the image of the artificial speckle pattern looks like the image of a laser speckle pattern. Since the white-light speckle pattern is random, the cross-correlation algorithms can be used to determine the positional changes of the white-light speckle pattern due to object deformation or movement. Since the artificial speckles are larger than the real laser speckles, white-light speckle correlation, which is

also called digital image correlation (DIC), is not as sensitive as laser speckle correlation (SC). The lower sensitivity of DIC can be a positive attribute, since it can be used in noisy conditions. More about this technique can be found in the chapter Image Correlation Chap. 20 in this Handbook.

23.2.2 Speckle Interferometry An alternative speckle measuring technique is speckle interferometry (SI). Unlike the speckle correlation, which measures the positional changes of speckles, the M

Laser

Monitor Semitransparent mirror

Nondeformation

Reference beam Image processing x

ks Object beam Aperture lens

Deformation a

υ z

CCD sensor

Object

Fig. 23.8 Speckle interferometer for measuring the displacements of the object which occur mainly in the direction

vertical to the surface of the object

Speckle Methods

phase changes carried by intensity changes of speckles are studied in speckle interferometry. Principles of Speckle Interferometry A schematic setup of a speckle interferometer is shown in Fig. 23.8. A reference beam and the object beam, which is scattered from the surface of the object, superimpose and generate an interfered speckle pattern in the image plane, which is known as a speckle interferogram. These speckle patterns therefore contain essential information about the random phase. A point P(x, y) in the speckle interferogram can be expressed as a product of complex and conjugate complex terms by using  ∗  I = ao eiΘo + ar eiΘr ao eiΘo + ar eiΘr

= ao2 + ar2 + 2ao ar cos(Θo − Θr ) = ao2 + ar2 + 2ao ar cos(φo r ) ,

(23.9)

where φo r = Θo − Θr is the random phase; ao eiΘo and ar eiΘr are a temporally constant object beam and reference beam, respectively; and ao , ar , and Θo , Θr are the amplitude and phase of the object beam and reference beam, respectively. When the object is deformed, the interferogram changes due to the changes in the object wave. The intensity I  at the same point P in the interferogram after loading can be described as

661

can be observed in quasi real time. The intensity at a point P(x, y) can be interpreted as Is = |I − I  | = 2ao ar |cos(φor ) − cos(φor + Δ)|  Δ Δ   sin = 4ao ar sin φor + . 2 2

(23.11)

The above equation represents a high-frequency carrier sin(φ + Δ/2) modulated by a low-frequency factor sin(Δ/2), which depends on small local deformations of the object due to loading. If each speckle covers 1 pixel, the pixels will be black (Is = 0) when the intensity of both images are identical, namely Δ = k2π (k = 0, 1, 2, 3. . .). The brightness of the pixel in the resulting image Is will increase as the difference between the intensities I and I  increases. Pixels with the same brightness generate macroscopic lines (fringes) in the resulting image Is . Fringe Interpretation The visible fringes of an interferogram describe the distribution of relative phase changes. Figure 23.9 shows the geometric relation between the illumination vector k1 , the observation vector k2 , and the deformation vector of a point P on an object surface d. The relative phase difference can be shown to be

Δ = k2 d − k1 d = ks d ,

(23.12)

where ks = k2 − k1 is called the sensitivity vector. In x, y, z-coordinates, (23.12) can be expressed as

I  = a o + ar2 + 2a o ar cos(Θo − Θr ) 2

(23.10)

where Δ = φo r − φo r is the phase change between the two states. Since the change of the amplitude in the object wave can usually be neglected (i. e., ao ≈ ao ), the difference between the intensities I and I  is only a phase change Δ. In classic speckle interferometry, two speckle patterns corresponding to the object before and after deformation of object are recorded on a single photographic plate. During reconstruction, the developed photographic plate is illuminated by the same reference beam to observe and evaluate the correlation fringes. In recent speckle interferometry, which is also called electronic speckle pattern interferometry (ESPI), two speckle patterns are recorded by a CCD chip [or a complementary metal–oxide–semiconductor (CMOS) chip] separately and stored in a computer. The fringes can be generated numerically by digital subtraction and

Δ = ks (uex + ve y + wez ) = uks ex + vks e y + wks ez , (23.13)

where u, v, and w are the components of the deformation vector and ex , e y , and ez are the unit vectors in the x-, y-, and z-directions, respectively. The sensitivity L k1 P' P

d k2 0

Fig. 23.9 Geometric relation of the light source L, observation position O and the object P

Part C 23.2

2 = a o + ar2 + 2a o ar cos(φor + Δ) 2  = a o + ar2 + 2a o ar cos(φor ),

23.2 Speckle Metrology

662

Part C

Noncontact Methods

Direction of illumination

Laser

Laser Object

Image plane

1

Object ks1

1

ks2 2

θxz+

θ1xz θ2xz

2 1

Camera θxz–

2 x z

y

x

Observation direction

y

Fig. 23.10 Sensitivity vector and point of the object surface

z

Laser

Fig. 23.12 Optical arrangement for measuring in-plane dis-

placement

vector ks lies along the bisector of the angle between the illumination and viewing directions and is theoretically not the same for each point of the object surface due to the changing illumination and observation directions (Fig. 23.10). However, it can be assumed that the sensitivity vector ks at each point of the investigated surface is equal when the dimensions of the object are small compared with the distances between the laser and the object or the camera and the object. To determine the sensitivity vector ks , the center point of the object is chosen (Fig. 23.11). Its magnitude can be expressed as |ks | = (4π/λ) cos(θxz /2) ,

(23.14)

Part C 23.2

where λ is the wavelength of the laser light and θxz is the angle between the illumination and observation vectors in the x, z-plane. If the illumination and observation vectors are perpendicular to the object surface, namely ks ez = 1 and ks e y = ks ex = 0, the relative phase difference is therefore Δ=

4π w. λ

The second in-plane displacement Δ yz can be measured similarly when both laser beams illuminate the object in the y, z-plane.

23.2.3 Shearography Laser

Object θxz ks

ex

Camera

ez x z

(23.16)

(23.15)

Direction of illumination

y

Since only the out-of-plane displacement is measured, this kind of interferometer is known as an out-of-planesensitive interferometer. For measuring in-plane displacements, the optical layout illustrated in Fig. 23.12 is often used. Two laser beams are arranged symmetrically on each side of the object with identical angles of incidence. The angle between the illumination vector and the observation vector + = θ − = θ. When the obis the same for both beams, θxz xz ject is illuminated by two laser beams simultaneously, the relative phase change or the displacements in the x-direction can be written as 2π  + − u(sin θxz + sin θxz ) Δxz = λ 4π + − + w(cos θxz − cos θxz ) = u sin θ . λ

Observation direction

Fig. 23.11 Simplified sensitivity vector ks

Speckle pattern shearing interferometry (SPSI), which is more easily called shearography, was first investigated for direct measurement of the first derivative of deformation by Leendertz and Butters [23.14] and Hung [23.15]. Unlike other methods of speckle interferometry, e.g., ESPI, the interference in shearography is generated by two identical, laterally sheared object beams. This effect can be realized by a shear element which is located between the object and camera. Since no additional reference beam is needed, the setup for shearography is simple. Furthermore, compared with ESPI, shearography is relatively insensitive to distur-

Speckle Methods

23.2 Speckle Metrology

663

Expanding lens Nondeformation Amplifier Mirror 2 L

Θ Object

a s e r

ks

x Mirror 1

z y

1

Deformation

1'

2 2'

PC

D/A transf. Frame grabber

CCD camera Measuring device

ks: sensitivity vector

Fig. 23.13 The experimental setup of digital shearography using single-beam illumination

bances from the environment and is very suitable for measurement under industrial conditions.

  I  = a1 ei(Θ1 +Δ1 ) + a2 ei(Θ2 +Δ2 )  i(Θ +Δ ) ∗ a1 e 1 1 + a2 ei(Θ2 +Δ2 ) = a12 + a22 + 2a1 a2 cos(Θ1 − Θ2 + Δ1 − Δ2 ) = a12 + a22 + 2a1 a2 cos(φ1 2 + Δ12 ) ,

(23.18)

where Δ12 = Δ1 − Δ2 is the relative phase difference between the two phase changes at points P1 and P2 .

Part C 23.2

Principle of Digital Shearography The experimental setup of digital shearography with single-beam illumination is shown in Fig. 23.13. The tested object is illuminated by an expanded laser beam. The light reflected from the object surface is focused on the image plane of an image CCD camera, where a modified Michelson interferometer is implemented in front of its lens instead of the optical shearing element. A pair of laterally sheared images of the tested object is generated on the image plane of the CCD camera by turning mirror 1 of the Michelson interferometer to a very small angle from the normal position. Two images with a certain shearing δ interfere with each other and generate a shearing speckle pattern interferogram. The rays from P1 = P1 (x, y) on the object are mapped into two points P1 = P1 (x, y) and P1 = P1 (x + δx  , y) in the image plane (Fig. 23.14). Similarly, the point P2 = P2 (x + δx  , y) on the object surface is mapped to two points P2 = P2 (x + δx  , y) and P2 = P2 (x + 2δx  , y), etc. At the point (x + δx  , y) in the shearographic interferogram P2 and P1 are superposed. The intensity at this point can be expressed as:  ∗  I = a1 eiΘ1 + a2 eiΘ2 a1 eiΘ1 + a2 eiΘ2

ment of the object. This optical path change induces a relative phase change between the two interfering points. Thus, the intensity distribution of the speckle pattern is slightly altered and is mathematically represented by

Mirror 1

Beam splitter 50/50 δx

P2 P1 Mirror 2

= a12 + a22 + 2a1 a2 cos(Θ1 − Θ2 ) = a12 + a22 + 2a1 a2 cos(φ1 2 ) ,

(23.17)

where φ12 = Θ1 − Θ2 is the random phase, and a1 eiΘ1 and a2 eiΘ2 are temporally constant object beams from P1 and P2 , respectively. When the object is deformed, an optical path change occurs due to the surface displace-

P1'' P1' P2'' P2'

Image plane

δx' δx'

Fig. 23.14 Schematic of shearing interferometry

664

Part C

Noncontact Methods

Pixel-by-pixel digital subtraction of the two intensity distributions yields a macroscopic fringe pattern, (i. e., the shearogram or shearographic correlogram). Fringe Interpretation in Shearography As mentioned earlier, in shearography the deformation derivatives can be measured directly. Using (23.13) the relative phase difference Δ12 can be written as

Δ12 = Δ1 − Δ2 = u 1 ks ex + v1 ks e y + w1 ks ez − u 2 ks ex + v2 ks e y + w2 ks ez = δuks ex + δvks e y + δwks ez .

(23.19)

When the images are sheared in the x-direction, (23.19) is represented by  Δx = δx

 δu δv δw ks ex + ks e y + ks ez , δx δx δx

(23.20)

vector lies in the x, z-plane as well. From Fig. 23.11 and (23.14) it follows that ks ex = |ks | sin(θxz /2) = (4π/λ) cos(θxz /2) sin(θxz /2) = (2π/λ) sin θxz , ks e y = 0, and ks ez = |ks | cos(θxz /2) = (4π/λ) cos(θxz /2) cos(θxz /2) = (2π/λ)(1 + cos θxz ) . Given (23.22) and (23.23), it follows that   ∂u ∂v ∂w ks ex + ks e y + ks ez Δx = δx ∂x ∂x ∂x

 ∂u ∂w 2πδx sin θxz + (1 + cos θxz ) , = λ ∂x ∂x 

where δx is the amount of shear in the x-direction on the object surface. Similarly, for shearing in the y-direction, the relative phase change is

(23.24)



∂u ∂v ∂w ks ex + ks e y + ks ez ∂y ∂y ∂y

 ∂u ∂w 2πδy sin θxz + (1 + cos θxz ) . = λ ∂y ∂y

Δ y = δy

(23.25)



 δu δv δw ks ex + ks e y + ks ez , Δ y = δy δy δy δy

(23.21)

Part C 23.2

where δy is the amount of shear in the y-direction on the object surface. For very small shears ∂x and ∂y, (23.20) and (23.21) can be rewritten as  Δx = δx

∂u ∂v ∂w ks ex + ks e y + ks ez ∂x ∂x ∂x

Similarly, when the direction of illumination is in the y, z-plane, the resulting equations are

 ∂u 2πδx ∂w sin θ yz + (1 + cos θ yz ) , Δx = λ ∂x ∂x

 (23.22)

Δy =



(23.26)

∂v 2πδy ∂w sin θ yz + (1 + cos θ yz ) . λ ∂y ∂y (23.27)

and 

 ∂u ∂v ∂w Δ y = δy ks ex + ks e y + ks ez . ∂y ∂y ∂y

(23.23)

Equations (23.22) and (23.23) are the fundamental equations for shearography and describe the wholefield correlation fringes as contours of constant first derivative of deformation instead of constant contours of deformation as in ESPI. For this reason, the shearogram depicts the strain concentrations directly and it is therefore suitable for nondestructive testing (NDT). Equations (23.22) and (23.23) can be simplified by adjusting the direction of illumination. When the direction of illumination is in the x, z-plane, the sensitivity

In these equations the illumination angles are characterized by θxz or θ yz , and the shearing displacements by Δx or Δ y . As shown in (23.24)–(23.27) a shearogram usually contains the in-plane as well as the out-of-plane terms of the strain tensor. When the illumination direction is adjusted normal to the object surface, the angle of illumination θ becomes zero; thus sin θ = 0 and cos θ = 1. The out-of-plane components ∂w/∂x and ∂w/∂y can be obtained from the following equations: 4πδx ∂w , λ ∂x 4πδy ∂w Δy = . λ ∂y

Δx =

(23.28) (23.29)

Speckle Methods

a)

23.2 Speckle Metrology

665

b)

c)

50

50

0

0

–50

0.0001

–50

0.00005

0.004

0 0.002 –0.00005 0

–0.0001

–50

–50

0

0 50

50

d)

e)

0

∂w/∂x = –50 μm/mm

Fig. 23.15a–e Comparison shearography and ESPI for measuring the same out-of-plane deformation. (a) A shearogram, (b) a ESPI interferogram (c) evaluated shearogram and ESPI interferogram in 3-D view, (d) sectional plane ∂w/∂x, (e) w

To demonstrate the differences between the characteristic fringe pattern of digital shearography and that of ESPI, a circumferentially fixed circular aluminium plate, with a diameter of 150 mm and a thickness of 3 mm, was evaluated under the same loading condition by means of both speckle interferometric methods. Figure 23.15 shows the comparison of the results.

23.2.4 Quantitative Evaluation (SI) Relative phase change can be determined numerically from a single fringe pattern by various methods of fringe analysis such as fringe tracking and Fourier transformation. However, these methods can not be automated due to the unavoidable need for human– computer interaction. The phase-shifting technique

Part C 23.2

w = 6 μm

666

Part C

Noncontact Methods

introduces the possibility of automatic evaluation. Actually, relative phase change between two states can be calculated directly by using Δ = φ − φ , when the phase distributions φ and φ are calculated by applying the phase-shifting technique. The phase distribution φ is the first state of loading (no load or preload) of the object. The phase distribution φ corresponds to the second state of loading (or the second state of deformation). Different methods of phase shifting can be applied for determining the phase distribution of an object state numerically and automatically. These methods can generally be divided into two categories: time-dependent (temporal) and spatial phase-shifting techniques. The temporal phase-shifting method is the most powerful technique for determining the phase distribution. With this method, N (N ≥ 3) frames of the same deformation state are stored sequentially to obtain a solvable system of equations consisting of three unknowns. a)

With the phase-shifting technique, however, a single frame is sufficient to determine the phase distribution. Thus, phase-shifting can be applied for vibration analysis, especially for measuring transient processes with a double-pulse laser. It is important to note that the determination of phase distribution by this technique is not as accurate as with the temporal method. Temporal Phase Shifting (TPS) Phase shifting can be executed by a wedge-shaped or parallel glass plate in front of a camera lens by moving a mirror a predetermined amount in the reference beam of an ESPI arrangement (Fig. 23.8) or by moving mirror 1 by using a piezoelectric transducer (PZT) in the digital shearographic setup (Fig. 23.13). By means of a diffraction grating or a liquid-crystal cell the phase shifting technique can also been realized without the mechanical movement.

b) I1

I2

I'1

I'2

I3

I4

I'3

I'4

Part C 23.2

c)

Before loading

After loading

φ

d)

φ' Δ = φ– φ'

Fig. 23.16 (a) Four stored intensity distributions I1 to I4 , and the calculated phase value φ of the undeformed or preloaded state. (b) Four stored intensity distributions I1 to I4 and the calculated phase value φ of the deformed or second loaded state. (c) The phase distribution φ and φ as a result of the measured intensities. (d) The phase map of the relative phase change Δ obtained by digital subtraction of φ from φ

Speckle Methods

In temporal phase-shifting speckle interferometry, the measured intensity of a point P(x, y) on the object surface as recorded on the image plane of an optical system is given by  Ii (x, y, ti ) = I0 (x, y) 1 + γ (x, y) cos[φ(x, y)  + φi (ti )] , i = 1, . . ., N , (23.30) where φ(x, y) is the random phase of interest and φi (x, y) is the additional phase shift in the i-th frame. There are three unknowns, φ(x, y), I0 (x, y), and γ (x, y), in (23.30). Therefore, three or more equations are required to solve for the phase value φ(x, y) when the phase shift φi (x, y) is known. Most of the algorithms assume a known phase shift. However, the phase shift can also be an unknown. In cases with N > 3, the phase value φ(x, y) is overdetermined. However, experimental investigations have demonstrated that, in these cases, the measurement process is less sensitive to phase shifting if calibration errors are present. Numerous algorithms have been developed over the last two decades for various applications. In most of the experiments where phase-shifting speckle interferometry is used, the so-called four-step algorithm [23.16] is usually applied. For this approach, four intensity speckle patterns are recorded to determine the experimental value of the phase shift φ(x, y). Using this technique, four phase shifts φi (x, y) with i = 1, . . ., 4, (i. e., φ1 (x, y) = 0◦ , φ2 (x, y) = 90◦ , φ3 (x, y) = 180◦ , and φ4 (x, y) = 270◦ ) are introduced so that an equation system of order four can be solved by I4 (x, y) − I2 (x, y) . I3 (x, y) − I1 (x, y)

(23.31)

The same procedure is followed when the object is loaded with the following phase distribution of the alternate state of the object: φ (x, y) = arctan

I4 (x, y) − I2 (x, y) . I3 (x, y) − I1 (x, y)

(23.32)

Since the phase distributions φ(x, y) and φ (x, y) are calculated in modulo 2π, the relative phase difference Δ = φ − φ can be determined easily. The procedure is illustrated in Fig. 23.16. Spatial Phase Shifting (SPS) An alternative phase-shifting technique is spatial phase shifting (SPS). Instead of the three or more exposures that are required in TPS for calculating the phase distribution of each object state, only a single frame is

667

required in SPS. In SPS, the phase distribution can either be encoded by a spatial carrier frequency on one image plane or be recorded by separate cameras with an appropriate static phase shift for each image. The first approach is attractive, because it does not require expensive optics, unlike the second approach. When two wavefronts with a small angle interfere on a CCD chip, a linear phase change is generated from one pixel to another. Thus, a spatial carrier frequency can be introduced in SI by tilting the mirror M in the Michelson interferometer (Fig. 23.8). The tilting angle should be adjusted so that three to five pixels on the sensor correspond to a carrier period. The mean speckle size in the carrier direction, which can be changed by turning the aperture size in the imaging system, must be about three to five pixels to assure sufficient spatial correlation [23.17, 18]. Therefore, one single speckle should cover at least three to five supporting positions (pixels) in order to determine the phase relation. Various types of SPS methods and their theoretical descriptions are described in [23.19]. Filtering and Unwrapping The relative phase change determined by using phaseshifting technique is a noisy, sawtooth phase function exhibiting phase jumps of 2π (modulo 2π). Such a phase map is the starting image for all the image processing procedures to follow. In image processing, a filter algorithm with modulo 2π is first used since the phase fringe patterns show a low signal-to-noise ratio. Median or average filtering algorithms are frequently used in speckle interferometry. Next, one-dimensional (1-D) and two-dimensional (2-D) demodulation of phase fringe patterns is performed (phase unwrapping); this consists of integrating or adding the phase values along a line or path. Each time the phase value jumps from zero to 2π, the actual phase value is increased by 2π. Similarly, 2π is subtracted from the actual phase value when the phase value jumps from 2π to zero. The proper detection of the 2π-phase jumps is an important issue during the demodulation of phase in an interferogram. For a filtered phase interferogram a simple demodulation algorithm (scanning the folded interferogram line by line and adding or subtracting 2π for every phase jump) is sufficient as long as the phase values are continuously distributed over the entire image. D. W. Robinson gives the following summary of phase unwrapping-algorithms for path-dependent and path-independent methods:

Part C 23.2

φ(x, y) = arctan

23.2 Speckle Metrology

668

Part C

Noncontact Methods

Path-dependent phase demodulation algorithms:

• • • • •

Sequential linear scanning Multiple scan directions Spiral scanning Counting around defects Pixel ordering/queuing

• •

Unwrapping by regions Tile processing

Path-independent phase demodulation algorithms:

• •

Cellular automata methods Band-limit approach (global feedback)

23.3 Applications Speckle methods such as ESPI, digital shearography, and DSC/DIC discussed above, enjoy the advantages of being full-field noncontact methods. Therefore they have already been accepted by industry as a tool for nondestructive testing or evaluation (NDT/NDE), contour measurement, strain measurement, and vibration analysis, gaining more and more applications. The following applications have been measured by using digital shearography. More applications of ESPI and DSC/DIC are demonstrated in the chapters on Holographic Interferometry and Digital Image Correlation Chaps. 24 and 20.

23.3.1 NDT/NDE

Part C 23.3

The demands for greater quality and product reliability have created a need for better techniques of nondestructive testing (NDT/NDI), in particularly, techniques for online inspection. Basically, we distinguish between three deformation processes that occur according to the nature of the loading: static, nonstationary (e.g., thermal, impact), and dynamic deformation processes. Static Loading In the case of static loading, the temporal phase-shifting method is always used. Four intensities are recorded during the constant maximum loading stage and four intensities in the minimum loading state, and the phase positions φ and φ are calculated. A cylinder of the kind actually used in conveying engineering for the transportation of paper and similar materials is described as an example. This roller basically consists of an outer rubber layer glued to a metal cylinder. During production, impurities on the metal surface can lead to partial delamination defects between the metal and the rubber. In this application, the cylinder is evacuated in a vacuum chamber. The environmental pressure between the metal and the rubber in the area of the delaminations causes an out-of-plane deformation of

the rubber at those points where there is no bonding between the metal and the rubber. These deformations are visible in the phase image (Fig. 23.17). In order to detect possible disbonds the cylinder should be scanned completely. Therefore, the cylinder has been divided into separate test areas. In every test area a test cycle according to the alternate loading and unloading is executed. From a segment of the cylinder a phase image recording φ1 is made under loading pmax , followed by a phase image recording φ1 in the unstressed state pmin . Then the cylinder and the camera are moved nearer to each other without altering the state of stress and a new testing area is set up. Without changing the load, a phase image recording φ2 is made of the new test area and, following reloading, a further phase image recording φ2 is taken. It should be noted that this procedure saves one load change at every gauging. Thermal Loading Another type of deformation that is relatively difficult to control and which allows the deformation of the test specimen under investigation is thermal loading. Since the warming and cooling process is time dependent, the spatial phase-shifting technique is suitable for this kind of application.

Buckling in the rubber layer around delaminations

Fig. 23.17 Demodulated phase image of a section of a conveying cylinder under vacuum

Speckle Methods

23.3 Applications

669

Bulging resulting from different thermal expansion coefficients of the rubber layer compared with steel core

Fig. 23.18 Cylinder segment without flaws

Dynamic Loading Damages and defects can be identified accurately under dynamic loading. Due to defects the regions of defects occur different stiffness, damping and therefore the different resonance frequencies as the entire structure. If the vibration mode can be observed or evaluated in real time, the defects can also be detected, when the test object is excited by a slow sine sweep. In order to map vibration mode, a recording technique allowing the realtime observation of the dynamic deformation of a tested specimen using the loading method is required. The recording technique that permits real-time observation synchronized with video is the real-time subtraction described in [23.20–26].

Fig. 23.19 Cylinder segment with flaws

92

93

94

95

96

71

72

73

74

75

60

61

62

63

64

49

50

51

52

53

38

39

40

41

42

27

28

29

30

31

Fig. 23.20 Honeycomb panel (size: 1000 × 1200 × 12 mm3 )

size of the grid: 100 × 100 mm2

When the tested object vibrates in a stationary state at a frequency f that is higher than the video rate, the (n − 1)-th plotted image is a time-averaged shearogram recorded during a single video frame. The following n-th image is then digitally subtracted from the previous (n − 1)-th image i. e., the continuously renewed reference image technique. This makes the recording technique relatively insensitive to outside influences such as environmental disturbances using the continuous refreshment of the reference image. The result

Part C 23.3

The task is to detect the delaminations and to make them visible using digital shearography. In the laboratory experiment the rubber layer is heated using a hot-air drier. The rubber is deformed relatively homogeneously with steady thermal loading and there is no areal expansion of the rubber where the lamination is intact. The stationary radial expansion becomes noticeable in the phase image through a constant fringe or greyish distribution (Fig. 23.18). Only where the lamination is missing does the rubber layer show signs of bulging, due to the areal expansion of the rubber layer in view of the greater thermal expansion coefficient of rubber compared with steel to balance the different change of length with respect to the steel body. The bulging is interpreted as an area of disbond in the phase image (Fig. 23.19). Figure 23.18 shows the longitudinal length of a cylinder without any signs of delamination. Figure 23.19 shows the longitudinal length of a cylinder displaying disbonding, visible in the form of buckling of the rubber layer in the phase map as a result of the thermal expansion.

670

Part C

Noncontact Methods

is a simple time-averaged digital shearogram showing a fringe pattern that depends on the vibrating object. By using real-time subtraction with the continuously refreshed reference frame, the resonant frequencies of

the vibrating object (Fig. 23.20) as well as the natural frequency of the oscillating defect (Fig. 23.21) can be determined simply and rapidly.

23.3.2 Strain Measurement Upon the deformation of a stained body, the displacement vector d at a individual point can be split into the components u, v, and w in the x-, y-, and z-directions, respectively. Figure 23.22 shows the elastic deformations of a small solid rectangle. With the assumption of small angles, the shear strain can be expressed approximately by using the derivatives of the displacement components. Thus, three normal strains and three shear strains can be written as ∂u ∂v ∂w (23.33) , ε yy = , εzz = , εxx = ∂x ∂y ∂z and Fig. 23.21 Time-averaged digital shearogram using real-

time subtraction. The test object (Fig. 23.20) was excited by frequency f = 514 Hz ∂u dy ∂y

C' D'

∂υ dy ∂y D

C

βy

Part C 23.3

dy B'

Θ

∂υ dx ∂x

βx A

B dx

∂x ∂y

∂u dx ∂x

Fig. 23.22 A small volume dx, dy, dz within a body is

chosen to illustrate the deformation of the body area ABCD to AB C D

F

∂u ∂v ∂v ∂w + , γ yz = + , ∂y ∂x ∂z ∂y ∂w ∂u (23.34) + . γzx = ∂x ∂z Since usually only the surface of the tested object can be measured, no information can be obtained normal to the surface (or in the z-direction) of the object. The deformation gradient ∂u/∂z, ∂v/∂z, ∂w/∂z cannot be measured directly. Only in a few cases can the outof-plane strain ∂w/∂z be calculated indirectly. For this reason it is possible to measure only the following six components directly: ⎞ ⎛ ∂u ∂u ⎜ ∂x ∂y ⎟ ⎟ ⎜ ⎜ ∂v ∂v ⎟ ⎟. ⎜ (23.35) H=⎜ ⎟ ⎜ ∂x ∂y ⎟ ⎝ ∂w ∂w ⎠ γxy =

F

Fig. 23.23 Araldite B tensile bar with a hole in the middle

With (23.24)–(23.29) in Sect. 23.2.3, six components of the matrix H can be generated by varying the illumination planes and shearing directions. An application of measuring in-plane strain and out-of-plane components solely using a tensile bar manufactured from Araldite B with a hole in the middle is shown in Fig. 23.23, and loaded to 136 N in Fig. 23.24a–g. Figure 23.24a and d show the fringe patterns of the in-plane strains εxx and ε yy , while Fig. 23.24b,c show ∂u/∂y and ∂v/∂x. As an additional result, the out-of-plane deformation gradients ∂w/∂x and ∂w/∂y can be measured directly (Fig. 23.24e,f) and represented by their pseudo-3-D

Speckle Methods

a1)

b) ∂u/∂y

c)

d) εyy = ∂υ/∂y

d1)

e) ∂w/∂x

f)

g) γxy = ∂u/∂y+ ∂υ/∂x

g1)

∂υ/∂x

∂w/∂y

Fig. 23.24a–g Strain phase images for the bar shown in Fig. 23.23: (a) ∂u/∂x, (b)∂u/∂y, (c) ∂v/∂x, (d) ∂v/∂y, (e) ∂w/∂x, (f) ∂w/∂y, and (g) γxy . (a1), (d1), (g1), corresponding 3-D plots

plots. The measurement results indicate that the out-ofplane strain components in the area near the hole are not

671

equal to zero, although the tensile bar is loaded under plane-stress conditions.

Part C 23.3

a) εxx = ∂u/∂x

23.3 Applications

672

Part C

Noncontact Methods

23.4 Bibliography • • • • • •

J. W. Goodman: Statistical Optics (Wiley, New York 1985) C. M. Vest: Holographic Interferometry (Wiley, New York 1979) R. S. Sirohi (Ed.): Speckle Metrology (Marcel Dekker, New York 1993) T. Kreis: Holographic Interferometry: Principles and Methods (Akademie Verlag, Berlin 1996) K. P. Rastogi (Ed.): Holographic Interferometry: Principles and Methods (Springer, Berlin, Heidelberg 1994) W. Osten: Digitale Verarbeitung und Auswertung von Interferenzbildern (Akademie Verlag, Berlin 1991)

• • • • •

R. Jones, C. Wykes: Holographic and Speckle Interferometry (Cambridge Univ. Press, Cambridge 1989) Chap. 2 W. Steinchen, L. X. Yang: Digital Shearography – Theory and Application of the Digital Speckle Pattern Shearing Interferometry (SPIE, Washington 2003) P. K. Rastogi (Ed.): Optical Measurement Techniques and Applications (Artech House, Boston 1997) D. W. Robinson, G. T. Reid (Eds.): Interferogram Analysis: Digital Fringe Pattern Measurement Analysis (Institute of Physics, Bristol 1993) E. Hecht, A. Zajac: Optics (Addison-Wesley, Reading 2003)

References 23.1

23.2 23.3

23.4

Part C 23

23.5

23.6 23.7

23.8

23.9

23.10

23.11

K. Exner: Über die Frauenhofer’schen ringe, die Quetelet’schen streifen und verwandte erscheinungen, Sitzungsberichte der Akademie der Wissenschaften Wien. II(76), 522–550 (1877) P. Hariharan: Speckle patterns: A historical retrospect, J. Modern Opt. 19(9), 791–793 (1972) M. von Lauer: Die Beugungserscheinungen an vielen unregelmäßig verteilten teilchen, Sitzungsberichte der preussischen Akademie der Wissenschaften zu Berlin 47, 1144–1163 (1914) G. Groh: Proc. Engineering Uses of Holography, ed. by E.R. Robertson, J.M. Harvey (Cambridge Univ. Press, Cambridge 1970) pp. 483–494 J.W. Goodman: Statistical properties of laser speckle patterns. In: Laser Speckle and Related Phenomena, ed. by J.C. Dainty (Springer, New York 1975) pp. 9–75 J.W. Goodman: Some fundalmental properties of speckle, J. Opt. Soc. Am. 66(11), 1145–1150 (1976) E.N. Leith, J. Upatnieks: Reconstructed wavefronts and communication theory, J. Opt. Soc. Am. 52, 11232–1130 (1962) J.N. Butters, J.A. Leendertz: Speckle pattern and holographic techniques in engineering metrology, Opt. Laser Tech. 3, 36–30 (1971) A.E. Ennos: Speckle Interferometry. In: Laser Speckle and Related Phenomina, ed. by J.C. Dainty (Springer, New York 1975) pp. 203–253 J.H. Tiziani: A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately, Opt. Commun. 5, 271–276 (1972) J. M. Burch, J. M. J. Tokarski: Production of multiple beam fringes from photographic scatterers, Opt. Acta 15(2) 101–111 (1968)

23.12

23.13

23.14

23.15

23.16 23.17

23.18

23.19

23.20

23.21

E. Archbold, A.E. Ennos: Displacement measurement from double-exposure laser photographs, Opt. Acta 19(4), 253–271 (1972) Th. Fricke-Begemann: Optical Measurement of Deformation Fields and Surface Processes with Digital Speckle Correlation, Dissertation. University of Oldenburg 2002 J.A. Leendertz, J.N. Butters: An image-shearing speckle-pattern interferometer for measuring bending moments, J. Phys. E: Sci. Instrum. 6, 1107– 1110 (1973) Y.Y. Hung, R.E. Rowlands, I.M. Daniel: Speckleshearing interferometric technique: a full-field strain gauge, Appl. Opt. 14(3), 618–622 (1975) K. Creath: Phase shifting speckle interferometry, Appl. Opt. 24(18), 3053–3058 (1985) J. Burke, H. Helmers: Performance of spatial versus temporal phase shifting in ESPI, Proc. SPIE 3744, 188–199 (1999) T. Bothe, J. Burke, H. Helmers: Spatial phase shifting in electronic speckle pattern interferometry: minimization of phase reconstruction errors, Appl. Opt. 36(22), 5310–5316 (1997) M. Kujawinska: Spatial phase measurement methods. In: Interferogram Analysis, Digital Fringe Pattern Measurement, ed. by D.W. Robinson, G.T. Reid (IOP, Bristol 1993) W. Steinchen, L.X. Yang, G. Kupfer: Vibration analysis by digital shearography, Proc. SPIE 2868, 426–437 (1996) R.L. Powell, K.A. Stetson: Interferometric hologram evaluation and real-time vibration analysis of diffuse objects, J. Opt. Soc. Am. 55, 1694–1695 (1965)

Speckle Methods

23.22

23.23

23.24

R.L. Powell, K.A. Stetson: Interferometric vibration analysis of 3-D-objects by wavefront reconstruction, J. Opt. Soc. Am. 55, 1593–1598 (1965) J.D.R. Valera, O.J. Løkberg: Composite vibration analysis with modulated electronic speckle shearing interferometry, Opt. Rev. 4(2), 261–264 (1997) J.D.R. Valera, J.D.C. Jones, O.J. Løkberg: Exact vibration amplitude derivative measurement with TV shearography, Meas. Sci. Technol. 7, 918–921 (1966)

23.25

23.26

References

673

W. Steinchen, Y. Gan, G. Kupfer, P. Mäckel: Digital shearography using stroboscopic illumination additional to time average method, 6-th AIVELAConference (Ancona 2004) W. Steinchen, Y. Gan, G. Kupfer: Comparison of vibration measurements – Laser Doppler Vibrometry and Digital Shearography, Workshop for final meeting LAVINYA (Ancona, 2005)

Part C 23

675

Holography 24. Holography

Ryszard J. Pryputniewicz

The intent of this chapter is to present a self-contained text on fundamentals and representative applications of holography (concentrating on those of current interest). As such, the Chapter begins with an overview of basic concepts of holography, then mathematical relationships and definitions sufficient to understand and apply these concepts are presented as, e.g., in optoelectronic holography (OEH) and in its implementation via optoelectronic laser interferometric microscope (OELIM) for studies of micro-electromechanical systems (MEMS) and microelectronics, both of which are, of great current interest. This is complemented by a description of systems for automated quantitative determination of displacements and deformations of objects. Finally, the description is illustrated with representative examples of applications in (experimental) micromechanics, concentrating on the emerging technologies addressing state-of-the-art developments in MEMS.

24.2 Fundamentals of Holography ................ 24.2.1 Recording of Holograms ................ 24.2.2 Reconstructing a Hologram............ 24.2.3 Properties of Holograms ................

677 677 678 679

24.3 Techniques of Hologram Interferometry . 679 24.3.1 Optoelectronic Holography ............ 681 24.3.2 Quantitative Interpretation of Holograms ............................... 683 24.4 Representative Applications of Holography ...................................... 24.4.1 Determination of the Absolute Shape of Objects .... 24.4.2 Determination of Time-Dependent Thermomechanical Deformation Due to Operational Loads .............. 24.4.3 Determination of the Operational Characteristics of MEMS .................

685 685

687 688

24.5 Conclusions and Future Work................. 695 References .................................................. 696

Since its invention, the field of (optical) holography has grown tremendously and found numerous uses in such diverse applications as aerospace, automotive, electronics, equipment, dental/orthodontics, machinery, medical, packaging, transportation, safety and security as well as frontiers of emerging technologies (ET). All these applications are well documented in countless publications too numerous even to be listed herein. And the field of holography is still growing. New ways are constantly being found to apply holography to solve problems that were previously unsolvable. So far, the author used holography in a very broad sense including also related fields of laser speckle and, in particular, holographic interferometry which is most important in experimental mechanics. Hence, holography will be used to refer to the broad field

Part C 24

The development of science and technology is based on discoveries which are often made because the time is right. Some discoveries have been made almost simultaneously by inventors who have worked independently of each other. This was not so with holography. Currently, credit for invention of holography is given to one man, Professor Dennis Gabor [24.1–3] of the University of London Imperial College of Science and Technology, who in 1971 received the Nobel Prize for his invention and development of the holographic method. Actually, in 1948 Gabor rediscovered and formulated the work done in late 1800s by Gabriel Lippmann [24.4] on a technique for encoding a color object on a thick emulsion photographic plate based on wavelength sensitivity, which was followed by Yuri Denisyuk’s reflection holography [24.5].

24.1 Historical Development ......................... 676 24.1.1 Hologram Interferometry............... 676

676

Part C

Noncontact Methods

of holography while hologram interferometry will refer to interferometric aspects of the field, as will be apparent from the context. Quantitative hologram interferometry, especially as practiced in experimental mechanics, is rather computationally intense. As new mathematical procedures for quantitative interpretation of holograms are being developed, it becomes increasingly easier to apply the state-of-the-art (SOTA) holographic methods of nondestructive testing (NDT) and metrology to the applications of interest in experimental mechanics. Effective operation of processes, systems, components, etc. depends on reliable validation of the results obtained during the development of a particular/relevant design. Today, there are a number of methods that can be used for such validation. A very prominent place among these methods is taken by quantitative holography. In fact, holographic methods are the only means that allow quantitative evaluation of certain designs [24.6]. However, the problem of quantitative interpretation of holograms is not trivial. Even today, some 40 years after the invention of hologram interferometry [24.7] and nearly 60 years after the fundamentals of holography were formulated [24.3–5] there is not yet a general method available that can be used reliably to interpret holographic fringe patterns quantitatively to obtain information on displacements and/or deformations of arbitrary objects. Nevertheless,

a number of methods and systems have been developed for specific applications [24.8]. Results that are being obtained using these methods are contributing to the further growth of hologram interferometry; the most promising of these methods are those allowing automated interpretation of holograms [24.9, 10]. All approaches to quantitative interpretation of holograms rely on mathematical relationships that govern conversions of data from a holographically produced image to results that are meaningful to a user. Developments in the quantitative interpretation of holograms show that most of the relationships between the involved parameters can be best described by matrix transformations [24.11]. These transformations map fields of three-dimensional vectors in one space into corresponding fields of vectors in another space [24.12]; that is, they allow conversion of holographic data from an image space into results that can be readily interpreted and understood by a user. Furthermore, the solutions of the resulting equations, to obtain quantitative results, can be achieved in a straightforward manner following established computational procedures for the solution of matrix equations [24.12]. In Sect. 24.1 we begin with a historical account of the beginnings of holography and follow in Sect. 24.1.1 these developments up to and including the discovery of hologram interferometry.

24.1 Historical Development

Part C 24.1

There are many individuals who have contributed to development of holography that was formalized by Gabor in 1948 [24.1–3]. Detailed accounts of these developments are given by Caulfield [24.13] and Johnston [24.14]. As the advances in holography were made, reconstructions from these holograms were so faithful that it was difficult to determine by optical means alone whether one was observing the object or its holographically produced reconstruction. It is this high quality and realism of the three-dimensional images obtained by holography that, to a large extent, have been responsible for the popular and scientific interest in the wavefront reconstruction technique and in its applications.

24.1.1 Hologram Interferometry One of the most spectacular applications of holography is in the field of interferometry. Development of the off-axis method and the invention of the laser have greatly improved the visibility of the reconstructed image and, at this stage, holography permitted the extension of classical interferometry, used up to this date to measure the small path-length differences of optically polished and specularly reflecting surfaces, to three-dimensional diffusely reflecting objects with nonplanar surfaces. This method, known as hologram interferometry, was invented in 1965 [24.7, 15]. Hologram interferometry was soon further advanced [24.16–21].

Holography

One of the major applications of hologram interferometry is in the area of experimental mechanics for displacement/deformation analysis, where this highly sensitive method is used to study the small threedimensional displacements and deformations of objects.

24.2 Fundamentals of Holography

677

The following sections present procedures for recording, reconstruction, and quantitative interpretation of images reconstructed from holographic interferograms; interpretation is illustrated with representative applications of contemporary interest.

24.2 Fundamentals of Holography The background knowledge in optics, including light interaction with materials, interference, diffraction, as well as optical spatial filtering, all necessary to understand recording and reconstruction of holograms as well as their properties are detailed by Cloud [24.22]. Fundamentals of holography that form the basic concepts in experimental mechanics are well described by Ranson et al. [24.23]. In this section, processes for recording, or formation, and reconstruction of holograms are outlined, a description of the techniques of hologram interferometry is given, and the general properties of holograms are discussed. During these discussions we can, of course, introduce almost any amount of mathematics, but essentials can be examined and understood from physical arguments aided with a few equations. More detailed analysis can be found in [24.24–31].

24.2.1 Recording of Holograms

Front surface mirror

Laser Beam splitter

Object Object beam

Reference beam

Rr

y jˆ

kˆ

Ro iˆ

Photographic plate

x

z

Fig. 24.1 A typical setup for recording of a hologram

Spatial filter

Part C 24.2

The surface of a diffuse object, no matter how complex, can be considered as a conglomeration of points. When such an object is illuminated with a light beam, every point on its surface reflects the light and scatters it. Therefore, each of the points may be considered a source of emanating waves. If a photosensitive material (e.g., a high-resolution photographic film or a charge-coupled device (CCD) array [24.32]) is exposed in such a way that the light reflected from an object is incident on this material without any intervening lens, a sensing surface of the material will be uniformly exposed. That is, no image will be recorded because the light from every point on the object will fall on every point of the photosensitive material. Thus, each part of the material will be subjected to the same energy and a uniform exposure of the medium will be produced. The propagating light energy, like all other forms of electromagnetic radiation, is a wavefront train having both amplitude and phase. Energy detection systems sense the intensity, i. e., the amplitude squared, of a light

wave, but not the amplitude directly. Photographic film, being such a square-law detector, faithfully records the intensity. Therefore, a conventional photograph is a two-dimensional (2-D) mapping of the point-by-point brightness of an object. Clearly, only partial information about a light wave from an object is recorded. If, on the other hand, a photosensitive material could record both the amplitude and the phase, all information contained in a wavefront modulated by interaction with an object would be preserved. Holography is a technique that allows such a recoding of the entire wavefront on a suitable recording medium [24.32]. Recording the amplitude of the waves poses no problem because photosensitive media record irradiance by converting it to the corresponding variations in the opacity of the sensing surface. The phase, however, is recorded by using the principle of interferometry. Figure 24.1 shows that recording of a hologram is accomplished by providing a second beam incident on the photosensitive surface. This second beam is called the reference beam, in distinction from the object beam,

678

Part C

Noncontact Methods

which is modulated by interaction with an object. The reference beam is usually split off the main source of illumination because the reference and the object beams must be mutually coherent, i. e., able to interfere. Although the absolute phase of the light wave cannot be measured, it is possible to determine the phase difference between two beams of light, and the second beam is, in fact, the reference for this determination. If these beams are allowed to overlap with each other in some region of space, then, due to their interference, fringes will form over the region of their overlap. The spatial frequency, f , of these interference fringes is entirely dependent on the angle α between the two beams and the wavelength, λ, of the light used. This dependence can be expressed as sin α (24.1) . λ Since the wavelength is usually expressed in microns, (24.1) shows that the spatial frequency of the interferometric fringes has units of μm−1 . In practice, a hologram is constructed with a setup similar to that shown in Fig. 24.1. In this setup, a highly coherent and monochromatic light from a (laser) source is split into two beams by means of a beam-splitter (BS). One of these beams is directed by mirrors, expanded by means of a spatial filter (a microscope objective and a pinhole assembly), and is used to illuminate the object to be recorded. This is the object beam. The other beam, known as the reference beam, is sent around the object to overlap, at the hologram recording (photographic/photosensitive) plate/plane (i. e., on a photosensitive medium/surface), with the object beam. For quantitative interpretation of holograms, beam propagation directions are required. These directions will be determined (in Sect. 24.3.2) in terms of position vectors defining illumination and observation geometry of a holographic setup. Location of any point within these two beams, composing a setup, can be specified by the position vectors Ro and Rr defined as f =

Part C 24.2

Ro = xo i + yo j + z o k ,

(24.2)

Rr = xr i + yr j + z r k ,

(24.3)

and

where x, y, and z represent Cartesian coordinates, while i, j, and k denote the unit vectors corresponding to the Cartesian coordinates defining the hologram recording setup, and the subscripts ‘o’ and ‘r’ identify the object beam and the reference beam, respectively.

Superposition of the object and the reference beams results in an interference pattern that is recorded in a photosensitive medium [24.32]. This interference pattern constitutes an interferogram. Basic theory of fringe formation is given by Ranson et al. [24.23]. Superposition of the two light fields, corresponding to the object and the reference beams, results in interferometric fringes forming a fringe pattern that is recorded by a photosensitive medium [24.12, 32] producing a hologram. Visual observation of a hologram quickly convinces one that it bears absolutely no resemblance to the original object. That is, when examined visually, a hologram is quite unintelligible and gives no hint whatsoever of the image recorded. A superficial examination of it may tempt one to identify the visible structures (concentric rings, specks, lines, etc.) with portions of an object. However, such an identification would be quite incorrect. The visible structures are purely extraneous and arise from dust particles and other scatterers present in the optical system that produced a hologram. In fact, the interferometric fringes (sometimes also called primary interferometric fringes; the primary fringes constitute a hologram) that are recorded in the medium [24.32] are not visible to an unaided eye. It is so because of very fine interfringe spacings (of approximately 1 μm, or roughly one wavelength λ), as can be determined using (24.1), for typical geometries employed in recording of holograms. The fringe pattern, seen under magnification, consists of highly irregular bright and dark bands that bear no apparent relation to the object that was holographed/recorded. It is quite unlikely that one could readily learn to interpret them visually without actually reconstructing the image/hologram.

24.2.2 Reconstructing a Hologram In practice, a hologram can be reconstructed with the original setup used in recording, only now the hologram is illuminated with the reference beam alone (Fig. 24.2). During reconstruction of a hologram, the variation in transmission of the holographic recording, due to the recorded fringes, diffracts light into the various orders of the hologram reconstruction. As a result of this, the diffracted wavefronts propagate in such a way that they resemble, in every characteristic, the original object beam used to construct the hologram. Therefore, the first-order diffracted waves produced by the hologram are each a replica of the wave that issued from

Holography

the original object subject to distortions imposed by a holographic medium.

24.2.3 Properties of Holograms The three-dimensional imagery produced by a hologram is quite pronounced and a viewer quickly realizes that much more information about the object is furnished by a hologram than by any other form of a three-dimensional process such as stereophotography or three-dimensional (3-D) photos. While observing a holographically produced image, the viewer can inspect the three-dimensional object not just from one direction as in stereophotography, but from many different directions and with the ability to focus on all planes of the image. This is possible because when the viewer, looking through the hologram as if it were a window, changes the viewing direction, the perspective of the image changes as if the viewer was observing the original object. This effect, known as parallax is easily observed when the viewer moves its head sideways or up and down while looking through the hologram. The property of parallax constitutes one of the most convincing proofs of realism. Also, if a lens is included in the recorded scene, a motion toward and away from the object provides the expected enlargement or contraction of the object located behind the lens. To summarize, an image reconstructed holographically has all the visual properties of the original object and there is no visual test that can be made to readily distinguish the two. Holography, initially called by Gabor photography by wavefront reconstruction, does not produce nega-

24.3 Techniques of Hologram Interferometry

CW-laser

679

Stop Beam splitter

Front surface mirror

Image of the object

Beam expander spatial filter Reconstructing beam

Hologram

First order wave

First order wave

Zero order wave

Fig. 24.2 A typical setup used for reconstruction of a hologram

tives. The hologram itself would normally be regarded as a negative, but the image it produces is a positive. In addition to the above, several different images can be superimposed on a single photosensitive medium [24.32] during successive exposures and each image can be recovered without being affected by the other images. In fact, this capability was used, in early years of holography, to record a sequence of images that could be played back in rapid succession by simply turning/indexing the hologram (usually a photosensitive plate) to positions corresponding to original locations/stages in the setup [24.12].

24.3 Techniques of Hologram Interferometry small path-length differences of optically polished and specularly reflecting flat surfaces. In the early 1960s, with the advent of a laser, holography has permitted this shortcoming to be circumvented in a very elegant manner. The method, based on the ability of a hologram to simultaneously record the phase and the amplitude of any wave at different states and to produce, during reconstruction, interference fringes relating to changes in these states, is known as hologram interferometry [24.16]. Hologram interferometry has allowed the extension of (classical) interferometry to three-dimensional diffusely reflecting objects with nonplanar surfaces.

Part C 24.3

Interferometry is a process in which two waves emitted from, say, the same source (note that multiple sources have also been used successfully, although they required more extensive preparation of the corresponding setups [24.12]) at the same time, interact with each other because of differences in their path lengths. As a result of this interaction, a pattern of alternating bright and dark bands (i. e., fringes) can be observed. These fringes relate directly to changes in the optical path lengths of either or both of the beams. Unfortunately, the coherence and monochromaticity of the light sources used in interferometers prior to 1960s were poor. As a result, classical interferometry was limited to measurements of

680

Part C

Noncontact Methods

There are three basic variations of hologram interferometry, each possessing certain advantage over the others in particular test situations/applications [24.12]: 1. double-exposure hologram interferometry, 2. real-time hologram interferometry, and 3. time-average hologram interferometry. The time-average hologram interferometry method can be further subdivided into: 1. stroboscopic time-average hologram interferometry, and 2. continuous time-average hologram interferometry.

Part C 24.3

In the double-exposure hologram interferometry method, two exposures of an object are made with the state of the object changed due to loading (by pressure, stress, temperature, velocity, frequency, etc.) between the two exposures. As a result, the configuration of the object is slightly different during the second exposure when compared with its configuration during the first exposure. While reconstructing a double-exposure hologram the two object beams (each corresponding to a different configuration) are faithfully reconstructed in phase and amplitude. Therefore two three-dimensional images of the object are formed, corresponding to the initial and the final configurations, respectively. Since both of these images are reconstructed in coherent light they interfere with each other. Thus, in any region of space where the two reconstructed beams overlap, an image of the object is seen covered with a set of alternating bright and dark fringes – these are frozen fringes in that the fringe pattern is fixed and cannot be altered/changed once a double-exposure hologram has been recorded. The frozen fringes are the so-called cosinusoidal fringes because their brightness varies cosinusoidally across the image. The fringes seen during the reconstruction of a hologram are a direct measure of changes in the position and/or deformation that an object experienced between the two exposures. Obviously, a unique displacement/deformation of the object, for a given illumination/observation geometry (i. e., the setup), does produce a unique three-dimensional interferogram during hologram reconstruction. The only problem that now remains is to decode the information contained in a particular fringe pattern [24.33]. This problem is introduced in Sect. 24.3.1. The real-time hologram interferometry method consists of taking a single exposure of an object in an initial state of stress/deformation (typically unstressed/undeformed), processing the medium in-place using a so-called liquid-gate plate holder (or a pho-

tothermoplastic camera), and reconstructing the hologram using a reference beam identical to the one used during the recording step. The reconstructed image is now superimposed onto the original object, which is also illuminated with the same light as when the single-exposure recording was made. Interference fringes can now be seen if the object is even slightly displaced/deformed. The interferometric comparison between the original state (i. e., the holographically reconstructed image from the single exposure) and the new state of the object is made at the instant it occurs (hence in real time). This comparison is manifested by a fringe pattern which is live in that it changes as the state of the object is changed. The particular advantage of the real-time hologram interferometry method is that different motions (static as well as dynamic) can be studied with a single holographic exposure rather than having to make a hologram for each new position (i. e., each new state) of the object being investigated. The stroboscopic time-average hologram interferometry (STAHI) method is really an extension of the double-exposure hologram interferometry method where a continuous-wave (CW) laser beam is chopped into short pulses synchronized with the vibration frequency of the object (it should be noted that, with advances in photonic technology, pulsed light sources are now available, which greatly simplifies recording of STAHI-type images [24.34–39]). To use the stroboscopic time-average hologram interferometry method effectively, object vibration should be monitored to assure proper synchronization of the illuminating pulsed beam of light. This synchronization should be maintained over many vibration cycles in order to provide sufficient exposure of the photosensitive medium [24.32]. Although interference fringes produced during the reconstruction of STAHI-type images are cosinusoidal and are straightforward to analyze, the electronic apparatus needed may be complex and depends on the specific application [24.12]. In the continuous time-average hologram interferometry method a single holographic recording of an object undergoing cyclic vibration is made [24.40]. As the (continuous) exposure time is long in comparison with one period of the vibration cycle the hologram effectively records an ensemble of images corresponding to the time-average of all positions of an object during its vibration. While reconstructing such a hologram, interference occurs between the entire ensemble of images, with the images recorded near zero velocity (i. e., at the maximum displacement) contributing most strongly to the reconstruction process, and a fringe pat-

Holography

tern corresponding to the vibratory motion of the object is produced. This fringe pattern however, is characterized by fringes of unequal brightness. In fact, they vary according to the square of the zeroth-order Bessel function of the first kind, i. e., J02 [24.12, 41]. Therefore, fringes observed during reconstruction of the continuous time-average hologram interferometry images are known as Bessel fringes or J0 fringes. Section 24.3.1 describes the contemporary optoelectronic implementation of hologram interferometry, whereas Sect. 24.3.2 provides rudimentary relationships for quantitative interpretation of holograms.

Continued advances in laser, computer, fiber optics, and photonic detector technologies have led to electronic implementation of holography. One of the results of this implementation is known as optoelectronic holography (OEH) [24.42, 43]. The basic configuration of an OEH system is shown in Fig. 24.3. In this configuration, laser light is launched into a single-mode optical fiber by means of a microscope objective (MO). Then, a single-mode fiber is coupled into two fibers by means of a fiber-optic directional coupler (DC). One of the optical fibers comprising the DC is used to illuminate an

LD

MO

OI

DC PZT1

IP

PZT2 K1 K2

OL CCD

Object

(24.4)

where K is the sensitivity vector defined in terms of vectors K 1 and K 2 identifying directions of illumination and observation, respectively, in the OEH system (Fig. 24.3). Quantitative determination of structural displacements/deformations due to the applied loads can be obtained by solving a system of equations similar to (24.4) to yield [24.33]  T −1 T ( K˜ Ω) , (24.5) L = K˜ K˜ where K˜ represents a transpose of the matrix of the sensitivity vectors K . Equation (24.5) indicates that deformations determined from interferograms are functions of K and Ω, which have spatial, i. e., (x, y, z), distributions over the field of interest on the object being investigated. The fringe-locus function is determined by recording intensity patterns of the object and the reference beams. For example, the intensity patterns of the first and the second exposures, i. e., In (x, y) and In (x, y), respectively, for the n-th frame/image in the doubleexposure sequence can be represented by the following equations: T

In (x, y) = Io (x, y) + Ir (x, y)  1/2 + 2 [Io (x, y)][Ir (x, y)]   cos [ϕo (x, y) − ϕr (x, y)] + θn (24.6)

Fig. 24.3 Single-illumination and single-observation geometry of a fiber-optic-based OEH system: LDD is the laser and diode driver, LD is the laser diode, OI is the optical isolaIn (x, y) = Io (x, y) + Ir (x, y)  tor, MO is the microscope objective, DC is the fiber-optic  1/2 + 2 I (x, y) I (x, y) o r directional coupler, PZT1 and PZT2 are the piezoelectric   fiber-optic modulators, IP is the image-processing comcos ϕo (x, y) − ϕr (x, y) + θn puter, IT is the interferometer, OL is the objective lens,  CCD is the camera, while K 1 and K 2 are the directions + Ω(x, y) , of illumination and observation, respectively

(24.7)

Part C 24.3

IT

681

object along the direction K 1 , while the output from the other fiber provides a reference against which signals from the object are recorded. Both, the object and reference beams are combined by the interferometer (IT) and recorded by the system camera (CCD). Images recorded by the CCD are processed by the image-processing computer (IP) to determine the fringe-locus function, Ω, constant values of which define fringe loci on the surface of object being investigated. The values of Ω relate to the system geometry and to the unknown vector L, defining displacements/deformations, via the relationship [24.33] Ω = (K 2 − K 1 ) · L = K · L ,

24.3.1 Optoelectronic Holography

LDD

24.3 Techniques of Hologram Interferometry

682

Part C

Noncontact Methods

where Io and Ir denote the object and reference beam intensities, respectively, with (x, y) denoting the spatial coordinates, φo the random phase of the light reflected from the object, φr the phase of the reference beam, θn the n-th applied phase step, and Ω the fringe-locus function relating to the displacements/deformations incurred by the object between the first and second exposures; Ω is what we need to determine. When Ω is known, it is used in (24.5) to find L [24.33]. In the case of the five-phase-step algorithm with θn = 0, π/2, π, 3π/2, and 2π (although other magnitudes of phase steps may be used, increments of 90◦ result in the simplest/fastest evaluation of the algorithms employed [24.12,33]) the distribution of the values of Ω can be determined using −1



Ω(x, y) = tan

   2 I2 (x, y) − I4 (x, y) . 2I3 (x, y) − I1 (x, y) − I5 (x, y) (24.8)

OB

XYZ

K1

BC OL

OI

K2

CCD

RB

RS

Results produced by (24.8) depend on the capabilities of the illuminating, imaging, and processing subsystems of the OEH system/setup. Developments in laser, fiber optics, CCD cameras, and computer technologies have led to advances in the OEH methodology; in the past, these advances have almost paralleled the advances in the image recording media [24.32]. A fiber-optic-based OEH system incorporating these developments is depicted in Fig. 24.4. In addition to being able to measure static and dynamic deformations of objects subjected to a variety of boundary, initial, and loading (BIL) conditions (just as the system displayed in Fig. 24.3), the system shown in Fig. 24.4 is also able to measure the absolute shape of the objects using multiple-wavelength optical contouring [24.44]. This dual use is possible because of the rapid tuning of the laser and the real-time monitoring of its output characteristics by a wavelength meter (WM) and a power meter (PM), both integrated into the OEH system and actively controlled by the system computer. In the configuration shown in Fig. 24.4, the imageprocessing computer (IP) controls all the functions of the OEH system. Furthermore, in response to the very demanding needs of the emerging MEMS technology, an optoelectronic laser interferometric microscope (OELIM) system for studies of objects with micron-size features has been developed [24.43, 45]. In the OELIM setup (Fig. 24.5) a beam of collimated light is brought

FA IP LD

CCD camera

OI FCA

LDD

WM Laser and beam expanding optics

PM

Fig. 24.4 Fiber-optic-based OEH setup configured to per-

Part C 24.3

form high-resolution surface shape and deformation measurements: LDD is the laser diode driver, WM is the wavelength meter, PM is the optical power meter, LD is the laser diode, OI is the optical isolator, FCA is the fiber coupler assembly, IP is the image-processing computer, FA is the single-mode fiber-optic directional coupler assembly, RB is the FC-connected reference beam fiber, CCD is the charge-coupled device camera, RS is the rotational stage, BC is the beam combiner, OL is the objective lens, XYZ is the X–Y –Z translational stage, OB is the FC-connected object or illumination beam fiber, OI is the object under investigation, while K 1 and K 2 are the vectors defining illumination and observation directions, respectively

DBS PZT controller LMO O Loading device

Fig. 24.5 Contemporary implementation of the OELIM system based on a commercially available microscope chassis

Holography

24.3 Techniques of Hologram Interferometry

into the system (in some versions of OELIM systems, this is facilitated by the use of fiber optics, enhancing miniaturization and the flexibility of different system configurations) and redirected by a directional beam-splitter (DBS) through a long-working-distance microscope (LMO) to illuminate an object (O). A beam modulated by an interaction with the object propagates back through the DBS onto a sensing element of the CCD camera, where it recombines with the reference beam, resulting in interference patterns, the formation of which is facilitated by the use of the lead zirconiumtitanite (PZT)-controlled LMO. These patterns are, finally, transferred to the system computer for subsequent quantitative processing and display of the results. Using systems such as those shown in Figs. 24.4 and 24.5, issues relating to the sensitivity, accuracy, and precision, of the algorithm defined by (24.8), have been studied while evaluating the effects that the use of modern high-spatial/digital-resolution cameras would have on the results [24.46]. In addition, the development of optimum methods for driving/controlling light sources is conducted concurrently. This development is closely coupled with the development of fiber-optic couplers and corresponding subsystems for the efficient delivery of light beams to ensure the effective functional operation of optoelectronic holography.

illumination and observation directions used during recording and reconstruction of holograms. These directions are defined by illumination and observation vectors, K 1 and K 2 , respectively, as shown in Fig. 24.6. More specifically, the projection matrices are based on the unit vectors defining the directions of illumination and observation [24.50, 51]. The vectors K 1 and K 2 are defined as propagation vectors of light from the point source of illumination to the object and from the object to the observer (i. e., a photosensitive medium or a sensing element of a CCD), respectively (Fig. 24.6). Therefore, K 1 and K 2 can be readily described in terms of position vectors, R1 , Rp , and R2 , which are defined with respect to the origin of a Cartesian x–y–z coordinate system describing the hologram recording/reconstruction geometry; the location of the origin of the coordinate system should be chosen in such a way as to facilitate the determination of pertinent parameters [24.12]. Following (24.2) and (24.3) the position vectors, defining the locations of a point source of illumination, a point on the object, and a point of observation, are mathematically described as follows:

24.3.2 Quantitative Interpretation of Holograms

and

(24.10)

R2 = x2 i + y2 j + z 2 k ,

(24.11)

(24.9)

respectively, where i, j, and k represent unit vectors that are parallel to the axes of the coordinate system. Arbitrary object y

P K1

P

jˆ

Rp

K2

iˆ x K1

kˆ z

R1

R2

K = K2 – K1 – K1

K2 Observation point

Point source of illumination

Fig. 24.6 Illumination and observation geometry in hologram interferometry; a graphical representation of the sensitivity vector, defined by (24.15), is shown in the upper right

Part C 24.3

Quantitative interpretation of holograms is based on one’s ability to delineate various parameters that characterize the recording and reconstruction processes of hologram interferometry [24.47]. These parameters, being vectorial in nature, can be most clearly defined by matrix transformations. The transformations that are pertinent to these analyses map fields of threedimensional vectors from one space into corresponding fields in another space by means of projection matrices [24.11, 48, 49]. The matrix transformation which is of primary interest in the quantitative interpretation of holograms is that which transforms a vector into its shadow on a surface. This transformation may fall into either of two categories: if the direction from which the shadow is cast is parallel to the surface normal, the operation is called a normal projection; if it is not, it is called an oblique projection. The development of matrix transformations and, in general, any procedure for the quantitative interpretation of holograms depends on knowledge of the

R1 = x1 i + y1 j + z 1 k , Rp = xp i + yp j + z p k ,

683

684

Part C

Noncontact Methods

Using (24.9) to (24.11) and the geometry shown in Fig. 24.6, the vectors K 1 and K 2 can be defined (using vector loop equations, VLE) as Rp − R1 |Rp − R1 | (xp − x1 )i + (yp − y1 ) j + (z p − z 1 )k = k√ (xp − x1 )2 + (yp − y1 )2 + (z p − z 1 )2   = k Kˆ 1 = k (| Kˆ 1x |)i + (| Kˆ 1 y |) j + (| Kˆ 1z |)k ,

K 1 = K 1 x i + K 1 y j + K 1z k = k

(24.12)

and R2 − Rp |R2 − Rp | (x2 − xp )i + (y2 − yp ) j + (z 2 − z p )k = k√ (x2 − xp )2 + (y2 − yp )2 + (z 2 − z p )2   = k Kˆ 2 = k (| Kˆ 2x |)i + (| Kˆ 2 y |) j + (| Kˆ 2z |)k .

K 2 = K 2 x i + K 2 y j + K 2z k = k

(24.13)

In (24.12) and (24.13), Kˆ 1 and Kˆ 2 are the unit vectors defining the directions of illumination and observation, respectively, and the parallel bars (i. e., |·|) indicate the magnitudes of the Cartesian components of these unit vectors, while k is the magnitude of the K 1 and K 2 vectors defined as 2π |K 1 | = |K 2 | = k = (24.14) , λ with λ being the wavelength of the light used while recording/reconstructing a hologram. Finally, the definition of the sensitivity vector K as a difference between the observation and the illumination vectors (Fig. 24.6) yields K = K x i + K y j + Kz k = K2 − K1  = k (| Kˆ 2x − Kˆ 1x |)i + (| Kˆ 2 y − Kˆ 1 y |) j  + (| Kˆ 2z − Kˆ 1z |)k ,

(24.15)

Part C 24.3

where K 1 and K 2 are as defined by (24.12) and (24.13), respectively. Determination of Displacements Hologram interferometry is used to measure the displacements and/or deformations of objects that are subjected to static and/or dynamic loads. Depending on the loading method used, the method used to record the hologram, and the method used for reconstructing/readout of the interferometric information, fringe patterns produced during these reconstructions can be classified as either cosinusoidal fringes or Bessel (J0 ) fringes, as discussed in Sect. 24.3.1.

The cosinusoidal fringes are of equal brightness across the image, regardless of the fringe order and are normally associated with a response of the object to a static load (it should be realized that cosinusoidal fringes can also be obtained while using a stroboscopic illumination). Bessel fringes are of unequal brightness, which decreases as the fringe order increases, across the image. These fringes are normally associated with a response of an object to a dynamic loading, especially due to periodic excitation causing an object to vibrate. In addition to displacements, strains (i. e., deformations) can also be determined quantitatively from holograms. In fact, the determination of strains from holograms was of great interest in the early phases of hologram interferometry. As a direct outcome of this need, numerous methods were developed to perform the necessary/anticipated diagnosis. Some of these methods were based on the principles of hologram interferometry which, with its high sensitivity, has found many applications in the analysis of small strains. This becomes obvious when we realize that deformations (on the order of a fraction of a wavelength) of an object, cause fringes to be formed during the reconstruction of a hologram that recorded this deformation. In this way, any change in the state of the object that causes a change in its shape and/or deformation can be measured. Hologram strain analysis was introduced in a classic paper by Ennos in 1968 [24.52]. Since then, a number of applications for the measurement of strains have been developed [24.53–57]. For example, Dändliker et al. [24.54, 55] developed an optoelectronic fringe interpolation method, Schumann et al. [24.11, 56] advanced a theory of fringe localization together with a corresponding apparatus of coupled telescopes, while Stetson [24.57] presented a theory based on a concept of a fringe vector that permitted the determination of strain when it can be assumed to be homogeneous over a sufficiently large region of a sufficiently three-dimensional object. Stetson’s approach is called the fringe-vector theory of holographic strain analysis. This approach recognizes that any combination of a homogeneous strain, shear, and rotation of an object yields fringes on the surface of an object that can be described by a single vector, the fringe vector. The fringe-vector method was generalized by Pryputniewicz and Stetson [24.58, 59] by introducing a procedure which accounts for variations of a sensitivity vector across the object, which therefore allows the application of the method even in the presence of a perspective variation in the illumination and the observation directions.

Holography

24.4 Representative Applications of Holography

685

24.4 Representative Applications of Holography Holography, and especially its optoelectronic implementations as described in Sect. 24.3.1, has been applied to measurements of shape as well as displacements and deformations of objects ranging in size from 300 mm down to several microns (although the literature in the field also reports on holographic analysis of much larger objects, such as automobiles, aircraft, large antennas, bridges, buildings, as well as other macrosize structures/objects). The applications discussed in this section address the determination of the absolute shapes of objects for computer-aided design (CAD), computeraided engineering (CAE), and computer-aided manufacturing (CAM), the determination of displacements and deformations of objects subjected to thermal and mechanical loads, and the quantitative characterization of mode shapes of objects vibrating with submicron, or even, nanometer amplitudes. In the following, representative contemporary applications of holography are described; other applications of (specific) historic nature can be found in abundant literature.

24.4.1 Determination of the Absolute Shape of Objects

in a number of applications involving concurrent engineering, because these quantities are directly related to the functionality, performance, and integrity of objects of current interest [24.60]. Knowledge of the surface shape can be applied to many areas, which include, but are not limited to, reverse engineering, quality control, rapid prototyping, and process simulation. Measured shapes can be used to define CAD models from existing objects, so that these can, in turn, be applied to CAE analysis, or to the manufacturing aspects of these objects via the use of CAM. In addition, measured surface shape can be applied to determine the dimensional accuracy of manufactured objects and also to temporally characterize changes in their geometry when subjected to different types of environmental, tribological, and/or other conditions [24.61]. Current inspection and metrology evaluation relies on contact measurement methods, using, e.g., coordinate measuring machines (CMMs). These machines digitize, on a point-by-point basis, the shape of an a)

b) Z– axis (rad) 60 50 40 30 20 10

Determination of the absolute shape and changes in the state of deformation of objects is a very important issue

Fig. 24.8a,b Tile 1: (a) 3-D data in contour representation, (b) wireframe representation obtained by subsampling the data cloud of (a)

b) Z– axis (rad) 60 50 40 30 20 10

Fig. 24.7 Laboratory setup of a fiber-optic-based OEH sys-

tem for high-resolution surface shape measurements of arbitrary objects, based on the configuration shown in Fig. 24.4

Fig. 24.9a,b Tile 2: (a) 3-D data in contour representation, (b) wireframe representation obtained by subsampling the data cloud of (a)

Part C 24.4

a)

686

Part C

Noncontact Methods

a)

Fig. 24.10a,b Patch obtained by combining tiles 1 and 2: (a) 3-D data in contour representation, (b) wireframe representation obtained by subsampling the data cloud of (a)

b) Z – axis (rad) 25 20 15 10 5 0 –5

c)

a)

Column profile 500

400

b)

V

Smallest feature on the object: 78 μm wide 46 μm long

300

Fig. 24.11a–c Extraction of detail from the OEH-measured 3-D shape of the object: (a) the 3-D shape displayed by a full data cloud, (b) graphical representation of the surface detail within the highlighted area, (c) surface profile measured along the vertical line VV of (b)

200

100 V 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4

Part C 24.4

object into a computer model, rendering the digitized data in three dimensions. Although the contact methods are functional, they are slow and are limited in terms of the number of data points that can be acquired because of practical limitations. However, using recent advances in optoelectronic holography/metrology (Sect. 24.3.1), surface shape and deformation data compatible with computer-based methodologies can be obtained with high spatial and measurement accuracies [24.62]. For example, a relatively large object, characterized by an overall height of 311.3 mm and a largest diameter of 203.2 mm, can be measured using the system shown in Fig. 24.7. However, because of the size, curvature, and accuracy required for measurements of the shape of this object, the OEH system was set up to record interferometric data on small sections (i. e., tiles) of the surface of interest. Local surface geometry was measured on each individual tile (Figs. 24.8 and 24.9) and the individual (i. e., local) results were

patched/stitched together to generate a 3-D representation of a section of the surface that was imaged and measured (Fig. 24.10). Then, by patching/stitching of the remaining tiles, a 3-D representation of the shape of the entire object was obtained (Fig. 24.11a). This representation contains information about the shape of the object to within a specific accuracy; in this case the measurement accuracy was better than 10 μm [24.62]. As such, the 3-D result (Fig. 24.11a) can be used to extract detailed information about the object that was measured. For example, Fig. 24.11b,c shows a surface detail determined from the 3-D shape displayed in Fig. 24.11a. It is emphasized that the results shown in Fig. 24.11, as well as in other figures in this chapter, were obtained without touching (i. e., without any contact whatsoever with) the objects measured; rather the objects were remotely illuminated with light that, upon interaction with the objects being investigated, was also remotely sensed to produce the results displayed herein.

Holography

24.4 Representative Applications of Holography

687

a)

Fig. 24.12 Collision avoidance radar (CAR) system

b)

24.4.2 Determination of Time-Dependent Thermomechanical Deformation Due to Operational Loads The capability of the OEH system to measure temporal changes of the spatial deformation fields of modern, often complex, electronic packages is exemplified by studying thermomechanical (TM) deformations of the collision avoidance radar (CAR) system (Fig. 24.12) [24.63]. The compactness of the CAR system, developed to operate at 78 GHz, is assured by integrating the electronics of the control unit and transceiver into the same single package. Any electronic system, including the CAR, takes a finite time to heat up to its equilibrium temperature under a given set of operating conditions. During this time, systems undergo transient deformations, which should be known in order to determine the quality of performance of the particular system. In the case of the CAR system, the shape of its transceiver, via which the system interacts with its surroundings, is measured at a steady-state power-off condition, i. e., when the system is in thermal equilibrium with the surroundings at ambient temperature; this is a reference shape. Then, changes in its shape are measured as the temperature and the corresponding spatial distribution, i. e., shape, change (when compared b)

to the reference shape) during the power-on condition. Representative OEH fringe patterns corresponding to these states are shown in Fig. 24.13. Quantitative analysis of the OEH fringe patterns of the CAR system leads to results shown in Fig. 24.14. These results are particularly useful for measuring deformations of the transceiver caused by temperature changes due to transient heating (i. e., during poweron) and cooling (i. e., during power-off) of various components in the control unit of the CAR system. These temperature changes cause the deformations to vary in space and in time as the system undergoes various cycles during its functional operation. Obserc)

Fig. 24.14 Spatiotemporal changes of the displacement/deformation patterns of the transceiver as it undergoes various stages of its functional operation during the power-on cycle; time increases from (a) to (c)

Part C 24.4

a)

Fig. 24.13a,b OEH fringe patterns of a transceiver representing: (a) absolute (reference) shape at ambient temperature, (b) deformations during a power-on condition

688

Part C

Noncontact Methods

vation of the results shown in Fig. 24.14 indicates that the extent of the heat-affected zone (HAZ) and the corresponding magnitudes of the thermal deformations change as the time into operation (i. e., after the instant the system is either turned on or off) of the CAR system increases. Animation of the experimental data, such as those shown in Fig. 24.14, vividly indicates the dynamics of TM effects in electronic packaging. It should be noted that, although greatly needed, analytical and computational determinations of results equivalent to those shown in Fig. 24.14 are not possible at this time, because there are too many unknowns (material properties, interfacial and geometrical dimensions, etc.) that need to be provided reliably as inputs for the required analytical and/or computational calculations/ modeling. In fact, the development of other contemporary microelectronic devices and their packaging strongly depends on effective and economical thermal management during their functional operation, which is made feasible and affordable by optoelectronic holography [24.64–68].

24.4.3 Determination of the Operational Characteristics of MEMS

Part C 24.4

MEMS is a revolutionary emerging technology, which is based on manufacturing processes that have their roots in photolithographic processing used in microelectronics and integrated circuits (IC). As such it is used for the development of complex machines with micron feature sizes [24.69]. These machines are batch fabricated, without piece part assembly required, and are ready to be used at the end of the fabrication process. Recent advances in the fabrication technology based on sacrificial surface micromachining (SSM) allow monolithic integration of MEMS with driving, controlling, and signal processing electronics [24.70]. This integration improves the performance of micromechanical devices as well as reducing the cost of manufacturing, packaging, and interconnecting of these devices by combining them with an electronic subsystem in the same manufacturing and packaging process [24.71]. This new multilevel polysilicon process enables the fabrication of complex mechanical systems with intricate coupling mechanisms [24.72–74], as illustrated in the next section, which discusses a high-rotational-speed microengine. Effective development of MEMS, however, requires knowledge of design, analysis/simulation, materials,

fabrication (with special emphasis on packaging), and testing/characterization of the finished products. Educational programs that are being developed to facilitate advances in this emerging technology address pertinent multiphysics issues via an integrated use of analytical, computational, and experimental solutions (ACES) methodology [24.75], where the success of experimental micromechanics solutions is based on the use of optoelectronic holography. In the following section, applications of optoelectronic holography to facilitate the design, analysis, simulation, determination of material properties, packaging, and testing and characterization of representative/selected MEMS are described. This description illustrates the breadth of issues relating to MEMS that are being addressed by the versatile methods based on optoelectronic holography. The MEMS examples are taken from work done by the author. High Rotational Speed Microengine The microengine considered in this section is a part of a micro-optoelectromechanical systems (MOEMS) or, more specifically, microsystems implemented as micromirror devices, which in this case are actuated by electrostatically driven microengines (Fig. 24.15); the goal was to develop a microengine capable to operate at 1 000 000 rpm. These microsystems are fabricated at Sandia National Laboratories (SNL) using Sandia’s Ultraplanar MEMS Multilevel Technology R ) [24.72–74]. The entire micromirror sys(SUMMiT tem is made of polysilicon by SSM. The process does not rely on the assembly of the microsystem out of

Fig. 24.15 Sandia micromirror system actuated by an electrostatically driven microengine, capable of operating at 1 000 000 rpm

Holography

separately fabricated pieces, but produces the finished device by batch fabrication [24.76]. That is, at the end of the fabrication process, the microsystem is ready for use. Using a vector-based analytical model, the forces acting on the microengine during its high rotational speed operation were calculated [24.77]. More specifically, analytical results indicated that the magnitude of these forces increases from 4 nN when the microengine operates at 6000 rpm to 27 μN when the microengine runs at 500 000 rpm. Clearly, this force increases with the rotational speed of the microengine, as expected. The forces generated during the operation of the microengine load the drive/input gear (which has a diameter of about 60 μm with teeth smaller than red blood cells) and make it wobble as it rotates around its shaft. A unique capability to measure this wobble is provided by the OELIM methodology [24.78]. Typical results obtained for two different positions in a rotation cycle of the drive gear are shown in Fig. 24.16, where fringe patterns vividly display changes in the magnitude and direction of the displacements/deformations of the microgears as they rotate during an actual application (which can be readily done since the magnitude a)

b)

during a study of the dynamic characteristics of microengines, at two different positions in a rotation cycle. The white lines indicate the locations (a) and (b) where measurements of displacements referred to in the text were made

689

is proportional to fringe frequency and the direction is normal/perpendicular to fringe slope); incidentally, because of the high rotational speed operation, these images were made possible by stroboscopic illumination (Sect. 24.3) of the MOEMS [24.38]. Displacements of the drive gear, corresponding to the fringe patterns shown in Fig. 24.16, were determined to vary in magnitude from 0.8 μm to 1.7 μm; it should be realized that the thickness of the microengine gears is merely 2 μm (as measured in the direction of the above stated displacements). These variations are due to kinematics and kinetics caused by impulsive loading forces generated by the input signals [24.79]. In addition, experimental results show that the wobble depends on the instantaneous angular position of the gear in its rotation cycle, which can be related to the forces exerted on the drive gear by the pin during the cycle. Design, Analysis, and Simulation Effective development of MEMS requires understanding of the electromechanical and thermal characteristics of these structures and, as a consequence, leads to the need for the application of extensive computational modeling and simulation in their design and analysis [24.80]. This need is amplified by high prototyping costs, long product development cycles, and time-tomarket pressures [24.81]. In addition, the modeling of processes that influence the functional operation of MEMS devices requires a number of different capabilities in simulation software such as the treatment of rarefied/noncontinuum flows and multiphysics computations of coupled fluid–structure–thermal interactions. Although the fluid–structure problems can be analyzed using commercial flow and structures codes [24.82, 83], solutions of some of the problems may lead to convergence difficulties. These difficulties have been overcome in a new MEMS development environment, which has recently been developed for high-fidelity multiphysics simulations of MEMS [24.84]. This software package uses state-of-the-art numerical techniques coupled with user-friendly mesh generation, problem setup, and execution interfaces. As such, it is a great asset not only in the development of MEMS, but also in teaching of the processes used in the development of these devices to students and other prospective practitioners in this field [24.73]. Optoelectronic holography is particularly helpful in providing the material properties necessary for effective modeling of MEMS and other microcomponents as well as in the validation and verification of simulation results [24.6, 85–87].

Part C 24.4

Fig. 24.16 Representative OELIM fringe patterns recorded

24.4 Representative Applications of Holography

690

Part C

Noncontact Methods

a) y (x)

b)

c)

a) y (x)

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

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0.3

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– 0.5

0

z1 (x)

0 –1

b)

– 0.5

0

0.5

c)

1

z2 (x)

Part C 24.4

Fig. 24.17a–c MEMS material property determination by the ACES methodology – first bending mode of a 300 μmlong test sample: (a) analytical result, (b) FEM result, (c)OELIM fringe pattern

Fig. 24.18a–c MEMS material property determination by ACES methodology – second bending mode of a 300 μmlong test sample: (a) analytical result, (b) FEM result, (c) OELIM fringe pattern

Determination of Material Properties by Using Holography Accurate modeling, analysis, and simulation of MEMS (in fact, of all components) depends on knowledge of the mechanical properties of materials in geometries similar in size to those used in the design of the actual devices [24.87]. The determination of mechanical properties of microscale components, however, is not a trivial task because conventional test procedures usually do not apply [24.88]. Therefore, alternative experimental methods must be used. One of these methods is based on the recently developed OELIM methodology. To determine material properties, e.g., modulus of elasticity, E, we record OELIM images of test samples under a variety of loading conditions (Figs. 24.17 and 24.18). From these images, we determine the deformations as a function of position on the sample and the corresponding frequencies subject to the specific loading condition. These deformations, determined experimentally with nanometer accuracy, and the frequencies are then correlated with the analytically and computationally obtained results by using the ACES methodology. The degree of correlation is based on the analytical model of the corresponding uncertainties [24.89]. This model builds on partial differential equations relating specific material property to the sam-

ple geometry, its dimensions, boundary, initial, and loading conditions, and fabrication tolerances. In this way, we determine material properties with high accuracy and precision, using micron-sized samples, and we also estimate how good these properties are. For example, for the test samples shown in Figs. 24.17 and 24.18, the modulus of elasticity was determined to be 160±1.8 GPa [24.90]. The high accuracy of these measurements was required to satisfy product specifications [24.89]. Packaging of MEMS Packaging of MEMS constitutes one of today’s greatest challenges in their fabrication; it is also the most expensive part in their development and the greatest contributor to the cost of finished MEMS-based products [24.91]. Currently, there are a number of different means for attachment of MEMS devices in their packages, depending on the type of device used and its intended application; MEMS packaging is application specific [24.92]. The packaging example that follows considers the development of a MEMS inertial sensor that is of great interest currently as it facilitates advances in global positioning sensing (GPS) systems and may improve inertial measurement units (IMUs).

Holography

24.4 Representative Applications of Holography

Gold bumps

D

B A

691

Braze

C

Interposer

E

A B C D E

Microgyro Wire bond Ceramic chip carrier Metal lid Printed circuit board

Fig. 24.20 Cross sections of chip carrier packages that show gold bumps, gold/tin braze, and interposer attachments of a microgyro Fig. 24.19 Tuning fork microgyro with folded beam sus-

pension

Since their invention, a major drawback to the commercial use of inertial guidance systems has been their cost. However, recent advances in MEMS technology have led to the development of microgyroscopes (Fig. 24.19), which have advantageous size, reliability, and cost over competing technologies. The microgyroscope sensors are usually designed as electronically driven resonators [24.93]. In the configuration shown in Fig. 24.19, a microgyro sensor is actuated by comb drives and is attached to a substrate through suspension springs; the substrate and other intervening layers of materials are, in turn, attached to a chip carrier and all together comprise a package. Since the substrate as well as other components of a package are usually made of different materials than the sensor (it should be realized, at this point, that each material is characterized by a different coefficient of thermal expansion), any temperature change to which a package is subjected will induce stresses that may adversely affect the accuracy and reliability of the sensor outb)

c)

Fig. 24.21a–c OELIM fringe patterns representing deformations of the test samples of Fig. 24.20 subjected to the same temperature of 71 ◦ C: (a) gold bump, (b) gold/tin braze, (c) mechanical interposer

Part C 24.4

a)

put. Therefore, it is necessary to quantify any thermal deformations and corresponding effects that the resulting thermomechanical package stresses may have on the performance of the device in order to optimize its packaging. Currently, there are three different approaches to sensor attachment in a package: (1) gold bump, (2) gold/tin braze, and (3) mechanical interposer (Fig. 24.20). Functional operation of MEMS, regardless of their application, leads to displacements and deformations that are on a nanometer scale. These displacements/deformations are inherent functions of space and time and are characteristic of a specific device. Therefore, in order to understand the operation of MEMS and to optimize their design, it is necessary to know the characteristic displacements/deformations with high accuracy and precision as functions of the operating conditions. Optoelectronic holography is indispensable for obtaining this information. Representative OELIM fringe patterns corresponding to the MEMS package configurations of Fig. 24.20 exposed to the same temperature of 71 ◦ C, are shown

692

Part C

Noncontact Methods

a) 6.20 μm

Time: 130 min

5.79 5.38 4.97 4.56 4.16 3.75

Min 0 Max 6.332 Mean 1.844 Sigma 1.949

3.34 2.93 2.52 2.11 1.70 1.29

Y

0.88 0.47

X

b)

0.07 Min 0

Max 6.332

Mean 1.544

Sigma 1.949

Time: 130 min

Z – axis (μm) 7 6 5 4 3 2 1 0

Fig. 24.22a,b Deformations of the gold/tin brazed test sample based on the OELIM fringe pattern of Fig. 24.21b, recorded at the highest temperature in the thermal loading cycle: (a) contour representation, (b) wireframe representation

Part C 24.4

in Fig. 24.21. Since one of the goals while developing a finished product is to examine the effects of temperature on MEMS packaging (i. e., issues relating to thermal management), fringe patterns clearly indicate different responses of the different test samples subjected to identical temperature conditions; the current operational/functional requirement is that the thermomechanical displacements/deformations of a MEMS package should not exceed 1 μm. Fortunately, differences in the fringe patterns can be readily related to the structural design of MEMS packaging for specific microdevices. Interpretation of the fringe patterns shown in Fig. 24.21 yields quantitative results defining the deformations of the test samples. For example, quantitative analysis of the fringe patterns for the gold/tin braze attachment, when exposed to the highest temperature in the thermal loading cycle, yields the result displayed in Fig. 24.22, which indicates that the maximum de-

formation is 6.3 μm. Similar analysis of the gold bump attachment and of the mechanical interposer attachment yield deformations of 2.2 μm and 1.0 μm, respectively. These deformations are considerably lower than the deformation of the gold/tin braze sample and indicate that the gold bump attachment and the mechanical interposer attachment provide more stable configurations than that provided by the gold/tin braze attachment. The accuracy and precision of a microgyro depend greatly on the quality of its suspension, which is provided by folded springs attached, at one end, to proof masses (i. e., shuttles) and, at the other end, to posts forming a part of a substrate (Fig. 24.23a). Any deformations of the springs that are not in response to the functional operation of a sensor (e.g., those caused by residual stresses due to fabrication) will cause an incorrect output. For example, thermomechanical distortions of a package affect the shape of the posts and these, in turn, lead to undesired deformations of the springs that may adversely affect the response of a sensor, which may output an erroneous signal/result because of inadequate support by the springs. However, due to the nanoscale (magnitude) of the deformations and microsize objects over which these deformations take place, it was not until the advancement of optoelectronic holography that these deformations could be quantified for the first time. Figure 24.23 shows that the thermomechanical deformations of a post are on the order of 40 nm. Functional operation of MEMS accelerometers, just like the operation of microgyros or other highfrequency MEMS sensors/devices, depends on the motion of a proof mass in response to an applied acceleration. The proof masses for a microaccelerometer are suspended by folded springs. Because of advances in design, dual-axis microaccelerometers were developed with suspension springs that have several folds (or turns) (Fig. 24.24). These multifold spring configurations allow a compact sensor design, which facilitates a fast response and thus good performance. As a sensor is actuated its proof mass displaces (Fig. 24.25). Figure 24.25 shows a 1.45 μm displacement of the proof mass and the corresponding deformations of the folded springs at the end of an actuation cycle [24.60] (the thickness of the proof mass, in this case, is approximately 3 μm); for simplicity, because of the symmetric design of the sensor, only the lower right corner, corresponding to the view shown in Fig. 24.24d, is displayed. MEMS radiofrequency (RF) switches are a promising technology for high-performance reconfigurable

Holography

a)

24.4 Representative Applications of Holography

693

c)

Post

H

b) Left shuttle

Right shuttle

d)

H

Row profile

1

0.8 0.6 40 nm

0.4

Post

0.2

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0

Spring

0

200

400

600

800

Fig. 24.23a–d OELIM measurements of thermomechanical deformations of a post in a microgyro: (a) dual-shuttle configuration with the rectangle indicating one of the posts, (b) interference fringe pattern over the post section indicated in part (a), (c) contour representation of the deformations corresponding to the fringe pattern shown in part (b)– the horizontal line HH indicates the trace along which the deformations of the post were determined, (d) deformations determined along the line HH in (c)

a)

b)

c)

d)

50 μm

Fig. 24.24a–d Dual-axis microaccelerometer: (a) overall view of the package, (b)the sensor – highlighted by the square in the center – and electronics integrated on a single chip, (c)the sensor consisting of a proof mass suspended by four pairs of folded springs – the square in the lower right corner highlights one of the spring pairs, (d)view of the spring pair suspending the lower right corner of the proof mass

Part C 24.4

5 mm

694

Part C

Noncontact Methods

a) Z – axis (μm)

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4

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Fig. 24.25a–c Optoelectronic holography measurements depicting the displacements of a proof mass and deformations of the folded springs as a microaccelerometer is actuated, for proof mass displacements of (a) 0.15 μm, (b) 0.74 μm, and (c) 1.45 μm

Part C 24.4

microwave and millimeter wave circuits [24.94]. The low insertion loss, high isolation, and excellent linearity provided by MEMS switches offer significant improvements over the electrical performance provided by p-type intrinsic n-type (PIN) diode and metaloxide semiconductor field-effect transistor (MOSFET) switching technologies. These superior electrical characteristics permit the design of MEMS-switched highfrequency circuits that are not feasible with semiconductor switches, such as high-efficiency broadband amplifiers and quasi-optical beam-steering arrays. In addition, operational benefits arise from the lower power consumption of RF MEMS switches, their small size and weight, and their capability for high levels of integration. Based on their topologies, RF switches can be grouped into two categories [24.95]: (1) membrane type, i. e., capacitive, and (2) cantilever type, i. e., resistive/ohmic. In addition to good knowledge of signal propagation, the development of RF MEMS switches also requires good knowledge of switch dynamics [24.96]. Here, preliminary results relating to the dynamics of a contact RF MEMS switch are presented. These results, based on analytical and computational modeling and simulation, are intended to determine the effects that a contact design has on the performance of a switch. The switch geometry considered in this investigation is shown in Fig. 24.26. This geometry is based on a cantilever beam of active length L that is

fabricated parallel to a substrate in such a way that the initial separation/distance between the electrodes (one electrode being on the bottom of the cantilever and the other on the top of the substrate) do is greater than the contact gap distance dg . The values of do and dg are based on a relationship defining the instantaneous distance, d, between the electrodes, as the switch is being activated, and the corresponding value of the activation voltage as well as other parameters characterizing the switch. If this relationship is not satisfied, the system is unstable and the cantilever snaps down (uncontrollably) to close the switch at a voltage, called the threshold voltage. Operation of a RF switch is facilitated by providing a protruding contact tip beneath the cantilever end, which limits the deflection (i. e., provides a hard stop) of the cantilever displacement and deformation and limits it to satisfy the inequality dg ≤ do /3. In this way it is possible to control the deflection/deformation of the cantilever to such an extent that it no longer becomes unstable and is not hysteretic [24.97]. Effective computational analysis of a RF MEMS switch must simultaneously combine different loads,

L dg

do Substrate

Fig. 24.26 RF MEMS switch: cantilever contact configuration (electrical traces omitted for simplicity)

Fig. 24.27 Computational multiphysics modeling of an RF MEMS switch closure under atmospheric conditions: 3-D representation of air damping; arrows indicate flow of air in a package

Holography

a)

24.5 Conclusions and Future Work

695

b)

Fig. 24.28a,b Dynamic characteristics of a 225 μm-long Si cantilever contact: (a) the first bending mode at 160 kHz and 500 nm amplitude, (b) the second bending mode at 1 MHz and 4 nm amplitude

including, but not limited to: electromagnetic, electrostatic, thermal, mechanical, and aeroelastic. A representative result of such computational multiphysics modeling is shown in Fig. 24.27, which indicates the damping effects of the air/gas surrounding the switch. In some applications, these damping effects help control switch dynamics and facilitate tribological characteristics of contacts. In other applications, they slow the response time of the switch and must be controlled by appropriate/adequate packaging.

Prototype cantilever beams were fabricated and their dynamic characteristics were determined using optoelectronic holography [24.97]. Figure 24.28 shows representative results obtained for a 225 μm-long Si cantilever contact. These results indicate that, as actuation conditions change, the operational response of a switch also changes, as it should. For example, a cantilever vibrating at 160 kHz has a maximum amplitude of 500 nm, while the same cantilever vibrating at 1 MHz has an amplitude of only 4 nm.

24.5 Conclusions and Future Work well suited for testing and characterization of MEMS, as discussed in this chapter. Applications of optoelectronic holography, as described herein, are based on the ACES methodology. This methodology combines analytical, computational, and experimental solutions to provide results where they would not otherwise be obtainable, or at best, be difficult to obtain using analytical, computational, or experimental solutions alone. Results obtained using optoelectronic holography are directly compatible with CAD, CAE, and CAM software packages and facilitate the development and/or characterization of the finished products. Moreover, these results are obtainable under a variety of loading conditions, which can be either static or dynamic in nature. In fact, in a number of applications, optoelectronic holography provides the only means to obtain the spatial distributions of the time- and space-varying deformations, which can then be animated to demonstrate the temporal and spatial characteristics of the transient thermomechanical behavior of objects, as demonstrated herein. It should be noted that, in a number of cases considered in this chapter, computational animations are not possible because the information needed to run, e.g., finite element method (FEM), software packages is not available, or at best is incomplete and, as

Part C 24.5

Herein, advances in optoelectronic holography have been described with an emphasis on static and dynamic measurements of absolute shape as well as displacements and deformations of objects of contemporary interest. Because of its scalability, optoelectronic holography is particularly suitable for applications over a wide range of sizes of the objects, which need to be characterized with high accuracy and precision because of demanding requirements/specifications of the evolving emerging technologies. In fact, the optoelectronic holography is increasingly frequently the methodology of choice when it comes to the testing and characterization of products produced by emerging technologies such as MEMS. MEMS is a revolutionary enabling technology that collocates, on a single chip, functions of sensing, actuation, and control with computation and communication to affect the way people and machines interact with the physical world. MEMS does so on very small scales and uses very low power for operation in multiple environments. Advances in MEMS, however, depend not only on design, analysis, and fabrication, but also on the testing and characterization of the finished products. Because of the microscale features of these products and because of their submicron displacements and deformations, optoelectronic holography is particularly

696

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such, severely limits the modeling and simulation capabilities of the software packages available at this time. Representative applications shown in this chapter indicate that optoelectronic holography can be used to determine material properties, boundary, initial, and loading conditions, as well as displacements and deformations of objects ranging in size from the micron to the millimeter scale and beyond. As such it can be used both to provide the input information necessary to run analytical and computational modeling and to verify and/or validate the results produced by such modeling and simulation. The need for remote and noninvasive measurements across the full field of view (FFV), providing

data/results in three dimensions and in real time, as provided by optoelectronic holography, will continue to increase as emerging technologies evolve into mature technologies. This need will continue to be over multiple scales ranging from millimeters to nanometers and even down to picometers as the size of the building blocks out of which large structures will be made in the future shrinks. This decreasing size will be driven by advances in nanotechnology, which will undoubtedly be closely followed by corresponding demands for experimental mechanics solutions [24.98]. In order to satisfy the testing and characterization demands generated by emerging technologies, the development of metrology, especially optoelectronic holography, should be continued.

References 24.1 24.2 24.3 24.4

24.5

24.6

24.7

24.8

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24.10

24.11 24.12 24.13

D. Gabor: A new microscopy principle, Nature 161, 777–778 (1948) D. Gabor: Microscopy by reconstructed wavefronts, Proc. Soc. A 197, 454–487 (1949) D. Gabor: Microscopy by reconstructed wavefronts: II, Proc. Soc. B 64, 449–453 (1951) M.G. Lippmann: La photographie des couleurs, Comptes Rendus Hebdomadaires des Séances de l’Academie des sciences 112, 274 (1891) Y.N. Denisyuk: Photographic reconstruction of the optical properties of an object in its own scattered radiation field, Sov. Phys. Doklady 7, 543 (1962) R.J. Pryputniewicz: Heterodyne holography applications in studies of small components, Opt. Eng. 24, 849–854 (1985) R.L. Powell, K.A. Stetson: Interferometric vibration analysis by wavefront reconstruction, J. Opt. Soc. Am. 55, 1593–1598 (1965) C.M. Vest, R.J. Pryputniewicz: Applications of Holography, SPIE Milestone Series (SPIE, Bellingham 2007) R.J. Pryputniewicz: Quantification of Holographic Interferograms: State-of-the-art Methods, Technical Digest (OSA, Washington 1986) R.J. Pryputniewicz: Automated systems for quantitative analysis of holograms, Holography: Commemorating the 90-th Anniversary of the Birth of Dennis Gabor, ed. by P. Gregus, T.H. Jeong (Tatabána, 1990) pp. 215–246 W. Schumann, J.P. Zürcher, D. Cuche: Holography and Deformation Analysis (Springer, Berlin 1985) R.J. Pryputniewicz: Holographic Numerical Analysis (Worcester Polytechnic Institute, Worcester 1992) H.J. Caulfield (Ed.): The art and science of holography: a tribute to Emmett Leith and Yuri Denisyuk (SPIE Press, Bellingham 2004)

24.14 24.15

24.16 24.17 24.18

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24.22 24.23

24.24 24.25 24.26 24.27 24.28

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Holography

24.29 24.30 24.31 24.32

24.33

24.34

24.35

24.36

24.37

24.38

24.39

24.40

24.42

24.43

24.44

24.45

24.46

24.47

24.48

24.49

24.50

24.51

24.52

24.53

24.54

24.55

24.56

24.57

24.58

24.59

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C. Brown, R.J. Pryputniewicz: Holographic microscope for measuring displacements of vibrating microbeams using time-average electro-optic holography, Opt. Eng. 37, 1398–1405 (1998) C. Furlong, J.S. Yokum, R.J. Pryputniewicz: Sensitivity, accuracy, and precision issues in optoelectronic holography based on fiber optics and high-spatial and high-digital resolution cameras, Proc. SPIE 4778, 216–223 (2002) R.J. Pryputniewicz, K.A. Stetson: Fundamentals and Applications of Laser Speckle and Hologram Interferometry (Worcester Polytechnic Institute, Worcester 1980) R.J. Pryputniewicz: Projection matrices in specklegraphic displacement analysis, Proc. SPIE 243, 158–164 (1980) K.A. Stetson: The use of projection matrices in hologram interferometry, J. Opt. Soc. Am. 71, 1248– 1257 (1981) R.J. Pryputniewicz: Determination of sensitivity vectors directly from holograms, J. Opt. Soc. Am. 67, 1351–1353 (1977) R.J. Pryputniewicz, K.A. Stetson: Determination of sensitivity vectors in hologram interferometry from two known rotations of the object, Appl. Opt. 19, 2201–2205 (1980) A.E. Ennos: Measurement of in-plane strain by hologram interferometry, J. Phys. E (Sci. Instrum.) 1, 731–746 (1968) E.A. Ennos: Strain measurement. In: Holographic Nondestructive Testing, ed. by R.K. Erf (Academic, New York 1974) pp. 275–287 R. Dändliker, B. Ineichen, F.M. Mottier: High resolution hologram interferometry, Opt. Commun. 9, 412–416 (1973) R. Dändliker, B. Ineichen: Strain measurement through hologram interferometry, Proc. SPIE 99, 90–98 (1976) W. Schumann, M. Dubas: On direct measurement of strain in holographic interferometry using the line of complete localization, Opt. Acta 22, 807 (1975) K.A. Stetson: Homogeneous deformations: determination by fringe vectors in hologram interferometry, Appl. Opt. 14, 2256–2259 (1975) R.J. Pryputniewicz, K.A. Stetson: Holographic strain analysis: extension of fringe vector method to include perspective, Appl. Opt. 15, 725–728 (1976) R.J. Pryputniewicz: Holographic strain analysis: an experimental implementation of the fringe vector theory, Appl. Opt. 17, 3613–3618 (1978) C. Furlong, R.J. Pryputniewicz: Optoelectronic characterization of shape and deformation of MEMS accelerometers used in transportation applications, Opt. Eng. 42, 1223–1231 (2003) C. Furlong, J.S. Yokum, K.W. Franklin, G.C. Cockrell, R.J. Pryputniewicz: Optoelectronic method for high accuracy measurements of absolute shape of objects, Proc. Internat. Congress on Experimental

697

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W.E. Kock: Engineering Applications of Lasers and Holography (Plenum, New York 1975) R.K. Erf (Ed.): Holographic Nondestructive Testing (Academic, New York 1974) C.M. Vest: Holographic Interferometry (Wiley, New York 1979) R.J. Pryputniewicz: Hologram interferometry from silver halide to silicon and . . . beyond, Proc. SPIE 2545, 405–427 (1995) R.J. Pryputniewicz: Quantitative determination of displacements strains from holograms. In: Holographic Interferometry, Springer Series in Sciences, Vol. 68, ed. by P. Rastogi (Springer, Berlin 1995) pp. 33–72 R.J. Pryputniewicz, C.J. Burstone: Pulsed laser holography for orthodontics, J. Dental Res. 62, 346–352 (1983) J.W. Forbes, R.J. Pryputniewicz: Automated fringe analysis in pulsed laser holography. In: Hologram interferometry and speckle metrology, ed. by K.A. Stetson, R.J. Pryputniewicz (Soc. for Exp. Mech., Bethel 1990) pp. 515–522 G.C. Brown, R.J. Pryputniewicz: Investigation of submillimeter components by heterodyne holographic interferometry and computational methods, Proc. SPIE 1755, 120–130 (1992) R.J. Pryputniewicz: Vibration studies using heterodyne hologram interferometry, Proc. SPIE 955, 154–161 (1988) G.C. Brown, R.J. Pryputniewicz, M.P. deBoer, S.L. Miller: Characterization of MEMS microgears rotating up to 360000 rpm by stroboscopic optoelectronic laser interferometry microscope (SOELIM) methodology, Proc. SPIE 4101B, 592–600 (2000) R.T. Marinis, A.R. Klempner, S.P. Mizar, P. Hefti, R.J. Pryputniewicz: Stroboscopic illumination using LED light source, Proc. 32nd Annual Symp. and Exhibition of IMAPS-NE (Boxboro 2005) pp. 392– 396 R.J. Pryputniewicz: Quantitative interpretation of time-average holograms in vibration analysis. In: NATO Advanced Science Institute Series on Optical Metrology, ed. by O.D.D. Soares (Martinus Nijhoff, Dordrecht 1987) pp. 296–316 R.J. Pryputniewicz: Time-average holography in vibration analysis, Opt. Eng. 24, 843–848 (1985) R.J. Pryputniewicz, K.A. Stetson: Measurement of vibration patterns using electro-optic holography, Proc. SPIE 1162, 456–467 (1990) G.C. Brown: Laser interferometric methodologies for characterizing static and dynamic behavior of MEMS. Ph.D. Thesis (Worcester Polytechnic Institute, Worcester 1999) C. Furlong: Hybrid, experimental and computational. approach for the efficient study and optimization of mechanical and electro-mechanical components Ph.D. Thesis (Worcester Polytechnic Institute, Worcester 1999)

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and Applied Mechanics for Emerging Technologies (Portland 2001) pp. 701–704 J.S. Yokum, C. Furlong, K.W. Franklin, G.D. Cockrell, R.J. Pryputniewicz: Fiber optic based laser optoelectronic holographic system for shape measurements, J. Pract. Fail. Anal. 1, 63–70 (2001) C. Furlong, R. J. Pryputniewicz: Novel experimentalcomputational method for quantitative applications in electronic packaging, Paper No. IMECE2001/EPP-24732 (ASME – Am. Soc. Mech. Eng., New York 2001) C. Furlong, R.J. Pryputniewicz: Computational and experimental approach to thermal management in microelectronics and packaging, J. Microelectron. Int. 18, 35–39 (2001) R.J. Pryputniewicz, S.A. Weller, S.R. Shaw, W.L. Herb, D.R. Pryputniewicz: Thermal management considerations of SFP system for high speed telecommunications, Proc. Photonics Materials Reliability Symposium (PhoMat 2003) (San Francisco 2003) pp. 101–105 R.J. Pryputniewicz, D.A. Rosato, C. Furlong, D.R. Pryputniewicz: Hybrid approach to thermal management of a FET power amplifier, Proc. 36th Internat. Symp. on Microelectron. (Boston 2003) pp. 841–846 R.J. Pryputniewicz, D.A. Rosato, K.A. Nowakowski: Thermal management in packaging of microscale systems, Proc. 32nd Annual Symp. and Exhibition of IMAPS-NE (Boxboro 2005) pp. 385–391 R.J. Pryputniewicz: Thermal Management of RF MEMS Relay Switch, European Conference on Fracture (2006), in review E.J. Pryputniewicz: Study of Design, Analysis, Fabrication, Characterization, and Reliability of MEMS at Sandia (Sandia National Laboratories, Albuquerque 2000) J.H. Smith: Micromachined sensor and actuator research at Sandia’s Microelectronics Development Laboratory, Sensors Expo ’96 (Anaheim 1996) pp. 119–123 J. H. Smith, S. Montague, J. J. Sniegowski, J. Murray, P. J. McWhorter: Embedded micromechanical devices for the monolithic integration of MEMS with CMOS, Proc. IEDM ’95, 609–615 (1995) M.S. Rogers, J.J. Sniegowski: 5-level polysilicon surface micromachine technology: application to complex mechanical systems, Proc. Solid State Sensor and Actuator Workshop (Hilton Head 1998) R. J. Pryputniewicz: MEMS design education by case studies, Paper No. IMECE2001/DE-23292 (ASME – Am. Soc. Mech. Eng., New York 2001) R.J. Pryputniewicz, E. Shepherd, J.J. Allen, C. Furlong: University – National Laboratory alliance for MEMS education, Proc. 4-th Internat. Symp. on MEMS and Nanotechnology (4-th-ISMAN) (Charlotte 2003) pp. 364–371

24.75

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R. J. Pryputniewicz: Integrated approach to teaching of design, analysis, and characterization in micromechatronics, Paper No. IMECE2000/DE-13 (ASME – Am. Soc. Mech. Eng., New York, NY 2000) E.J. Garcia, J.J. Sniegowski: Surface micromachined microengine, Sens. Actuator. A 48, 203–214 (1995) E.J. Pryputniewicz: ACES Approach to the Study of Electrostatically Driven MEMS Microengines (Worcester Polytechnic Institute, Worcester 2000), MS Thesis R.J. Pryputniewicz, C. Furlong: Novel optoelectronic methodology for testing of MOEMS, Proc. SPIE – Internat. Symp. on MOEMS and Miniaturized Systems III, Vol. 4983 (2003) pp. 11–25 E.J. Pryputniewicz, S.L. Miller, M.P. deBoer, G.C. Brown, R.R. Biederman, R.J. Pryputniewicz: Experimental and analytical characterization of dynamic effects in electrostatic microengines, Proc. Internat. Symp. on Microscale Systems (Orlando 2000) pp. 80–83 R.J. Pryputniewicz, X.G. Tan, A.J. Przekwas: Modeling and measurements of MEMS gyroscopes, Proc. Homeland Security Conf. on Inertial Sensing Technology (Monterey 2004), IEEE-PLANS2004:111-119 A.J. Przekwas, M. Turowski, M. Furmanczyk, A. Hieke, R.J. Pryputniewicz: Multiphysics design and simulation environment for microelectromechanical systems, Proc. Internat. Symp. on MEMS: Mechanics and Measurements (Portland 2001) pp. 84–89 A. Klein, S. Matsumoto, G. Gerlach: Modeling and design optimization of a novel micropump, Proc. Internat. Conf. on Modeling and Simulation of Microsystems (Santa Clara 1998) pp. 506–511 J. Ulrich, R. Zengerle: Static and dynamic flow simulation of KOH-etched microvalve using the finite element method, J. Sens. Actuator 53, 379–385 (1996) M.M. Athavale, H.Q. Yang, H.Y. Li, A.J. Przekwas: A high-fidelity simulation environment for thermo-fluid-mechanical design of MEMS (Final Report for Baseline Project, DARPA Contract 1998) R.J. Pryputniewicz: Quantitative holographic analysis of small components, Proc. SPIE 604, 71–85 (1986) R.J. Pryputniewicz (Ed.): Unification of analytical, computational, and experimental solution methodologies in micromechanics and microsystems, Proc. 11-th Internat. Invitational UACEM Symp. (SEM, Bethel 1993) R.J. Pryputniewicz, P. Galambos, G.C. Brown, C. Furlong, E.J. Pryputniewicz: ACES characterization of surface micromachined microfluidic devices, Int. J. Microelectron. Electron. Packag. (IJMEP) 24, 30–36 (2001)

Holography

24.88

24.89

24.90

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24.92

W.N. Sharpe Jr., B. Yuan, R.L. Edwards: A new technique for measuring the mechanical properties of thin films, JMEMS 6, 193–199 (1997) R. J. Pryputniewicz, C. Furlong, E. J. Pryputniewicz: Optimization of contact dynamics for an RF MEMS switch, Paper No. IMECE2002-39504 (ASME - Am. Soc. Mech. Eng., New York, NY 2002) D.R. Pryputniewicz, C. Furlong, R.J. Pryputniewicz: ACES approach to the study of material properties of MEMS, Proc. Internat. Symp. on MEMS: Mechanics and Measurements (Portland 2001) pp. 80–83 D. Hanson, T. F. Marinis, C. Furlong, R. J. Pryputniewicz: Advances in optimization of MEMS inertial sensor packaging, Proc. Internat. Congress on Experimental and Applied Mechanics for Emerging Technologies, Soc. Exp. Mech. (Portland 2001) pp. 821–825 T. F. Marinis, J. W. Soucy, D. S. Hanson, R. J. Pryputniewicz, R. T. Marinis, A. R. Klempner: Isolation of MEMS devices from package stresses by use of compliant metal interposers, Proc. ECTC, 56, 1108–1117 (2006)

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Part C 24

701

Photoelastici 25. Photoelasticity

Krishnamurthi Ramesh

An overview of photoelasticity and its several variants ranging from conventional transmission photoelasticity to digital photoelasticity is presented. A representative fringe pattern for each of the variants is provided to give a glimpse of the range of problems it can solve. The technique basically provides the difference of principal stresses/strains and their orientation at every point in the model domain. It is the only whole-field technique which can study the interior of a threedimensional model. The advancements in digital photoelasticity have made photoelastic analysis more efficient and reliable for solving engineering problems. Photoelasticity is useful as a design tool, to understand complex phenomenological issues, and as an excellent teaching aid for stress analysis. It can be used to study models made of transparent plastics, prototypes made of different materials, and also directly on end products such as glass components. With developments in rapid prototyping and novel methods for fringe plotting from finite element results, the technique is ideally suited for hybrid analysis of complex problems.

25.1 Preliminaries ....................................... 25.1.1 Polarization............................... 25.1.2 Birefringence............................. 25.1.3 Retardation Plates......................

704 704 704 705

25.2.3 Fringe Contours in a Plane Polariscope ................ 25.2.4 Jones Calculus............................ 25.2.5 Ordering of Isoclinics .................. 25.2.6 Ordering of Isochromatics............ 25.2.7 Calibration of Model Materials .....

706 707 708 709 709

25.3 Variants of Photoelasticity..................... 25.3.1 Three-Dimensional Photoelasticity 25.3.2 Dynamic Photoelasticity .............. 25.3.3 Reflection Photoelasticity ............ 25.3.4 Photo-orthotropic Elasticity......... 25.3.5 Photoplasticity...........................

710 710 714 715 717 718

25.4 Digital Photoelasticity ........................... 25.4.1 Fringe Multiplication and Fringe Thinning ................... 25.4.2 Phase Shifting in Photoelasticity .. 25.4.3 Ambiguity in Phase Maps ............ 25.4.4 Evaluation of Isoclinics ............... 25.4.5 Unwrapping Methodologies......... 25.4.6 Color Image Processing Techniques 25.4.7 Digital Polariscopes ....................

719 719 720 723 725 727 729 730

25.5 Fusion of Digital Photoelasticity Rapid Prototyping and Finite Element Analysis. 732 25.6 Interpretation of Photoelasticity Results. 734 25.7 Stress Separation Techniques................. 25.7.1 Shear Difference Technique ......... 25.7.2 Three-Dimensional Photoelasticity 25.7.3 Reflection Photoelasticity ............

735 735 736 736

25.8 Closure ................................................ 737 25.9 Further Reading ................................... 737

25.2 Transmission Photoelasticity ................. 705 25.2.1 Physical Principle ....................... 705 25.2.2 Stress–Optic Law ........................ 706

25.A Appendix ............................................. 738

Analytical or numerical methods for stress analysis require the formulation of the problem in general as differential equations and subsequent solution of this is obtained either analytically or numerically with suit-

able approximations depending on the complexity of the problem. The results can then be postprocessed to present them in the form of contours of displacement/strain/stress or a combination of these components

References .................................................. 740

Part C 25

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Noncontact Methods

using computer graphics. On the other hand, most whole-field optical methods provide certain specific contours when the model is loaded and viewed with appropriate optics. Thus, all whole-field methods are optical computers to provide instant contours of a certain physical variable. The physical principle on which the experimental method is based determines what these contours represent. Let us consider the problem of a beam under fourpoint bending (Fig. 25.1a) for which a closed-form solution exists even from strength of materials approach. The central section of the beam away from the loading points will have stresses governed by simple beam theory. The stress tensor at any point is given by   − MIzzz y 0 . (τij ) = 0 0 The principal stresses for this problem are simply σ1 = σx and σ2 = 0. How will the contours of (σ1 − σ2 ) be for the middle section of the beam? A contour is nothing but a collection of points having the same magnitude of the variable. Since σx varies linearly over the depth of the beam and is the same along a horizontal line, the contours of discrete values will be horizontal lines parallel to the x-axis. If we have a color plotting facility available, it will show a smooth variation of colors. Now let us consider a model of a beam made of photoelastic material viewed in photoelastic equipment a)

b)

y P

P

– x

P

+

P

c)

d)

e)

f)

Part C 25

3 2 1 0 1 2 3

Fig. 25.1 (a) Beam under four-point bending. (b) Stress variation over the depth. (c) Theoretical representation of (σ1 − σ2 ) contours. Experimentally recorded images using: (d) monochromatic light source, (e) white light source, (f) enlarged view of the tile selected in (e) with fringe orders marked

with a monochromatic light source and appropriate optical arrangement. The resulting contours are shown in Fig. 25.1d, which are again horizontal lines. The one difference is that the contours appear as a band rather than a line. This is due to the deficiency of the recording medium. If the light source is white light then the contours are nicely colored (Fig. 25.1e,f). Thus if an unknown model is viewed with the same optical arrangement, then the contours could be physically interpreted as representing contours of principal stress difference. Such contours are known as isochromatics in the photoelasticity literature. In the specific example illustrated, the contours of principal stress difference are the same as the contours of the stress component σx . However, this is only a special case and should not be confused with the information generally provided by photoelasticity. Apart from this, photoelasticity also provides the contours of principal stress orientation, known as isoclinics; one needs to choose another optical arrangement to obtain these. Photoelasticity comes under the category of common path interferometers and vibrations do not have much effect on these. Furthermore, the coherence requirements are not stringent. Hence, photoelasticity has gained wide acceptance and established itself as an excellent tool in visualizing/quantifying stress fields and as a teaching aid for stress analysis. Though the phenomenon of temporary birefringence, the physical principle on which photoelasticity is based, was discovered by Brewster [25.1] in 1816, it was only in the 1930s through the works of Coker and Filon in the UK that the technique gained popularity. Oppel in Germany was the first to introduce the concept of frozen stress photoelasticity to analyze three-dimensional models. Frocht and Durelli and their innumerable students further advanced the technique in the USA. With the availability of epoxy resins in the 1950s, the extension of photoelasticity to the analysis of prototypes became a reality and in 1960 Zandman introduced a unique procedure for preparing contourable plastics for coating complicated geometries of industrial components. Even a qualitative study of isochromatic fringe patterns can provide sufficient information for comparative evaluation of various designs. Shape optimization can be effected by altering the boundary of the component such that the fringes on the boundary follow the boundary contour rather than cutting it. Such fillets are known as streamline fillets. The evaluation of the stress concentration factor (SCF), the stress intensity factor (SIF), and the contact stress parameters only demands knowledge of the isochromatic fringe order at selected points

Photoelasticity

in a variety of specimen materials such as aluminium alloys, concrete, ceramic, bone, composite materials, and rubber. The technique has also been found to be useful for detecting assembly stresses/residual stresses. In view of their complexity, no mathematical modeling is easy for these classes of problems. In this chapter the interpretation of transmission photoelasticity fringe patterns based on knowledge of crystal optics is initially discussed. Issues relating to fringe ordering and the calibration of the photoelastic model material is also discussed. The methodologies of several variants of photoelasticity are then presented briefly. The use of fibre-reinforced composites as structural materials saw the birth of photo-orthotropic elasticity. A desire to extend photoelastic analysis for solving metal-forming operations saw the birth of photoplasticity. An outline of these methods is also given. The application of photoelasticity to solve industrial problems had a temporary setback between the mid 1960s and the 1980s due to the euphoria generated by the development of the numerical method of finite elements. In any problem of practical interest, the simulation of boundary conditions is a key issue and, if not done properly, one is actually solving a difa)

b)

c)

d)

Fig. 25.2a–d Load transfer in dry granular media by force chains

forming an inhomogeneous contact network. Experimental isochromatics: (a) low shear stress, (b) high-stress isotropic compression. Theoretical simulation where forces lower than 10% of the mean force are not computed: (c) low shear stress, (d) high-stress isotropic compression (after [25.2])

Part C 25

in the field. Furthermore, in two-dimensional problems, if the principal stresses have opposite signs, then the isochromatic fringe contour is equivalent to the Tresca yield contour for zones where the yield condition is satisfied. In three-dimensional problems, evaluation of von Mises stress at a point of interest can be determined from photoelastic data alone. Thus, for a wide variety of design problems, photoelasticity provides direct quantitative information. Photoelasticity has always been a leading method for gaining phenomenological understanding. The examples include the need for multiple parameters for crack-tip stress field modeling, various issues related to stress wave propagation, crack–stress wave interactions, etc. Photoelasticity is not only an experimental tool to evaluate stresses but has contributed significantly to the development of solution methods (analytical/numerical) for stress analysis. During the early stages of the development of the theory of elasticity, the solutions obtained by the theory were verified by using the experimental results of photoelasticity. Rapid strides in the field of fracture mechanics were possible only because of the experimental verification provided by photoelasticity. The recent use of conventional transmission photoelasticity for contact force measurements and stress-induced anisotropy in granular materials (Fig. 25.2) has reinforced the utility of photoelasticity for solving complex phenomenological problems even today [25.2]. Three-dimensional photoelasticity has been used to solve a wide variety of industrial problems spanning different disciplines that include the determination of the stress distribution in screw threads and gears; the determination of the SCF in nuclear reactors and solid propellant grains; the design of nozzles, turbines, engines, dams, tunnels, and bridges; and the modeling of layered soil in geotechnical engineering etc. Integrated photoelasticity, and scattered light photoelasticity, which are variants of three-dimensional photoelasticity, are routinely used in glass industries for process monitoring and quality control of bulbs, tumblers, cathode-ray tubes, plate glass, fibre-optic glass preforms etc. Dynamic photoelasticity has been used with success for manufacturing applications such as water jet machining, identifying the sequence of explosive welding in the fabrication of nuclear heat exchangers etc. Over the years, photoelastic coatings have found applications in various fields ranging from aerospace to biomechanics. Since the technique is strain based, it has been applied with success to analyze strain/stress fields

703

704

Part C

Noncontact Methods

ferent problem. Problems relating to assembly stress, residual stress, and contact stress are quite complex to model numerically. Experiments are the best choice for such class of problems and it may be worthwhile combining both experimental and numerical techniques judiciously. For a parametric analysis, numerical methods are the best, as they are cost effective. However, it is desirable to verify the choice of elements, discretization scheme, imposed loading, and boundary conditions of the numerical model for at least one configuration by experimental means. Modern photoelasticity no longer looks at the basic data as fringe patterns but as intensity data. Voloshin and Burger [25.3] were the first to exploit the intensity data and developed half-fringe photoelasticity (HFP). The use of digital cameras replaced the human eye, and digital hardware was then used to mimic conventional data augmentation methods such as fringe multiplication and fringe thinning for improved accuracy. Though such developments simplified human effort and are quite useful in certain applications, whole-field automated photoelasticity was possible only with the development of phase-shifting techniques. Several innovative tech-

niques to use color information for quantitative analysis have also been developed. A detailed account of these developments can be found in [25.4]. Digital photoelasticity is researched worldwide, notably in the UK, Italy, India, Singapore, Japan, the USA, and France and is a fast developing technology. Autonomous fringe ordering is one of its goals and this chapter provides a brief comprehensive overview of recent advances in this area. With advances in digital photoelasticity, newer and sleeker equipment for photoelasticity has been developed. Advances in rapid prototyping have made a significant impact on photoelastic model making and newer initiatives are needed to carry out quick comparisons of the results. Brief summaries of these developments are given later in this chapter. With these developments, photoelasticity is now quite competitive with numerical methods and is being used more widely in industry. As photoelasticity is basically done using plastics, how these results can be interpreted for physical prototypes is then presented. The maximum information obtainable by using only photoelasticity and the techniques for stress separation are presented towards the end of the chapter.

25.1 Preliminaries Typically in photoelasticity, for a known incident light, the emergent light is analyzed for stress/strain information. Rather than impinging natural light on the model, polarized light is usually used. The formation of fringes in all variants of photoelasticity is due to the phenomenon of birefringence exhibited by the models/coatings. The photoelastic model and elements forming a photoelastic equipment can be thought of as a retarder or a series of retarders. Concepts related to these are discussed next.

25.1.1 Polarization

Part C 25.1

Most light sources consist of a large number of randomly oriented atomic or molecular emitters. The light rays emitted from such a source will have no preferred orientation and the tip of the light vector describes a random vibratory motion in a plane transverse to the direction of propagation. If the tip of the light vector is forced to follow a definite law, the light is said to be polarized. For example, if the tip is constrained to lie on the circumference of a circle, it is said to be circularly po-

larized. If the tip describes an ellipse, it is said to be elliptically polarized. If the light vector is parallel to a given direction in the wavefront, it is said to be linearly or plane polarized.

25.1.2 Birefringence The crystalline media are optically anisotropic and are characterized by the fact that a single incident ray will give rise to two refracted rays, ordinary (‘o’) and extraordinary (‘e’), thus exhibiting birefringence or double refraction (see the animation on the enclosed DVD [25.5, 6]). The ordinary and extraordinary rays are plane polarized and their planes of polarization are perpendicular to each other. One of the rays is extraordinary in the sense that it violates Snell’s Law under suitable circumstances. This ray need not be confined to the plane of incidence. Furthermore, its velocity changes in a continuous way with the angle of incidence. The two indices of refraction for the ‘e’ and ‘o’ rays are equal only in the direction of an optic axis. An isotropic medium can transmit common light, whereas light traveling through a crystal is always polarized.

Photoelasticity

25.2 Transmission Photoelasticity

705

25.1.3 Retardation Plates h

Consider a crystalline plate of thickness h. Let planepolarized light be incident normally, as shown in Fig. 25.3a. Let the polarizing axes of the plate be oriented at angles θ and θ + π/2 to the horizontal. Since the incident ray is perpendicular to the optic (polarizing) axes, two rays (namely, the ordinary and the extraordinary rays) travel in the same direction but with different velocities, v1 and v2 . These two rays will have a net phase difference of δ upon emergence from the crystal plate. Since the velocities of propagation within the crystal are different for these two rays, they will take respective times of h/v1 and h/v2 to traverse the plate. This time difference contributes to the phase difference. Let the frequency of the light be f , then     h 1 c 1 h − − = 2πh δ = 2π f v1 v2 λ v1 v2 2πh = (n 1 − n 2 ) . λ The emerging light is, in general, elliptically polarized. If the thickness is such as to produce a phase difference of π/2 radians, then, it is known a quarter-wave plate (λ/4); if the phase difference is π radians, then it is called a half-wave plate. If the retardation is 2π, then, one obtains a full-wave plate and the incident light is unaltered. Usually, wave plates are made of mica or quartz and λ/4 plates are of the thickness of a millimeter. It is to be noted that a crystal plate can be called a quarter-wave plate, half-wave plate, or a full-wave plate only for a particular wavelength. Wave plates or retarders have two polarizing axes; one of these axes

F

P

Source

y

θ

S x δ F

a)

b)

y

θ

x

S

Fig. 25.3 (a) Linearly polarized light incident on a retarder. (b) Inside the retarder two plane-polarized lights, whose planes of polarization coinciding with the fast and slow axes of the retarder, travel with different velocities

is labeled the fast axis (F) and the other as the slow axis (S). Plane-polarized light can be produced from a natural source by the following ways: (1) (2) (3) (4)

Reflection Scattering Use of Polaroid sheets Nicol’s prism

Using wave plates, linearly polarized light can be changed to circular or elliptically polarized light. If the tip of the light vector describes a counterclockwise motion, then it is said to be right-handedly polarized; if the path is traversed in a clockwise direction, then it is said to be left-handedly polarized.

25.2 Transmission Photoelasticity

25.2.1 Physical Principle Certain noncrystalline transparent materials, notably some polymeric plastics, are optically isotropic under normal conditions but become doubly refractive or bire-

fringent when stressed. This effect normally persists while the loads are maintained but vanishes almost instantaneously or after some interval of time depending on the material and conditions of loading when the loads are removed. This is the physical characteristic on which photoelasticity is based. The photoelastic model behaves like a retarder with different retarder characteristics at different points in the model governed by the induced stress field. The light has to hit the model at normal incidence and the principal stress directions act as polarizing axes at the point of interest. The ordinary and extraordinary rays are polar-

Part C 25.2

Photoelastic phenomenon can be best understood by a study of transmission photoelasticity. In this, usually plane or circularly polarized light is used to analyze a transparent model for stress information. Since polarized light is used, the basic equipment used is called a polariscope.

706

Part C

Noncontact Methods

ized with their polarizing axes mutually perpendicular and travel in the same direction but with different velocities. When the rays emerge, there exists a relative retardation between the rays, which contributes to the formation of fringes.

difference. Equation (25.3) shows the dependence of Fσ on the wavelength. Equation (25.2) implicitly indicates that Fσ and (σ1 − σ2 ) are linearly related. However, at higher stress levels, the relationship is nonlinear and (25.2) cannot be used. Therefore, (25.2) should be used with care.

25.2.2 Stress–Optic Law Consider a transparent model made of a high polymer subjected to a plane state of stress. Let the state of stress at a point be characterized by the principal stresses σ1 , σ2 and their orientation θ with reference to a set of axes. Let n 1 and n 2 be the refractive indices for vibrations corresponding to these two directions. Following Maxwell’s formulation, the relative retardation in terms of principal stress difference can be written as 2πh 2πh (25.1) (n 1 − n 2 ) = C(σ1 − σ2 ) , λ λ where C is the relative stress–optic coefficient, which is usually assumed to be a constant for a material. However, various studies have shown that this coefficient depends on wavelength and should be used with care. Equation (25.1) can be rewritten in terms of the fringe order N as C δ = h (σ1 − σ2 ) , N= 2π λ which can be recast as NFσ σ1 − σ2 = (25.2) , h where λ Fσ = , (25.3) C which is known as the material stress fringe value with units of N/mm/fringe. Equation (25.2) is famously known as the stress–optic law as it relates the stress information to an optical measurement. The principal stresses are labeled such that σ1 is always algebraically greater than σ2 . In view of this, the left-hand side of (25.2) is always positive. Thus, in photoelasticity, the fringe order N is always positive. However, there are methods to find the sign of the boundary stress. It can be determined by a nail test wherein a compressive load is applied to the free edge using the nail or a sharp edge. The change in the fringe order (usually the change in color) is noted to decide the sign of the boundary stresses. If the sign of the boundary stress is positive the fringe order will increase and vice versa. If the fringe order and the material stress fringe values are known, then one can obtain the principal stress δ=

25.2.3 Fringe Contours in a Plane Polariscope One of the simplest optical arrangements possible is a plane polariscope (Fig. 25.4). Let the light source be a monochromatic one. A circular disk under diametral compression is kept in the field of view. The light incident on the model is plane polarized. As it passes through the model, the state of polarization changes from point to point depending on the magnitude of the principal stress difference and the principal stress direction. Information about the stress field can be obtained if the state of polarization of the emergent light is studied. This is easily achieved by introducing a polarizer at 0◦ . Since this optical element helps to analyze the emergent light, it is known as an analyzer. With the introduction of the analyzer, fringe contours that are black appear on the screen. The fringe contours correspond to those points where the intensity of transmitted light is zero. Since the analyzer is kept at 0◦ , this is possible only if the emergent light from the model has its plane of polarization along the vertical. Thus, to give a physical meaning to the fringe contours, one has to identify the conditions under which the incident plane-polarized light is unaltered as it passes through the model. The incident light is unaltered at all those points where the model behaves like a full-wave plate. This happens when the principal stress difference (σ1 − σ2 ) is such as to cause a relative phase difference of 2mπ

y

P Specimen

90°

F

S

Source

x

Polarizer

θ

Part C 25.2

Analyzer A

θ - Principal stress direction

Fig. 25.4 Arrangement of optical elements in a plane polariscope set to dark-field arrangement

Photoelasticity

(m = 0, 1, 2, . . .), where m is an integer. Since stress is continuous, one observes a collection of points forming contours that satisfy the above condition and the corresponding fringe field is known as the isochromatics. The term isochromatics is more appropriate to use when white light is used as a source. It has been pointed out earlier that wave plates are wavelength dependent. Hence, when white light is incident on the model, at any point only a single wavelength is cut off. In view of this one observes white light minus the extinct color over the field. Thus isochromatics (iso means constant and chroma means color) represent contours of constant color (see Fig. 25.1f and animation on the enclosed DVD [25.5]). Another possibility in which the incident light is unaltered is when the polarizer axis coincides with one of the principal stress directions at the point of interest. In this case, light extinction is not wavelength dependent and one observes a dark fringe even in white light. These are known as isoclinics, meaning contours of constant inclination. Isoclinics are usually numbered with the angles they denote such as 0◦ , 10◦ , 15◦ , etc. The principal stress direction on all points lying on an isoclinic is a constant. Thus, in a plane polariscope, one has two sets of contours, namely, the isochromatics and isoclinics, superposed over each other. Except for the zeroth fringe order isochromatics, these two contours can be distinguished by using a white light. To distinguish an isoclinic from a zeroth fringe order, the polarizer and analyzer can be rotated in crossed combination, in which case the isoclinics will move whereas the zeroth fringe order will not. A purely physical argument was used to establish the presence of two contours in a plane polariscope. It is desirable to develop a mathematical approach, as once developed, it can be used to understand the outcome of a polariscope with more number of optical elements. One of the mathematical techniques available is Jones calculus, which is discussed next.

25.2.4 Jones Calculus

cos θ sin θ − sin θ cos θ

.

707

The usual convention of measuring angles is valid for constructing a rotation matrix and the angle is positive if it is measured counterclockwise. A doubly refracting medium introduces a relative retardation δ between the vibrating components along its axes. Using complex number notations, the emerging light can be obtained as a matrix operation of the incident light as [25.4]      e−iδ/2 0 a1 eiα1 u = Re eiωt . (25.4) v a2 eiα2 0 eiδ/2 The first matrix on the right-hand side of (25.4) is called the retardation matrix. On the explicit understanding that we deal with real parts only, the notation Re is usually left out. It is to be noted that, in the above representation, the u  axis is considered as the slow axis. As many elements of a generic polariscope can be considered as retarders, it is desirable to obtain a matrix representation of a retarder. Let the slow axis of the retarder be oriented at an angle θ with respect to the xaxis and let the total relative retardation introduced by the retarder be δ. The incident light has to be rotated by an angle θ, then the relative retardation of δ is to be introduced and finally the emerging light is to be represented with respect to the original reference axes. These operations can be easily accomplished by multiplying the rotation and retardation matrices appropriately. The retarder can be represented as a matrix as follows   −i sin 2δ sin 2θ cos 2δ − i sin 2δ cos 2θ . −i sin 2δ sin 2θ cos 2δ + i sin 2δ cos 2θ Analysis of Plane Polariscope Using Jones calculus, the components of the light vector along the analyzer axis and perpendicular to the analyzer axis for the plane polariscope arrangement are obtained as     cos 2δ − i sin 2δ cos 2θ −i sin 2δ sin 2θ Ex = −i sin 2δ sin 2θ cos 2δ + i sin 2δ cos 2θ Ey   0 (25.5) × k eiωt , 1

where k eiωt is the incident light vector, and E x and E y are the components of the light vector along the analyzer axis and perpendicular to the analyzer axis, respectively. In (25.5), the polarizer is represented as a vector and the model is represented as a retarder. The intensity of the transmitted light is the product E x E ∗x ,

Part C 25.2

In general, an optical element in a polariscope introduces a rotation and retardation. In Jones calculus, these basic operations are represented as matrices. A rotation matrix is useful to find the components of a vector if the reference axes are rotated by an arbitrary angle θ. It is given  by, 

25.2 Transmission Photoelasticity

708

Part C

Noncontact Methods

where E ∗x denotes the complex conjugate of E x and the intensity of transmitted light is δ Ip = Ia sin2 sin2 2θ . 2 The intensity equation is a function of both the magnitude of the principal stress difference (δ) and its orientation (θ). In conventional photoelasticity, with a plane polariscope, only the dark-field arrangement is valid and the isochromatic fringes are counted as 0, 1, 2, . . . etc. Analysis of Circular Polariscope In this device a circularly polarized light is used to reveal stress information in the model. The basic arrangement (Fig. 25.5) cuts off the light beyond the model boundary and the background appears dark; hence the arrangement is known as a dark-field arrangement. Inside the model boundary, the intensity of light transmitted is governed by the stress field and can be obtained by Jones calculus. To obtain the components of light along and perpendicular to the analyzer axis, the individual matrices of the various optical elements have to be multiplied in the order in which they are placed in the polariscope as     1 1 −i Ex = 2 −i 1 Ey   cos 2δ − i sin 2δ cos 2θ −i sin 2δ sin 2θ × −i sin δ sin 2θ cos 2δ + i sin 2δ cos 2θ    2 1 i 0 (25.6) × k eiωt . i 1 1 y

P I quarter wave plate F S ξ

Specimen F II quarter wave plate F

Part C 25.2

A Analyzer

Let the intensity of transmitted light in the dark field be denoted by Id , obtained as the product of E x E ∗x , which simplifies to Id = Ia sin2

δ , 2

where Ia accounts for the amplitude of the incident light vector. By rotating the analyzer and keeping it along the vertical, one obtains a background that is bright. The fringes in the model also shift when the analyzer is rotated (see the animation in the enclosed DVD [25.5]). In this case, the analyzer will transmit the E y component of (25.6) and, denoting the intensity of light by Il , one obtains Il = Ia cos2

δ . 2

It is to be noted that, for both dark and bright fields, the intensity equations are independent of θ and hence the extinction condition is only a function of δ; thus only isochromatics will be seen. This is a significant achievement in separating isoclinics and isochromatics. In the dark-field arrangement, the intensity is zero when δ = 2mπ (m = 0, 1, 2, . . .) and fringes correspond to 0, 1, 2, . . . etc. In the bright-field arrangement, the intensity is zero when δ = (2m + 1)π, i. e., when the retardation is an odd multiple of half-wavelengths. The fringes correspond to 0.5, 1.5, 2.5 etc.

Source

25.2.5 Ordering of Isoclinics 90°

x

Polarizer

S θ

S η

Upon simplification one gets     sin 2δ e−i2θ Ex = k eiωt . Ey cos 2δ

ξ Slow axis of first quarter wave plate with respect to x axis – 135 deg θ Principal stress direction η Slow axis of second quarter wave plate with respect to x axis – 45 deg

Fig. 25.5 Arrangement of optical elements in a circular polariscope

In order to determine the direction of the principal stresses at a desired point in a stressed model, the model is viewed through a dark-field plane polariscope and the polarizer and analyzer are rotated in unison until the dark band representing the isoclinic passes through the point of interest. The orientations of the polarizer and analyzer coincide with the principal stress directions at the point. The orientation of the polarizer axis with the vertical is referred to as the isoclinic angle. It is to be noted that an ambiguity exists in identifying whether the isoclinic represents the σ1 or σ2 direction. To remove this ambiguity, the polariscope needs to be calibrated [25.4].

Photoelasticity

25.2.6 Ordering of Isochromatics

709

It is to be noted that there is no standard procedure to order fringes. Only certain broad guidelines are highlighted [25.4, 7].

25.2.7 Calibration of Model Materials Determination of the material stress fringe value is known as calibration of a photoelastic material. The material stress fringe value has to be evaluated to an accuracy of at least two or three decimal places, as it is the only parameter that links the optical information to the stresses. The stress fringe values of model materials vary with time and also from batch to batch. Hence, it is necessary to calibrate each sheet or casting at the time of the test. Calibration is performed on simple specimens for which a closed-form stress field solution is known. Although the stress fields for simple tension or a beam under pure bending are known, the use of a circular disk under diametral compression is preferred for calibration. This is because the specimen is compact, easy to machine, easy to load, and the fringe order at the center can be accurately measured. A circular disk is to be cut from the sheet to be calibrated. The ratio of the diameter (D = 2R) to the thickness of the disk should be chosen so that the disk does not buckle under the load P. The principle stress difference (σ1 − σ2 ) at any point in the disk can be expressed as [25.4] (σ1 − σ2 ) =

4PR R2 − (x 2 + y2 ) . (25.7) πh (x 2 + y2 + R2 )2 − 4y2 R2

Using (25.7) at the center of the disk and (25.2), the material stress fringe value is obtained as Fσ =

8P . π DN

(25.8)

In the conventional method the load is first increased and then decreased incrementally. A graph is drawn between the load P and fringe order N. A best-fit straight line is then drawn through the points (a graphical approach to least squares) and the slope of the line is substituted for P/N in (25.8) to evaluate the material stress fringe value. Sampled Linear Least-Squares Method Due to the spread of the applied loads, the agreement between the theoretical and experimental value at the center of the disk is off by about 4%. However, in the zone r/R = 0.3–0.5, the theoretical and experimental results are in good agreement. Hence, the use of field

Part C 25.2

Ordering of isochromatics is one of the crucial steps in the process of determining quantitative information from the fringe field. Experience and intuition have guided the numbering of isochromatic fringes and it is not an easy task to order the fringes from a given dark-/bright-field black-and-white photograph. Experimentalists in the past have attempted to use several auxiliary methods to assist fringe ordering. One of the simplest approaches is to use a white light source, as the color code (Fig. 25.1f) can be used to identify the fringe orders and also the fringe gradient direction. The method is advantageous up to the third-order fringe in the field, as beyond this the colors merge. Fringe ordering can be made simpler if one knows the fringe gradient direction and the order of at least one fringe forming the fringe packet. The fringe gradient directions can be established if one understands the various features of the isochromatic fringe field such as source, sink, or saddle point. Identification of the zeroth-order fringe is possible with knowledge of the isoclinic fringe field, namely isotropic and singular points. At free external corners, the stress tensor is zero and hence the fringe order is also zero. The load application points, stress concentration zones, and the tips of cracks are stress raisers and act like sources in the fringe field. As the load is increased, fringe orders appear at sources and propagate to other areas. A sink represents the point at which fringes vanish. It is at a lower fringe order compared to the neighborhood but not necessarily zero. One of the techniques (at least in static experiments) usually adopted is to increase the load gradually and to identify the points at which the fringes originate and vanish. A saddle point is bounded by two families of isochromatic fringes, one representing a fringe order slightly greater and the other slightly smaller than the one corresponding to the saddle point. An isotropic point is one through which all isoclinics pass through. The point is termed isotropic because, at these points, the loaded polymer does not behave like a crystal but rather like an optically isotropic material. An isotropic point lying on the free surface is termed a singular point. The sign of the boundary stress changes when it crosses a singular point. Thus at a singular point the isochromatic fringe order is zero. All the isoclinics also pass through the load application points but these are not isotropic points.

25.2 Transmission Photoelasticity

710

Part C

Noncontact Methods

data will be more suitable to evaluate the material stress fringe value accurately. The basic idea is to use as many data points as possible from the field to evaluate the material stress fringe value. It is desirable that the final results be nearly independent of the choice of data points from the field. In order to achieve this, the least-squares technique is usually combined with a random sampling process. It is worthwhile to account for residual birefringence also in the analysis. Assume that the residual birefringence expressed in fringe orders is a linear function of x and y [25.8] Nr (x, y) = Ax + By + C1 . The material stress fringe value is evaluated in a leastsquares sense by solving the following equation: bTbu = bTN ,

(25.9)

where⎛

⎞ ⎞ ⎛ ⎞ ⎛ S1 x1 y1 1 N1 1/Fσ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ S2 x2 y2 1⎟ ⎜ N2 ⎟ ⎜ A ⎟ ⎟ ⎟ ⎜ b= ⎜ , N= , u= ⎟ ⎜ ⎜ .. .. .. .. ⎟ ⎜ .. ⎟ , ⎝ B ⎠ ⎝ . . . .⎠ ⎝ . ⎠ C1 NM SM x M yM 1 and R2 − (x 2 + y2 ) 4PR . S(x, y) = 2 π (x + y2 + R2 )2 − 4y2 R2 The unknown coefficient vector u can be easily evaluated using the standard Gaussian elimination procedure. The construction of (25.9) is very simple from experimental data and hence the method has found wide acceptance. The least-squares technique by itself does not guarantee that the result is physically admissible. Evaluation of the parameter values is complete only if the theoretically reconstructed fringe patterns agree well with the experimentally obtained ones [25.4].

25.3 Variants of Photoelasticity Use of photoelasticity for diverse applications became possible only with the development of its several variants. Among the various whole-field experimental techniques, only photoelasticity offers techniques for measuring the stress field inside the model. Moderate velocities of stress waves in photoelastic materials compared to metals have made it attractive to study dynamic phenomenon using photoelasticity. Photoelasticity, being a strain-induced phenomenon, is exploited in reflection photoelasticity to study prototypes made of different materials, in orthotropic photoelasticity to study composites, and in photoplasticity to study strains beyond elastic limit.

25.3.1 Three-Dimensional Photoelasticity

Part C 25.3

The interpretation of fringe patterns in two-dimensional photoelasticity was simple, mainly because the model behaved like a simple retarder at the point of interest. In a general three-dimensional problem, the principal stresses vary continuously and also their directions need not remain in one plane. It is well established that only those stress components that lie in a plane perpendicular to the light path contribute to the photoelastic effect. These components are called secondary stress components and the pseudo principal stresses computed from these are known as secondary principal stresses and

they vary along the light path. One will observe the integrated effect of these variations as fringe patterns. Three approaches are used to analyze such problems. One of these uses concepts of two-dimensional photoelasticity for solving the three-dimensional (3-D) problem by mechanically slicing the model and is experimentally intensive. The second approach attempts to relate the integrated retardation pattern to the stress distribution along the light path and is mathematically intensive. The third approach attempts optical slicing but in general uses integrated photoelasticity for data interpretation. Conventional Three-Dimensional Photoelasticity In this, a model made of an appropriate epoxy is loaded and allowed to go through a thermal cycling process known as stress freezing [25.9]. At the end of the process, the loads are removed and the model is cut into thin slices for further analysis by using the principles of two-dimensional photoelasticity. The slice cut from the model will retain the same stress distribution as when it was part of the 3-D model, provided that the slicing operation is carried out carefully so that no machining stresses are introduced. This restriction is usually satisfied by using single-point cutting tools and providing appropriate coolant while cutting.

Photoelasticity

25.3 Variants of Photoelasticity

711

Stress-frozen threedimensional model Light path Ny, θy

y

y

x

y x

z

b) Slice

e) Integrated fringe

z

R/2

pattern

a)

⎛ σx τxz ⎛ ⎜ ⎜ ⎝ τzx σz ⎝

y Nx, θx

⎛ σy τyz ⎛ ⎜ ⎜ ⎝ τzy σz ⎝

z

f) Slice location Nz, θz

Secondary stress components

x

x

y

⎛ σx τxy ⎛ ⎜ ⎜ ⎝ τyx σy ⎝

x z

c) Sub-slice

d) Sub-sub-slice

g) Isochromatics of slice

Fig. 25.6 Three stages of slicing of a stress-frozen three-dimensional model. The figure also shows the direction of light incidence, the associated secondary stress components, and the nomenclature of the photoelastic data for these light incidences. It also shows fringe patterns observed for a problem of a sphere under diametral compression y

y

x x F Incident light ellipse

F S F

S

θ

θ +γ

Incident light ellipse

θ

S

2Δ Retarder

True 3-D model

+

γ

Intermediate light ellipse Rotator

Emergent light ellipse

Fig. 25.7 Principle of optical equiva-

lence

Part C 25.3

Emergent light ellipse

712

Part C

Noncontact Methods

The slicing plan for a three-dimensional model has to be carefully developed to suit the application. In general, three stages of slicing and recording of the data have to be executed (Fig. 25.6). In many instances, the fringe patterns in a carefully located and cut slice may give the requisite information for a designer. Problems of evaluating the stress concentration factor (SCF), the stress intensity factor (SIF), and contact stress parameters fall under this category. If a designer wants to evaluate the von Mises stress, it can be obtained directly. Consider Fig. 25.6d, in which the sub-subslice is analyzed by light incident along the x, y, and z directions, and let the respective fringe orders be denoted by N x , N y , and N z and the isoclinic angles be θx , θ y and θz . In terms of photoelastic data, the von Mises yield criteria can be written as [25.9]     N x Fσ 2 1 N z Fσ 2 (2 + sin2 2θz ) + (2 + sin2 2θx ) 2 h h

1   2 N y Fσ 2 2 + (2 + sin 2θ y ) = S y , h where S y is the yield stress of the model material. Integrated Photoelasticity For data interpretation, integrated photoelasticity (Fig. 25.6e) depends on the concept of optical equivalence (Fig. 25.7). The principle states that the optical effect introduced by a three-dimensional model to an incident light can be represented by a simpler system consisting of a retarder and a rotator. This means that, for the same incident light, the light ellipse coming out of the 3-D model and its corresponding optically equivalent model are the same (Fig. 25.7). The parameters representing the retarder and the rotator are the experimental parameters to be determined in integrated photoelasticity. These are 2Δ, the retardation of the retarder termed the characteristic retardation, θ, the orientation of the retarder termed the primary characteristic direction, and γ , the rotary power of the rotator termed the characteristic rotation. The Jones matrix representation M of the optically equivalentmodel is obtainedas

M=

Part C 25.3

P − iQ −R − iS R − iS P + iQ

where P = cos Δ cos γ , Q = sin Δ cos(2θ + γ ) , R = cos Δ sin γ , S = sin Δ sin(2θ + γ ) .

,

(25.10)

Experimentally one can find the characteristic parameters [25.10], and from these the matrix given in (25.10) can be constructed. Theoretically one can assume an appropriate stress distribution and the simplest of these is to assume them to be appropriate polynomial functions. The 3-D model can be represented as a train of retarders and, using Jones calculus, one can obtain the net effect of these. If the stress distribution is correctly modeled then the final matrix should be identical to the optically equivalent model. In other words, one should try to evaluate the polynomial coefficients such that these are same. This is very easily said, but in practice it becomes very difficult to solve even simple problems. A single experimental measurement may not be sufficient to find all the polynomial coefficients, and additional measurements along different light paths may have to be used. The model is usually immersed in a bath of matching liquid to ensure normal incidence. Further details can be found in [25.11]. Integrated photoelasticity has been successful in analyzing axisymmetric problems. It has also been widely used for evaluating residual stresses in glass components. Scattered Light Photoelasticity Scattered light photoelasticity provides a nondestructive approach for analyzing stresses in a three-dimensional model. Scattering of light inside the model is used either as a polarizer or an analyzer. In the methodologies described in this chapter, scattering is used as an analyzer. In view of the fact that only scattered light is recorded, the signal is quite weak and may require long exposure in the conventional approach or may require image-intensifier charge-coupled device (CCD) cameras for digital recording. The direction of observation is in a plane perpendicular to the direction of light propagation. In a three-dimensional model, in general, the secondary principal stress magnitude and their directions vary. A simple case to analyze is the situation in which the secondary principal stress directions do not change but their magnitudes change continuously over the length of the light path. A model in a plane state of stress, pure torsion, and planes of symmetry in axisymmetric objects fall under this category. These class of problems are considered as the nonrotational case. For this case, the stress–optic law is [25.12]

σ1s − σ2s = Fσ

dN . dz

(25.11)

Photoelasticity

Figure 25.8a shows a plane-polarized light incident on the model and that the secondary principal stress direction does not change along the light path. A disk under diametral compression is viewed for scattered light fringes as shown in Fig. 25.8c. The fringes are shown in Fig. 25.8d and for any point one can evaluate the slope of the fringe order variation from the graph in Fig. 25.8e. Since the state of stress along the light path a)

25.3 Variants of Photoelasticity

713

is quite simple, the secondary principal stress difference of (25.11) is simply the stress component σx . If the secondary principal stress directions change along the light path then one has to invoke the concept of optical equivalence (Figs. 25.7, 9). Instead of considering the fringe order slope, one has to evaluate the slope of the characteristic parameters. The secondary normal stress difference and shear stress at the point of

b)

c) σz

Sheet of linearly polarized light τzx

σy

B

σx

σ s1

P

C1 σ s2 σ s1

P

C2 σ s2

d)

B

45° C

e) Fringe order 20

C 16 12 8 4 0 C

B 0

0.5

1

1.5

2

2.5

3 3.5 4 Distance from B (cm)

Part C 25.3

Fig. 25.8 (a) Secondary principal stress directions do not change along the light path. (b) Stress components on an element. (c) The diametral line of a disk is investigated using scattered light. (d) Scattered light fringe pattern observed. (e) Variation of fringe order over the length of the light path [25.12]

714

Part C

Noncontact Methods

interest are obtained as [25.12]  2 dθ (σx − σ y )m = sin 2Δm sin(2θm + 2γm ) C dz  dΔ cos(2θm + 2γm ) , − dz  1 dθ (τxy )m = sin 2Δm cos(2θ + 2γm ) C dz  dΔ sin(2θm + 2γm ) . − dz

25.3.2 Dynamic Photoelasticity Dynamic photoelasticity is a logical extension of transmission photoelasticity to study time-varying problems. The stress-optic law is essentially the same except that the material stress fringe value needs to be obtained from a dynamic experiment and is usually 10–30% more than the static value for the model material [25.13]. σ1 − σ2 =

NFσd , h

where Fσd is the dynamic material stress fringe value. The calibration specimen is a long rod of square cross

section impacted by a projectile fired from an air gun. It is instrumented with strain gauges to measure the bar velocity CB and its fringe trace is also simultaneously recorded. Fringe ordering in dynamic photoelasticity is quite difficult. In wave propagation problems, the position of zero orders, peaks, valleys etc. may move. The isochromatic pattern associated with Rayleigh waves can be easily identified by its characteristic pattern, which is shown in Fig. 25.10. For impact studies, the speed of the stress waves determines the order of the exposure time required for dynamic photography. Commonly used photoelastic materials are Homolite 100 and epoxy. In these materials, the longitudinal wave speed lies in the range 1800–2150 m/s and the Rayleigh wave speed is in the range 960–1130 m/s. In order to record these highvelocity fringes adequately, exposure times less than a microsecond are required. Several dynamic cameras have been developed such as the rotating mirror camera and the Cranz Schardin camera etc. In the Cranz Schardin camera there are no moving mechanical parts, the light source is a set of electrical sparks, and the image separation is done by optical means. The decay time of the electrical spark dictates the exposure time and the time interval between the sparks determines the frame rate of the camera.

S Peak

Zero

Peak Zero

Zero

1.5

2Δm , θm , γm

2Δm +1, θm +1, γm +1

Peak

9

B

x G1

G2

D1 D2 1.5 0.5

Part C 25.3

z

0.5

Fig. 25.9 Scattered light measurement scheme when the

secondary principal stress directions change along the light path. The experimental parameters to be measured are characteristic parameters [25.12]

Fig. 25.10 Isochromatics associated with a Rayleigh wave traveling along a straight boundary [25.7]

Photoelasticity

25.3 Variants of Photoelasticity

Cameras that can record up to 20 frames of a dynamic event have been developed. Digital equivalents of the Cranz Schardin camera have also been attempted. The challenge in dynamic photoelasticity lies in properly synchronizing the dynamic event with the recording. Dynamic photoelasticity has significantly contributed to the phenomenological understanding of fracture [25.14, 15], stress wave propagation, crack– stress wave interaction [25.16], load transfer in granular media [25.17], etc.

well known that the retardation introduced by the birefringent material is a function of the wavelength λ and the strain–optic coefficient is usually expressed as λ , Fε = K where K is the strain coefficient, supplied by the manufacturer or to be determined by calibrating the coating. Expressions relating the coating stresses to the specimen stresses are obtained based on the following assumptions:

25.3.3 Reflection Photoelasticity

1. The thickness of the coating is very small; 2. Both the specimen and the coating have the same Poisson’s ratio; and 3. The specimen and the coating are in a state of plane stress. Later, correction factors are used to account for the role of the finite-thickness coating. Let a small element of a coating and specimen, far removed from the edges, be as shown in Fig. 25.11b. The reference axes are taken as the principal stress/strain axes. Let the surface strains of the specimen be transmitted to the coating through the adhesive without loss or amplification. The interrelationship between the difference in principal stresses of the specimen and the coating is obtained as  E s 1 + νc  c σ − σ2c . σ1s − σ2s = E c 1 + νs 1 The Poisson’s ratio of the coating and the specimen are indicated separately for generality. This does not mean that these equations fully account for the Poisson’s ratio mismatch. For a finite-thickness coating, the following points must be considered: 1. The coating reinforces the specimen; 2. A strain variation exists over the thickness of the coating; and 3. Mismatch of Poisson’s ratio between the coating and the specimen needs to be accounted for. For various types of loading situations, correction factors have been developed to account for factors (1) and (2) considering that there is no mismatch of Poisson’s ratio. The specimen principal stress difference is related to the photoelastic parameters as Es N λ , σ1s − σ2s = Rf 1 + νs 2h c K where Rf is the reinforcement factor, which is a function of the mechanical properties of the coating and the

Part C 25.3

Reflection photoelasticity is an extension of transmission photoelastic analysis for the analysis of opaque prototypes. A thin temporarily birefringent coating is pasted on the prototype with a reflective backing at the interface. The prototype is loaded appropriately and a reflection polariscope is used for collecting the optical information. The optical response achievable is an order of magnitude less than what is possible in transmission photoelasticity, which has necessitated the use of white light for reflection photoelastic analysis. Reflection photoelasticity is truly an engineering tool for solving practically important and complicated problems. Significant development in its industrial use is mainly due to the development of contourable plastics by Zandman et al. [25.18] in 1960. Several approximations are made in the interpretation of the optical information recorded. Using the principles of transmission photoelasticity, the optical response of the coating is initially related to the coating stresses. Based on the principles of mechanics of solids, the specimen stresses are determined from the coating stresses. The analysis is improved by the use of appropriate correction factors [25.19]. Commercially available reflection polariscopes do not provide precise normal incidence but only a shallow oblique incidence. The oblique angle is usually of the order of 4◦ , and its influence on the optical pattern interpretation is considered negligible. Thus, the engineering approximation used in reflection photoelasticity starts right from the data collection stage. The birefringence in the coating is introduced through the surface deformations at the interface and is useful to represent the photoelastic phenomena based on the strains developed. The strain–optic law is NFε , εc1 − εc2 = 2h c where Fε is the strain–optic coefficient and (εc1 − εc2 ) gives the principal strain difference in the coating. It is

715

716

Part C

Noncontact Methods

a)

c)

Coating

z

b)

Coating

Prototype 2

1

Specimen

Fig. 25.11 (a) Prototype coated with a suitable photoelastic coating. (b) A portion of the specimen and coating shown with axes of reference for analysis. (c) Assembly stresses revealed by photoelastic coatings [25.18]

specimen, as well as the geometry of the specimen and loading. Table 25.1 summarizes the expression of the correction factors for a variety of loading situations. The correction factors given in Table 25.1 are not applicable for stress concentration regions such as keyways and abrupt changes in the thickness of the specimen. Further they are not valid if the part is subjected to strains beyond the elastic limit. In such cases, thin coatings should be employed and they will reveal sufficient birefringence due to higher levels of stress and no correction factor needs to be applied. Mismatch of Poisson’s ratio introduces a strain variation through the thickness of the coating. Experiments have been conducted [25.20] to study the influence of Poisson’s ratio mismatch using glass-fibre epoxy tension specimens. A transition zone exists near the boundary where the fringe order lies between these two extremes. The length of the transition zone is found to

be a function of (νc − νs ), which is usually less than 0.06 for metallic specimens. Hence, the effect of the Poisson’s ratio mismatch is often neglected in the analysis of most metallic components. One of the most useful and simplest applications of photoelasticity is the evaluation of the stress concentration factor (SCF). Let the maximum fringe order observed at the boundary of the hole be Nmax and let the average fringe order be Nfar-field . The ratio of these does not directly give the SCF in reflection photoelasticity; rather one has to evaluate SCF by Nmax (1 + νs ) . Nfar-field (1 + νc )

SCF =

Evaluation of the strain coefficient K for a photoelastic coating is known as calibration of the coating material. A cantilever beam in bending is recommended for cal-

Table 25.1 Photoelastic coating correction factors for various loading conditions Nomenclature e=

Ec Es , 1−νs2 1−νc2

m=

Part C 25.3

P=

Plane stress Beam in bending

ro

rc

a=

g=

Type of problem hc hs

ri

ri ro ,

c=

em(c2 −1) (1−a2 )

rc ro

Correction factor (1+νs ) Rfa = 1 + hhcs EEsc(1+ν c)   2 )2  1+ν  (1+eg) s Rfb = (1+g) 4(1 + eg3 ) − 3(1−eg (1+eg) 1+νc

bp

Bending of thin or medium thick plates

Rf =

Torsion of circular shaft

Rft =

Long cylindrical pressure vessel

1 p Rf

=

(1+emg) (1+g)

2 (1+c)





  2 )2 4(1 + emg3 ) − 3(1−emg (1+emg)

1+

G c (c4 −1) G s (1−a4 )



2(1−2ν+c)(1−ν) (1−2ν+c2 )+P(1−2ν+a2 )

− (1−2ν) (1+P)



Photoelasticity

ibration purposes. The calibration apparatus is compact for the displacement-controlled arrangement Fig. 25.12. For calibration purposes, only a small portion of the beam is to be coated with the coating. This saves the precious coating material, and its influence on the deflection of the beam at the free end (yo = PL3s /3E s Izz ) will be minimal. The strain coefficient K is obtained as K=

Rfb L 3s λ N . 1 + νs 3Lh s h c yo

where h L is the length of the light path. For a transmission arrangement it is h c , and for a reflection arrangement it is 2h c .

25.3.4 Photo-orthotropic Elasticity

N=

σL σT − FL FT

 +

hs

A Ls

2τLT FLT

2 12 h.

Prabhakaran [25.27] developed a strain–optic law based on the concept of Mohr’s circle of birefringence and introduced three material strain fringe values, FεL , FεT , and FεLT . These are related to the observed fringe order in the model material by the relation     1 εL εT 2 γLT 2 2 N= − + h. FεL FεT FεLT Agarwal and Chaturvedi [25.28] derived an exact strain–optic law for orthotropic materials using Pockel’s theory of crystalline photoelasticity. In composites, the directions of principal stresses and strains are not identical. Hence, the stress–optic laws and the strain–optic laws are different for these model materials. The isochromatic fringes observed in such materials are representatives of neither contours of principal stress differences nor the principal strain differences. Typical isochromatics observed in a transparent orthotropic composite are shown in Fig. 25.13. The fringes are not as smoothly bent as in conventional model materials. The fibre direction plays a role in the fringe appearance. The optical isoclinics seen represent neither the principal stress directions nor the principal strain directions. In view of this, separation of prina)

b)

Fig. 25.13a,b Isochromatics observed in the neighborhood of a crack in mixed-mode loading in a glass-fibre polyester composite of 35% volume fraction [25.21]. (a) Crack parallel to fibre. (b) Crack perpendicular to fibre

Part C 25.3

Composite materials have established themselves as strong candidates for important structural applications. Pih and Knight [25.22] initiated the use of a transparent birefringent model of a composite with anisotropic elastic and optical properties for photoelastic stress analysis. One of the important requirements for this technique is the development of appropriate model materials. Several investigators [25.23–25] have addressed this. Unlike isotropic two-dimensional (2-D) photoelasticity, here one requires three material constants to relate the optical response to stress information. It is well established in the photo-orthotropic literature that the fringe order obtained in a transparent composite is a function of three material constants FL , FT , and FLT . The stress–optic law is [25.26] 2

A

L

ment

N λ , h L (1 + νc ) εc1



bs

717

Fig. 25.12 Displacement-controlled calibration arrange-

The value of K has a wide range from 0.16 to 0.009. For materials with higher values of K , the mentioned approach is feasible, but for lower values of K one needs to apply a large strain of the order of 10–15%. For such materials, it is generally recommended to perform the calibration test directly on the material itself by making a tension member out of the material. Since large strains are involved, the axial strain measurement can be carried out by an extensometer. If the coating is transparent, the fringe order can be measured in a transmission polariscope and, if the coating has a reflective backing, one has to use a reflection polariscope. The strain coefficient K can be obtained as K=

Section A–A hc

25.3 Variants of Photoelasticity

718

Part C

Noncontact Methods

cipal stresses or strains is not a simple task even in a two-dimensional problem. Prabhakaran [25.29] has summarized various techniques for separating principal stresses. Using tension specimens with fibres oriented at 0◦ and 90◦ , it is easy to determine FL and FT . On the other hand, determination of FLT is not easy since it requires loading the specimen in pure shear. An approximate method to find FLT is by finding Fπ/4 initially, by conducting a 45◦ off-axis coupon test, and then using this value in the following relation to obtain FLT    1  FL 2 2 FLT 1 FLT 2 1− = 1+ . Fπ/4 4 FL FT The error in equating FLT to Fπ/4 depends upon the ratios FLT /FL and FL /FT . This error is small (usually less than 3%). Strain fringe values can be determined by using the interrelationships between the stress fringe and strain fringe values. Curing of all types of epoxy or polyester resin is accompanied by a change of volume. Due to this shrinkage, residual birefringence is introduced. This is unavoidable while fabricating a transparent composite for conducting photo-orthotropic experiments. It has been found experimentally [25.31] that the principal direction of residual birefringence is identical with the direction of the fibres. Load-induced birefringence is given by   N = NT2 + Nr2 − 2NT Nr cos 2(θT − θr ) and tan 2θ =

NT sin 2θT − Nr sin 2θr , NT cos 2θT − Nr cos 2θr

where NT and θT are the total observed fringe order and isoclinic angle, respectively. Nr and θr are, respectively, the residual fringe order and residual isoclinic angle. θT and θr should be referred to the L-axis of the composite. In view of the several challenges involved, the technique has so far remained mostly of academic value.

Part C 25.3

25.3.5 Photoplasticity Photoplasticity has been used to evaluate the stress concentration factors under plastic loading conditions, the plastic zone in fracture problems, and modeling large deformations in metal-forming operations. Conventional polariscopes can be used to study the model.

Fig. 25.14 Polycarbonate strip plastically loaded and then unloaded. In the necking zone, one side is sanded to obtain a wedge showing several fringes [25.30]

It has been studied either under plastically loaded conditions or in unloaded conditions after plastically loading the model. Figure 25.14 shows a tensile strip of polycarbonate plastically deformed and unloaded retaining birefringence. Note the ellipses on the model, indicating large deformations. Due to plastic flow, the molecular structure of the permanently deformed polycarbonate is given preferred orientation corresponding to the direction of the principal strains. The resulting birefringence is permanent and is locked into the model on a molecular scale. The strain–optic law governing the behavior of the polycarbonate in the plastic zone was established by Dally and Mulc [25.30] to be p

(ε1 − ε2 )p = NFε /h , p

where N is the fringe order, Fε is the material fringe value in terms of permanent strain, and h is the model thickness. The isochromatics and isoclinics are related to the in-plane principal strain difference and directions. The stresses are not linearly related to strains.

Photoelasticity

One of the most persistent difficulties in obtaining strain information in photoplasticity is the selection of a model material with stress–strain characteristics that adequately simulate the real materials. The large number of methods in photoplasticity are largely due to differences in the mechanical and optical behavior of the various model materials. In metal-forming

25.4 Digital Photoelasticity

719

studies the suggested method for getting a suitable model material is to mix two mixtures of rigid and flexible polyester resin appropriately and compare its stress–strain behavior with that of the actual prototype material [25.32]. The industrial application of photoplasticity has remained limited due to difficulties in identifying and characterizing suitable model materials.

25.4 Digital Photoelasticity The advent of computers coupled with developments in personal computer (PC)-based digital image processing (DIP) has had a great influence in the development of modern photoelasticity. In the early development of digital photoelasticity, DIP systems were essentially used to automate the procedures that were done either optically or manually. A paradigm shift in data acquisition methodologies occurred when digital cameras could record intensity data over the model domain at video rates. The techniques could be broadly classified into spatial-domain and frequency-domain methods. The frequency-domain methods usually demand more images to be recorded (as many as 90 images in some cases) and are computationally very intensive. On the other hand spatial-domain methods require fewer images to be recorded (three to eight in most cases). Furthermore, they are computationally very fast. A detailed description of various methods can be found in [25.4, 34, 35]. In this section, for brevity only those a) 1 1

1 2

2

b) 0.5

c)

1.5

0.5

1.5

0.5

1.25 1.25 0.25

0.75 1.75

Fig. 25.15a–c Fringe multiplication by image subtraction of bright- and dark-field images: (a) dark-field image, (b) bright-field image, and (c) mixed image (after [25.33])

25.4.1 Fringe Multiplication and Fringe Thinning Although the focus is now on the evaluation of photoelastic parameters for the entire field, digital fringe multiplication and digital fringe thinning still have a place in solving practical problems. Techniques for fringe multiplication use DIP hardware as a paperless camera. The simplest of the fringe multiplication techniques is due to Toh et al. [25.36], who proposed that a simple digital subtraction of bright- and dark-field images could result in fringe multiplication by an order of two. In such a case one obtains g(x, y) ≈ Ia cos δ .

(25.12)

In (25.12), the extinction of light will occur when δ = (2n + 1)π/2. The resultant image is termed as a mixed image and the fringe orders are N = 0.25, 0.75, 1.25, . . .. Figure 25.15 shows the fringe multiplication obtained for the problem of a central crack in a bimaterial Brazilian disk [25.33]. Historically, one of the earliest approaches that used intensity information for fringe ordering was by Voloshin and Burger [25.3]. In this, the hardware feature of the black-and-white image processing system is effectively used to find the fractional fringe orders between 0 and 0.5 directly. In view of this, the technique came to be known as half-fringe photoelasticity (HFP). Since, for an 8 bit system, 256 gray level shades exist between pitch black and pure white, the method can be thought of to be a fringe multiplication technique with a factor of 512. One-to-one correspondence between the digitized value and the fractional fringe order is established by properly calibrating the polariscope–camera system.

Part C 25.4

0.75

techniques that have either initiated a new line of thinking in data acquisition or those that are time-tested and in wide use are presented.

720

Part C

Noncontact Methods

a)

b) g (–1, –1)

g (0, –1)

g (1, –1)

g (–1, 0)

g (0, 0)

g (1, 0)

g (–1, 1)

g (0, 1)

g (1, 1)

Left side elimination

Top side elimination

Right side elimination

Bottom side elimination

Fig. 25.16 (a) 3 × 3 pixel mask. (b) Pictorial representation of the eliminative conditions needed for scanning the image left to right, right to left, top to bottom, and bottom to top. In the figure, filled circles are fringe pixels, circles are nonfringe pixels, and filled triangles are the pixels that are considered for elimination

One of the simplest methods for fringe thinning is to treat the fringe patterns as a binary image and determine the fringe center lines by a process of erosion [25.37]. The fringe areas are initially identified by applying a suitable global threshold. The binary image obtained after thresholding is scanned from left to right, right to left, top to bottom, and bottom to top sequentially to eliminate border pixels forming the fringe band. During each such scan, for every pixel with a gray-level value below the threshold (a point on a fringe), a 3 × 3 pixel mask is considered (Fig. 25.16a) to eliminate the border pixel. In view of progressive thinning, the approach is basically iterative in nature. It is to be noted that, in broad fringes, the center lines may not be the actual fringe location.

0° Scan

45° Scan

Original image OR

90° Scan

25.4.2 Phase Shifting in Photoelasticity OR

AND

Part C 25.4

Fringe skeleton super-imposed image

Fig. 25.17 Global fringe thinning algorithm

Among the various intensity-based fringe thinning algorithms, the one proposed by Ramesh and Pramod [25.38] is the fastest and assures fringe skeletons of one pixel width [25.39]. It has two steps: the first is edge detection followed by identification of the minimum intensity. Since photoelastic fringes have high contrast, a simple global thresholding can help in finding the fringe areas. Once the fringe areas are identified, the image is scanned to locate the minimum-intensity points forming the fringe skeleton. In order to account for various fringe curvatures, a unique process of logical operators is proposed in which the edge detected image is scanned in four scanning directions to determine the points of minimum intensity. The scheme is illustrated in Fig. 25.17. Binary-based algorithms are convenient if the fringe pattern available is of a poor quality. For the evaluation of fracture parameters/contact stress parameters, the use of fringe multiplication in conjunction with fringe thinning techniques has been found to be quite useful [25.33].

135° Scan

Hecker and Morche [25.40] introduced the concept of phase shifting to photoelasticity for the determination of isochromatic parameter over the whole domain of the model. In photoelasticity, a change in phase between the beams involved is achieved by appropriately rotating the optical elements of the polariscope. Unlike in most other interferometric techniques [25.41], the phase information is affected by two parameters, viz. the difference in principal stresses and the orientation of the principal stress direction. This is referred to as isochromatic–isoclinic interaction and

Photoelasticity

affects the evaluation of isochromatics and isoclinics differently. Using Jones calculus, the intensity of light transmitted in a generic circular polariscope (Fig. 25.18) with ξ = 135◦ can be represented as [25.4] Ia Ia + [sin 2(βi − ηi ) cos δ 2 2 − sin 2(θ − ηi ) cos 2(βi − ηi ) sin δ] ,

I quarter wave plate F S ξ

Specimen F II quarter wave plate

(25.13)

(25.14)

Although δ can be obtained by several ways, the following one assures of a high modulation [25.45]:   −1 (I5 − I3 ) sin 2θc + (I4 − I6 ) cos 2θc δc = tan (I1 − I2 )   Ia sin δ (25.15) . = tan−1 Ia cos δ

F

721

Source

90°

x

Polarizer

S θ

S η

ξ Slow axis of first quarter wave plate with respect to x axis

β

θ Principal stress direction

A Analyzer

η Slow axis of second quarter wave plate with respect to x axis β Orientation of analyzer

Fig. 25.18 Generic arrangement of a circular polariscope

except for the ambiguity in its sign (Sect. 25.4.3). Usually the fractional retardation is represented in the form of a phase map. The phase map is obtained using the following relations [25.4]:  δc for δc > 0 (25.16) δp = for δc ≤ 0 2π + δc 255 g(x, y) = (25.17) δp , 2π where g(x, y) is the gray level of the pixel at the location (x, y). If the modulus of δc is used in (25.16) one obtains a simulated dark-field image. Figure 25.19a shows the phase map (corrected for ambiguity) of a disk under diametral compression obtained from experimentally recorded phase-shifted images. Figure 25.19b shows a simulated dark field and one can notice the similar features between Figs. 25.19a,b. Figure 25.19c shows the variation of fractional fringe order along the diametral line. One can observe that the fractional fringe order increases in the direction of increasing total fringe order. EquaTable 25.2 Six-step phase-shifting arrangement ξ

η

β

Intensity equation

3π/4 3π/4 3π/4 3π/4 π/4 π/4

π/4 π/4 0 π/4 0 3π/4

π/2 0 0 π/4 0 π/4

I1 = Ib + I2 = Ib + I3 = Ib + I4 = Ib + I5 = Ib + I6 = Ib +

Ia 2 (1 + cos δ) Ia 2 (1 − cos δ) Ia 2 (1 − sin 2θ sin δ) Ia 2 (1 + cos 2θ sin δ) Ia 2 (1 + sin 2θ sin δ) Ia 2 (1 − cos 2θ sin δ)

Part C 25.4

In (25.14) and (25.15) the subscript c indicates that the principal values of the inverse trigonometric functions are referred. It is recommended to use the atan function for evaluating θ and the atan2 function for evaluating the fractional retardation [25.4]. It can be seen that the fractional retardation depends on the value of θ. The evaluation of θ based on circular polariscope methods is not quite good. More discussion on this is presented in the chapter. Nevertheless the evaluation of the fractional retardation from these algorithms is very good

y

P

Ii = Ib +

where Ia accounts for the amplitude of light vector and the proportionality constant, Ib accounts for the background light intensity. In fact many of the algorithms reported in phase shifting technique (PST) failed experimentally due to improper accounting for the background light [25.42]. The choice of an appropriate set of optical arrangements to obtain a set of intensity equations and extraction of photoelastic parameters from these equations has been the study by several researchers [25.4]. Extending the work of Hecker and Morche, Patterson and Wang [25.43] proposed a six step phaseshifting technique for the determination of both isoclinic and isochromatic parameters. Later, Ajovalasit et al. [25.44] proposed a six-step phase-shifting algorithm (Table 25.2), which uses both left and right circularly polarized light. From the equations of Table 25.2, the values of θ can be obtained as   1 I5 − I3 θc = tan−1 2 I − I6  4  1 −1 Ia sin δ sin 2θ = tan for sin δ = 0 . 2 Ia sin δ cos 2θ

25.4 Digital Photoelasticity

722

Part C

Noncontact Methods

tional retardation along the diametral line may decrease in the direction of increasing total fringe order. Thus calibration of the generic polariscope is essential. A discussion on the optical arrangements used to get intensity equations in Table 25.2 is apt here. In the early developments of phase-shifting techniques, researchers proposed several optical arrangements [25.4]. Though they looked different at first sight, they yielded one of the equations reported in Table 25.2. Ramesh [25.4] reported that there could be multiple optical arrangements c) Fringe order from which the intensity equations given in Table 25.2 3.5 can be obtained. One of the important sources of error in experiments 3 is the mismatch of quarter-wave plates. Table 25.3 2.5 shows the various ways I1 can be obtained using left Theory 2 circularly polarized light and the intensity equation Wrapped phase in the presence of quarter-wave plate mismatch (er1.5 Unwrapped phase ror ε) [25.46, 47]. The table shows that the individual 1 relative orientations of the elements do play a role. The 0.5 error in intensity is a minimum for crossed quarter-wave plates and maximum for the parallel arrangements. 0 –1 –0.5 0 0.5 1 This is true for dark field (I2 ) too, which reconfirms x /R the wisdom of using crossed quarter-wave plates in Fig. 25.19a–c Disk under diametral compression. (a) Phase conventional photoelasticity [25.48]. When the input map. (b) Simulated dark field. (c) Wrapped and unwrapped handedness is changed to right circularly polarized phase along the diametral line. Total fringe order from light, the intensity equation with quarter-wave plates theory is also shown crossed (Table 25.3) is not altered but, the intensity equations for parallel and other optical arrangements tion (25.13) is derived by considering the slow axes differ. The sin ε term in these equations is changed to of the quarter-wave plates. There has to be a one-to- − sin ε when the input light is changed [25.48, 49]. one correspondence between the mathematical model of In the intensity equations I3 , I4 , I5 , and I6 of Tathe polariscope (Fig. 25.18) and the actual experimen- ble 25.2 when the input handedness is changed, the term tal setup. If a correspondence exists, use of (25.14) and sin ε is changed to − sin ε. Inspection of (25.14) and (25.15) will give a phase map (after correcting ambi- (25.15) indicates that a judicious use of left and right guity), as shown in Fig. 25.19a. If the slow axes of the circularly polarized light can help to minimize the error quarter-wave plates are not aligned properly, the frac- due to quarter-wave plate mismatch. Six arrangements a)

b)

Table 25.3 Intensity equations including quarter-wave plate error corresponding to multiple optical arrangements for

obtaining I1 using left circularly polarized light

Part C 25.4

ξ

η

β

λ/4 plates

3π/4

π/4

π/2

Crossed

3π/4 3π/4

3π/4 3π/4

0 π

Parallel

3π/4 3π/4

π 0

π/4 π/4

Others

3π/4

π/2

3π/4

Others

Intensity equation     Ib + Ia (1 − cos2 2θ sin2 ε) cos2 2δ + cos2 2θ sin2 ε    Ib + Ia sin2 2δ sin2 2θ + sin 2θ sin δ sin ε cos ε        + 1 − 2 sin2 2δ + sin2 2δ cos2 2θ cos2 ε  Ib + I2a 1 − sin 2θ cos 2θ(1 − cos δ) − sin ε cos ε sin δ(cos 2θ − sin 2θ)  +(cos δ + sin 2θ cos 2θ(1 − cos δ) cos2 ε)  Ib + I2a 1 + sin 2θ cos 2θ(1 − cos δ) + sin ε cos ε sin δ(sin 2θ + cos 2θ)  +(cos δ − sin 2θ cos 2θ(1 − cos δ) cos2 ε)

Intensity equation without including quarter-wave plate error I1 = Ib +

Ia 2 (1 + cos δ)

Photoelasticity

are possible [25.48] and one of these corresponds to the six-step algorithm proposed by Ajovalasit et al. (Table 25.2), which is a very important contribution to PST. In the presence of quarter-wave plate mismatch, the best values of θ and δ that can be obtained are [25.4]   Ia sin δ sin 2θ cos ε (25.18) , θc = tan−1 Ia sin δ cos 2θ cos ε   Ia sin δ cos ε  −1 δc = tan     . Ia cos δ 1−cos2 θ sin2 ε +cos2 2θ sin2 ε Though the error due to quarter-wave plate mismatch modifies the intensity, (25.18) indicates that the evaluation of θc is not affected by it as cos ε cancels out in both the numerator and the denominator. Here the quarter-wave plates are considered to be homogeneous and behaves identically for both left and right circularly polarized light [25.50]. If the response is different, the isoclinic evaluation is severely affected and is verified by experiments (see (25.24)). For total fringe order evaluation, the phase map needs to be unwrapped. This requires the information of total fringe order for at least one point in the domain. For phase unwrapping to be effective, the phase map should be free of ambiguities and noise. The phase map a)

25.4 Digital Photoelasticity

723

has to mimic the features of dark-field isochromatics. If there is any deviation one can conclude that ambiguity exists over the model domain and this needs to be corrected. The detail of what is ambiguity and how it can be corrected is discussed next.

25.4.3 Ambiguity in Phase Maps The principal value of θc evaluated using (25.14) lies in the range −π/4 to π/4 whereas the physical value of principal stress direction lie in the range −π/2 to π/2. Hence, θc calculated by (25.14) will not give either the σ1 or σ2 direction uniformly over the domain. For example, when the principal stress direction of σ1 is more than π/4, say (π/4 + a), (25.14) gives a value of (−π/4 + a). This causes a shift of π/2 in the calculated value of the σ1 direction; it actually represents the σ2 direction. Because of this error in θc , the sign of δc obtained from (25.15) will be reversed. The phase map of a circular disk obtained from theoretically simulated phase-shifted images obtained using (25.16) is shown in Fig. 25.20a. One can clearly a)

b)

c)

d)

b) Zone-2 Fractional retardation (radians) 0

Zone-1

0

1

2

3

4

5

6

7

50

c)

d)

y σ2 +π/4

–π/4

σ1

100

y σ1 +π/4

σ2

θc

θc

x

x –π/4

150 200 250 300

compression obtained from theoretically simulated phaseshifted images. (b) Representative line diagram of the phase map in (a) showing zone 1 and zone 2. (c) Orientation of the principal stresses σ1 and σ2 in zone 1. (d) Orientation of the principal stresses σ1 and σ2 in zone 2

Pixels

Fig. 25.21a–d Different classification of ambiguous zones in a phase map (a) ambiguous zone distinctly seen, (b) ambiguous zone boundary not well defined, (c) ambiguous zone boundary is subtle, (d) fractional retardation along the line shown in (c)

Part C 25.4

Fig. 25.20 (a) Phase map of a disk under diametral

724

Part C

Noncontact Methods

see two zones. In zone 1 (Fig. 25.20a), the fractional retardation increases in the direction of increasing total fringe order, and in zone 2 the fractional retardation decreases in the direction of increasing total fringe order. To simplify phase unwrapping, it is desirable that the fractional retardation varies in one uniform fashion in relation to the total fringe order over the complete model domain. Hence when the theoretical value of θ is outside the range of −π/4 to π/4, ambiguity exists on how to unwrap. This is simply referred to as ambiguity in the phase map. Various Classes of Ambiguous Zones and Their Correction Methodology Figure 25.21 shows a sample of different types of ambiguous zones that are possible in a phase map obtained from experimentally recorded images. The ambiguity zone is distinctly visible as a reversed intensity zone near the load application points for the problem of a ring under diametral compression (Fig. 25.21a). Figure 25.21b shows the phase map of a disk under diametral compression in which, near the load application points, one observes a black streak and the boundary a)

c)

b)

d)

Fractional retardation (radians) 0

0

1

2

3

4

5

6

7

50 100 150 200 250

Part C 25.4

300

Pixels

Fig. 25.22a–d Phase maps in Fig. 25.21 corrected for ambiguity by (a) image processing approach, (b) the interactive approach, and (c) monitoring the isoclinic value. (d) Fractional retardation along the line shown in (c)

of the ambiguous zone is not well defined. The existence of a black or white streak is attributed to the misalignment of optical elements [25.46]. Nevertheless, the poorly obtained phase map as observed needs to be corrected for this error before phase unwrapping. Figure 25.21c shows a portion of phase map obtained for a beam under three-point bending. The existence of an ambiguous zone in this case is not obvious. The ambiguous phase map shown in Fig. 25.21a can be effectively solved by an image processing approach [25.51]. In this image processing approach, initially the pixels forming the ambiguous boundary are identified. Then the intensity in the ambiguous zone is reversed. The resulting phase map is shown in Fig. 25.22a. The phase map in Fig. 25.21b can be solved by an interactive approach [25.4] in which the phase value is recalculated using all six phase-shifted images by a different formula which is independent of θ. The retardation is calculated by    2 2 −1 ± (I5 − I3 ) + (I4 − I6 ) , (25.19) δc = tan (I1 − I2 ) where I1 , I2 , I3 , I4 , I5 , and I6 are the intensities of the phase-shifted images corresponding to the six-step phase-shifting algorithm. In (25.19), for the positive square root, the value the of fractional retardation in fringe orders will always lie between 0 to 0.5, and for the negative square root it always lies between 0.5 and 1.0. The ambiguity in the sign of the fractional retardation is solved by knowing the fringe gradient direction in the problem domain. The results are shown in Fig. 25.22b. Identification of ambiguous zone in Fig. 25.21c is difficult. Figure 25.21d shows the variation of the fractional retardation for this case. In the lower half of the graph, the fractional retardation increases in the direction of the total fringe order. In the upper half it is the other way around and the transition occurs midway. This boundary can be obtained by monitoring the value of the isoclinic angle. Within the boundary, the value of the fractional retardation is recalculated using the modified θ to obtain the correct phase map, as shown in Fig. 25.22c. In many practical situations one may be required to use different ambiguity removal methodologies for different zones of the phase map. Combined use of these techniques has also been reported [25.52]. Load Stepping In load stepping, the fractional retardation is calculated independent of θ, thus avoiding the formation of ambiguous zones in the phase map of δ. Intensity data for

Photoelasticity

the first four optical arrangements of the six-step phaseshifting technique (Table 25.2) is recorded for small steps of loads on either side of the load of interest p (i. e., p + d p and p − d p). From the intensity equations of Table 25.2, one can define the quantities Is , I k , I l , and I m as [25.53] I1 + I2 Ia = Ib + , 2 2 I1 − I2 Ia k I = = cos δ , 2 2 Ia l I = Is − I3 = sin δ sin 2θ , 2 Ia m I = I4 − Is = sin δ cos 2θ . 2 The fractional retardation corresponding to load p for the whole field is obtained as [25.53]  k  (I p− d p − I kp+ d p )/ sin ∂δc −1 δc = tan (I kp− d p + I kp+ d p )/ cos ∂δc   Ia sin δc −1 = tan (25.20) , Ia cos δc Is =

where ∂δc is given by  k  I p+ d p + I kp− d p −1 . ∂δc = cos 2I kp

a)

b)

 gc (x, y) =

gp 2

255 −

gp 2

if (g p+ d p ≥ g p− d p ) . if (g p+ d p < g p− d p )

The wrapped phase is unwrapped by using a simple row-wise scanning algorithm; the resulting phase map is shown in Fig. 25.23b which is free of both ambiguity and noise. Load stepping is not applicable for stress-frozen models and to handle such problems the use of color stepping has been suggested [25.55]. Buckberry and Towers [25.56] proposed a three-wavelength approach in which the wavelengths are chosen such that they follow the relationship λ2 =

2λ1 λ3 . (λ1 + λ3 )

(25.22)

This is necessary to obtain a square-wave mask in order to yield a ramp variation of the phase map.

25.4.4 Evaluation of Isoclinics It is now established that, among the various phase shifting/polarization stepping methods, only those based on plane polariscopes yield accurate isoclinic estimation [25.57]. Brown and Sullivan [25.58] introduced the use of polarization stepping for isoclinic determination in 1990. In polarization stepping, conventional plane polariscope in dark field is used to record isoclinics. The polarization steps used by Brown and Sullivan and the appropriate intensity equations are summarized in

β

Intensity equation

0 π/8 π/4 3π/8

I1 = Ib + Ia sin2 2δ sin2 2θ I2 = Ib + I2a sin2 2δ (1 − sin 4θ) I3 = Ib + Ia sin2 2δ cos2 2θ I4 = Ib + I2a sin2 2δ (1 + sin 4θ)

Part C 25.4

Table 25.4 Optical arrangements for polarization stepping

Fig. 25.23a,b Phase map of a circular disk obtained by various load-stepping methods: (a) Conventional load stepping. (b) Improved load stepping

725

each load, the fractional retardation is calculated as per the normal phase-shifting procedure. Using this, noisefree dark-field images are obtained for the three loads, namely p, p + d p, and p − d p. Let g p+ d p , g p , and g p− d p be the gray levels of the pixel at the location (x, y) corresponding to loads p + d p, p, and p − d p, respectively. The phase map corresponding to load p is obtained as

(25.21)

Inspection of the phase map indicates that it does not contain any zone where the sign of δ is ambiguous as in phase shifting (Fig. 25.23a). However, the phase map has significant noise points. The existence of noise points is not just due to problems in recording intensity data but is significantly due to the nature of (25.20) and (25.21) [25.54]. A novel method, which combines the advantages of phase shifting and load stepping, was proposed by Ramesh and Tamrakar [25.54]. In this approach, for

25.4 Digital Photoelasticity

726

Part C

Noncontact Methods

Table 25.4. The isoclinic parameter is obtained as 1 θc = tan−1 4 1 = tan−1 4

 

I4 − I2 I3 − I1



by θ=

Ia sin2 2δ sin 4θ

Ia sin2 2δ cos 4θ

 for sin2

δ = 0 . 2 (25.23)

Equation (25.23) shows that the isoclinic parameter is undefined when δ = 0, 2π, 4π, . . .. This problem could be partly overcome by using white light as source, which is proposed by Petrucci [25.59]. The polarization-stepped images are recorded using a threeCCD color camera. The isoclinic angle is determined a)

b)

c)

d)

e)

f)

Part C 25.4

Fig. 25.24a–f Isoclinics in steps of 10◦ for the problem

of a ring under diametral compression using theoretically simulated phase-shifted images: (a) Ajovalasit six step (Table 25.2) and (b) Brown and Sullivan (Table 25.4). Experimentally recorded images: (c) Ajovalasit six step (Table 25.2), (d) Brown and Sullivan (Table 25.4), (e) Petrucci (Table 25.4). (f) Theory

  (I4,R+I4,G+I4,B )−(I2,R+I2,G+I2,B ) 1 tan−1 , 4 (I3,R+I3,G+I3,B )−(I1,R+I1,G+I1,B )

where Ii, j (i = 1, 2, 3, 4; j = R, G, B) correspond to pixel gray levels of the R-, G- and B-planes for the four analyzer positions of Table 25.4. Figure 25.24 shows a comparison of isoclinics plotted in steps of 10◦ by a few selected methods discussed in this chapter on both theoretically simulated phase-shifted images and experimentally recorded phase-shifted images for a ring under diametral compression. The plot obtained by the algorithms of Brown and Sullivan (Fig. 25.24d) and Petrucci (Fig. 25.24e) are closer to the theoretical plot of isoclinics (Fig. 25.24f). Minor kinks in the plot of Fig. 25.24d are smoothed out in Fig. 25.24e. Though, the algorithm of Ajovalasit using six-step PST based on a circular polariscope has worked well for theoretically phase-shifted images, it has failed badly when applied to experimentally recorded phase-shifted images (Fig. 25.24c). Ajovalasit recently pointed out that the quarter-wave plate error is different for left and right circularly polarized light. If one considers ε as the error for left circularly polarized light and ε as the error for right circularly polarized light then [25.50]  1 θc = tan−1 sin 2θ(cos ε + cos ε) sin δ + (cos2 2θ 2  + sin2 2θ cos δ)(sin ε − sin ε) cos 2θ(cos ε + cos ε) sin δ − (1 − cos δ) sin 2θ cos 2θ  (25.24) × (sin ε − sin ε) . From (25.24) one could clearly see that the error term is not cancelled when compared to (25.18), which means that the six-step algorithm using circular polariscope is not quite suitable for evaluating isoclinics if the quarterwave plates are not properly matched. Nevertheless,the six-step phase-shifting method [25.44] has stood the test of time in accurately determining the value of the isochromatic parameter over the domain. Reference [25.57] gives a comprehensive review of several other methods too. Among all the methods, the method proposed by Petrucci [25.59] is recommended for general use. In this, apart from isoclinics, it is also possible to obtain the isochromatic intensity field as Ij =

 (I3, j − I1, j )2 + (I4, j − I2, j )2

( j = R, G, B) . (25.25)

Photoelasticity

These are the R, G and B values to be used in three fringe photoelasticity (TFP) or RGB photoelasticity (RGBP) to find the total fringe order. If one wants to evaluate both isochromatics and isoclinics with reasonable accuracy one may have to use two different techniques to estimate these. It is interesting to note that efforts are already underway to combine the six-step method judicially with the plane polariscope methods to evaluate both isochromatics and isoclinics with satisfactory accuracy [25.60].

25.4.5 Unwrapping Methodologies The fractional fringe orders obtained by phase-shifting methods are usually free of random noise. However, the key problem in photoelasticity is one of ambiguity. The other issue to be tackled is on how to handle arbitrary model boundaries with cutouts. The simplest phase unwrapping methodology is to start from a seed point and scan the image horizontally, vertically or with judicious use of these scans, adding or subtracting 2π at phase discontinuities depending on the slope. This algorithm can work well if it does not encounter a noise point, geometric discontinuity or material discontinuity. Geometric and material discontinuities can easily be handled by segmenting the phase map into different zones by interactively drawing their boundaries using geometric primitives. To account for arbitrarily shaped boundaries, newer data structures for boundary information handling, viz. XBN for vertical scanning and YBN for horizontal scanning, has been developed [25.61]. The encoded boundary information is then effectively used for unwrapping. a) D

727

Let A be the primary seed point selected for unwrapping an arbitrary model boundary (Fig. 25.25a). Horizontal seeding yields secondary seed points along the line AB. The area to the right of line BD requires a seed point for unwrapping. To scan this area, the primary seed point is reassigned to C, which is the midpoint of the line BD. Horizontal seeding along CE and vertical scanning of the area BDFEB is then performed as usual. The process of changing the primary seed point continues until the new primary seed point coincides with the boundary pixel. To avoid a possible error in boundary pixel, a boundary tolerance is used. The phase map of a frozen slice with material and geometric discontinuities segmented into various zones to make it an assembly of simply connected regions is shown in Fig. 25.25b; the material property of zone 1 is slightly different than the other zones. The sequence of steps to unwrap is shown in Fig. 25.26a–d. At one location, a small phase streak can be seen in Fig. 25.26d, which is due to noise. In view of boundary delimiting, the unwrapped phase information is quite clear near the cutouts. In Fig. 25.26a, the portion shown has fractional fringe order decreasing in the direction of increasing total fringe order. Rather than correcting it by ambiguity removal methodology, the unwrapping algorithm takes care of this by a user-activated flag. If the phase map has random noise then one can attempt to use Goldstein’s branch cut algorithm or quality-guided phase unwrapping [25.62]. The focus in Goldstein’s approach is to go around the location of noise by providing suitable cuts in the phase map. To locate noise points, the concept of residues is used; the residues are suitably connected to provide the necessary b)

Primary seed point

25.4 Digital Photoelasticity

2

3

4

F C A

E

B

1

Fig. 25.25 (a) Autonomous autoseeding for arbitrary boundaries. (b) Phase map of a stress-frozen slice with cutouts dissected into simply connected regions

Part C 25.4

Scanning lines

728

Part C

Noncontact Methods

branch cuts. Figure 25.27a shows the residues obtained by Goldstein’s approach and the branch cuts are shown in Fig. 25.27b. The unwrapped fractional retardation is shown in Fig. 25.27c, which is far from satisfactory. In quality-guided phase unwrapping, the pixels in the phase map are graded depending on its quality where 1 refers to highest quality and 0 denotes lowest quality. Unwrapping starts from the highest-quality pixel and then proceeds to lowest-quality pixels, thereby assuring the best possible results. This method is computationally intensive and several quality measures are reported in the literature. Among these, phase derivative variance finds application in photoelasticity. Phase derivative variance gives the badness of the pixel as   

Δi,x j

x ¯ m,n −Δ

2

+

  

y y ¯ m,n Δi, j − Δ

k2

2 , (25.26)

where, for each sum, the indices (i, j) range over the k × k neighborhood of each center pixel (m, n) and Δi,x j

0

a)

b)

c)

d)

1

2

3

4

5

6

7

8

Part C 25.4

Fig. 25.26 Sequence of steps to unwrap a model with geometric and material discontinuities. The portion shown in (a) has optical properties that are slightly different from the other zones. Note that the segmented regions are not rectangular tiles but have arbitrarily shaped boundaries and their recognition and handling was possible due to a new data structure for boundary information handling [25.61]

y

and Δi, j are the partial derivatives of the phase (i. e., x , and ¯ m,n the wrapped phase differences). The terms Δ y ¯ Δm,n are the averages of these partial derivatives in the k × k windows. Equation (25.26) is a root-mean-square measure of the variances of the partial derivatives in the x- and y-directions. The variance is then negated to represent goodness. Figure 25.28 shows the quality map and the unwrapped phase for the stress-frozen slice by phase derivative variance. The unwrapped phase map does not show phase streaks, as in Fig. 25.26d. However, only one of the zones compare with Fig. 25.26d and unwrapping is affected by the choice of seed points (Fig. 25.28b,c). Both Goldstein’s approach and quality-guided phase unwrapping attempt autonomous phase unwrapping. Although with domain masking they work for geometries with cutouts, for complex geometries with distinct fractional retardation variation they fail. The combined use of delimiting various regions interactively followed by quality-guided phase unwrapping within each region would be more effective. This would involve only minimal user intervention, which in many cases is desirable [25.63]. In Sect. 25.4.3 it has been shown clearly that ambiguity in the sign of fractional retardation is due to the fact that the orientation of θ should reflect one of the principal stress directions uniformly over the domain. Recognizing this fact, there have been attempts to unwrap θ first and then use this value of θ for calculating the fractional retardation. The input, isoclinic value (θ) physically lies in the range −π/2 to +π/2, whereas the calculated isoclinic value (θc ) lies in the range −π/4 to +π/4. A gray scale θ plot is obtained by defining   π 255  θc + = INT(R) , g(x, y) = INT π 2 where g(x, y) is the gray level value at the point (x, y) and INT(R) is the nearest integer to R. This image representation is very helpful for identifying ambiguous zones as there would be a sudden jump from white to black in the mapped image. Figure 25.29 shows the problem of a disk under diametral compression with wrapped θ obtained using the algorithm of Petrucci [25.59]. The zones that need to be unwrapped are very clearly seen. To unwrap θ, a scanning method is developed. The boundary of the zone is identified by a user-selected variable (theta tol) and then unwrapping is effected. The scheme needed for a bottom to top scan

Photoelasticity

a)

b)

25.4 Digital Photoelasticity

729

c) 0 8 100

7 6

200

5 300

4 3

400

2 500

1 0

100

200

300

400

500

600

700

0

Fig. 25.27a–c Goldstein’s approach to the stress-frozen slice (a) Residues. (b) Branch cuts. (c) Unwrapped phase [25.63]

a)

b)

c)

0 50 100

8 7

150 200 250 300 350 400 450 500 550

6 5 4 3 2 1 0

100

200

300

400

500

600

700

0

0 50 100

8 7

150 200 250 300 350 400 450 500 550

6 5 4 3 2 1 0

100

200

300

400

500

600

700

0

Fig. 25.28a–c Quality map and unwrapped phase for the stress-frozen slice (a) Quality map. (b) Unwrapped phase with seed point in the top zone. (c) Unwrapped phase with seed point in the bottom zone [25.63]

is

25.4.6 Color Image Processing Techniques abs(θ j+1,i − θ j,i ) > theta tol , then  θ j+1,i − θ j,i < 0 θ −π θ j,i = j,i π2 , θ j,i + 2 θ j+1,i − θ j,i > 0

a)

b)

Fig. 25.29 (a) Wrapped phase of θ for a disk under diametral compression. (b) Unwrapped phase

Part C 25.4

where i, j correspond to the row and column, theta tol is the θ tolerance value. Likewise different checking conditions are evolved along different scanning directions [25.64]. The unwrapped θ by the above method is shown in Fig. 25.29b. If the model has cutouts then the boundary encoding discussed previously needs to be used for θ unwrapping too. Use of the quality-guided approach for θ unwrapping has also been reported [25.65]. At the isochromatic fringe locations θ is not defined mathematically. At these zones, suitable interpolations need to be done for obtaining a smooth variation of θ. Correct and accurate evaluation of θ is one of the challenges in digital photoelasticity and its solution is not far off.

Among the various experimental techniques, photoelasticity has a unique distinction of providing fringe data in color. Use of a color code to identify fringe gradient direction and to assign the total fringe order approximately has been in use in conventional photoelasticity. Before the advent of intensity-based

730

Part C

Noncontact Methods

a)

b)

c)

Fig. 25.30 (a) Color isochromatics of a disk under diametral compression. (b) Total fringe order evaluated using TFP, represented as an image. (c) Smooth variation of fringe order by RTFP

processing methods, extraction of total fringe order by processing the spectral content of the color image was analyzed [25.66]. Processing the intensity of the color image for extraction of data was then thought of. Use of the green plane of the color image recorded by a color CCD camera for phase shifting was proposed by Ramesh and Deshmukh [25.67]. Another approach which directly mimics the conventional approach of using color information of all the three planes (R, G, and B) was also proposed. The technique is variedly known as three-fringe photoelasticity (TFP) [25.68] as beyond three fringe orders colors tend to merge, or RGB photoelasticity (RGBP) [25.69] as information from the R, G, and B planes are used for fringe order evaluation. In TFP/RGBP, for any test data point, an error term e is defined as [25.4]  e = (Re − Rc )2 + (Ge − Gc )2 + (Be − Bc )2 , (25.27)

Part C 25.4

where, the subscript ‘e’ refers to the experimentally measured values for the data point and ‘c’ denotes the values in the calibration table. The calibration table is to be searched until the error ‘e’ is a minimum. For the Rc , Gc , and Bc values thus determined, the calibration table provides the total fringe order. The fringe order data thus obtained can be converted into a set of gray level values using the following equation [25.70]:   255 × f (x, y) = INT(R) , g(x, y) = INT Nmax where f (x, y) is the fringe order at point (x, y), Nmax is the maximum fringe order of the calibration table (three in most cases), g(x, y) is the gray level value at the point (x, y), and INT(R) is the nearest integer to R. The fringe order evaluated by conventional TFP for a circular disk under diametral compression is repre-

sented as a gray level image in Fig. 25.30b. Ideally one should obtain a smooth variation of gray levels, which is not the case here. The variation of principal stress difference is continuous and this has to be judiciously used to order fringes. In refined TFP (RTFP) [25.70], (25.27) is modified to e=



(Re−Rc )2+(Ge−Gc )2+(Be−Bc )2+(Np−N)2×K 2 , (25.28)

where Np is the fringe order obtained for the neighborhood pixel to the point under consideration in the generic specimen and N is the fringe order at the current checking point of the calibration table. A multiplication factor K is used to control the performance of (25.28). The magnitude of K has to be selected iteratively and a value of 100 is found to be suitable for most purposes. Smooth variation of fringe order is seen in Fig. 25.30c. There have been efforts to extend the range of RGBP beyond fringe order three [25.65, 71]. These efforts attempt to use more images and thereby compromise the elegance of TFP/RTFP. Use of a single image for data processing is attractive from an industrial standpoint and RTFP is a good candidate for such applications.

25.4.7 Digital Polariscopes If individual elements of a polariscope can be independently rotated then conventional polariscopes are equally suitable for recording phase-shifted images. The one difference is that the human eye is replaced by a suitable CCD camera and associated hardware. Furthermore, the polariscope needs to be calibrated as discussed in Sect. 25.4.2 before making any measurements for an unknown model. As rotation of individual

Photoelasticity

Fig. 25.31 Digital polariscope developed for residual stress measurement in glass [25.72]. The light source unit accommodates a stepper motor for changing the polarization of the incident light for phase-shifting measurements. The monitor shows a typical residual stress fringes over the wall thickness in a glass tumbler

elements can lead to slight optical misalignments and is also time consuming, efforts have been aimed at developing automated polariscopes. The first generation of these replaced manual rotation of the optical elements by stepper motors controlled by a PC. Usually four stepper motors are needed for such an approach to handle each element independently. Such polariscopes can have a large field. Andrei et al. [25.72] have developed a secondgeneration automated polariscope specifically for evaluating residual stress in glass components (Fig. 25.31). This has some very special features. Instead of rotating the output quarter-wave plate and analyzer combination, using the principle of reversibility of optical systems, here the polarization of the input light is modified. The light source is a photodiode and prealigned polarizer and achromatic quarter-wave plate combination is a)

b) Viewing lens

25.4 Digital Photoelasticity

731

brought in front of the light source by a stepper motor. This has provision for eight steps and in principle can accommodate a phase-shifting algorithm with up to eight steps. Only one stepper motor is used for automation. The one restriction of the polariscope is that the field of view is quite small, of the order of only 25 mm diameter. The polariscope is ideally suitable for analyzing stress-frozen slices too. A third generation of digital polariscopes aims at recording as many phase-shifted images as possible simultaneously, thus paving the way for digital dynamic photoelasticity. Hardware limitations forced researchers to come up with the least number of images for analysis. Since there are four unknowns (Ib , Ia , θ, and δ), a minimum of four images are required. Several four-step methods have been reported in the literature [25.44, 74–76]. Figure 25.32a shows the first of these efforts in recording simultaneously four phase-shifted images. This model requires four separate CCD cameras [25.73, 75]. In view of the use of prismatic beam splitters, the polarization behavior of these optical elements needs to be properly accounted for in the data interpretation. A compact unit which uses only one high-resolution camera has also been developed. Asundi modified a multispectral imager for PST in photoelasticity [25.77], but it is bulky and shares the same problem of prismatic beam splitters. Another piece of equipment has been developed by Lesniak et al. [25.78], christened the poleidoscope as it combines the features of a polariscope and a kaleidoscope. In this, a suitable lens is produced by sectioning a conventional lens into four quadrants and separating the quadrants by a small amount (Fig. 25.32d). Optically the result is four identical images that can be focused into corresponding quadrants of a CCD chip. Four sets of prealigned quarter-wave plates and analyzer are placed after the c)

d)

Light source

Four-way beamsplitter

CamerasX4

Fig. 25.32 (a) Overall view of the four-camera arrangement to record phase-shifted images. (b) Cut section view showing the inner arrangement. (c) Poleidoscope [25.73]. (d) Lens cut into four quadrants to get four images simultaneously (Courtesy: Product brochure of Poleidoscope)

Part C 25.4

Light source Viewing lens

732

Part C

Noncontact Methods

considered then the photoelastic parameters would be evaluated with error. The RMS error [(δ − δ)/2π] for δ = 0–2π for different values of θ is shown in Fig. 25.33, from which it is clear that the RMS error for the algorithm proposed by Ajovalasit is the lowest. This algorithm uses the first four arrangements of the six-step method (Table 25.2). The isoclinic value is obtained as   1 I1 + I2 − 2I3 θc = tan−1 2 2I − (I1 + I2 )  4  Ia sin δ sin 2θ 1 −1 for sin δ = 0 . = tan 2 Ia sin δ cos 2θ

RMS error in isochromatic parameter (fringe orders) 0.03 Ajovalasit 4 step Asundi 4 step Barone and Patterson 4 step

0.025 0.02 0.015 0.01 0.005 0

The following equation assures a high modulation in evaluating the fractional retardation: 0

15

30

45 Theta (deg)

Fig. 25.33 Plot of RMS error in the isochromatic parameter for various four-step algorithms as a function of θ [25.48]

lens, each set aligned to a particular quadrant, to get different phase-shifted images simultaneously. The proponents of these new hardware used their phase-shifting methods incorporated into these hardware. However, it is desirable to use an appropriate technique that gives the least error and also adopt a solution method that provides the highest modulation. Although many four-step methods have been developed, if the mismatch of quarter-wave plate is

  (I1+I2−2I3 ) sin 2θc+[2I4−(I1+I2 )] cos 2θc δc =tan−1 . (I1−I2 )

The accuracy of a four-step PST is lower than that of the six-step method. Hence only in those applications where dynamic recording is needed is the choice of four-step method recommended. There has been development of a tricolor light source for photoelastic analysis [25.79]. If this concept is further developed to have three wavelengths that satisfy (25.22), a simple color CCD camera can provide a wealth of information, and a phase map free of ambiguity and noise can be obtained as mentioned in Sect. 25.4.3. If the focus is only on total fringe order less than three then RTFP is ideally suited for dynamic studies too.

25.5 Fusion of Digital Photoelasticity Rapid Prototyping and Finite Element Analysis

Part C 25.5

For complex analyses, significant advantages can be gained through the integrated use of numerical and experimental techniques. Photoelasticity has the unique distinction that it is the only whole-field technique that can reveal stresses interior to the model. The traditional methods for model fabrication involve casting to shape or machining from solid blocks using thermosetting resins. These methods require skill and are time consuming. Model making may typically require several days for preparation [25.9]. Rapid prototyping (RP) is an advanced manufacturing technology and is widely used at present for the fabrication of conceptual and functional models. Three-dimensional complex models that are photoelastically sensitive can be built directly from computer-aided design (CAD) models in

a matter of hours. The CAD models generated can also be profitably used for a finite element analysis of the model under consideration. Thus a fusion of RP for model making, digital photoelasticity as a tool for experimental analysis, and finite elements (FE) as a tool for numerical analysis can improve confidence in the analysis of complex structures. Fused deposition modeling (FDM), which is a cost effective RP methodology and is also easily available, can be used to make models. Using rapid tooling (RT) one can make a mould from these models. This mould could be used to make models out of conventional threedimensional photoelastic materials or special quality RT resins. The mould has to be carefully designed so that residual birefringence is not introduced due to differ-

Photoelasticity

25.5 Fusion of Digital Photoelasticity Rapid Prototyping and Finite Element Analysis

ential cooling. Ramesh et al. [25.80] showed that RT resins (SG 95) do have stress-freezing properties but has a poor optical response compared to conventional model materials. Stereolithography (STL) is another RP technique, in which prototypes are built using a photosensitive monomer formulation usually an acrylic- or epoxybased resin. This technique is unique to photoelasticians in the sense that the models made can be directly used for stress freezing and slicing. However, sufficient care is required in the choice of photopolymer, build parameters, build style etc. [25.82]. Photopolymers typically shrink up to 8% by volume during polymerization in STL, which lead to the formation of residual stresses. An experimental investigation by Karalekas and Agelopoulos [25.83] showed that the shrinkage strains in acrylic-based photopolymer resin were of considerable magnitude whereas for the epoxy-based resin system the residual strains measured were of low magnitude. In order to produce models that are free of residual stress and birefringence it is essential to achieve homogeneous material properties. The standard build parameters employed in STL lead to substantially different material properties in the edge and core of the specimens. This can be avoided by producing photoelastic models in the STL without a border or skin. It is important that all parts of the component receive an equal dosage of energy from the laser, preferably above the critical value of ten times that required to achieve gelation. It is reported that the accurate clear epoxy solid (ACES) build style gave the most accurate a)

b)

y

733

parts when compared to the WEAVETM and QuickCast styles. For realistic FE modeling, the geometry of the problem under consideration needs to be precisely defined. This is taken care of by the CAD model used for making the RP model. The CAD model can be discretized by standard mesh generators and FE analysis can be carried out by one of the standard packages. FE analysis can provide the individual stress components directly. Photoelasticity by itself can provide only principal stress difference or normal stress difference and in-plane shear stress directly. Individual normal stress components can be obtained only by stress separation techniques, which require the use of auxiliary methods and are also time consuming. If the FE results can be postprocessed to plot photoelastic fringe contours then the comparison between experiments and FE results become a lot simpler. Ramesh et al. [25.84] were the first to present a practical way of plotting fringe contours from 2-D FE results by using a scanning scheme. Plotting of photoelastic contours from 3-D FE results to plot photoelastic contours is quite involved and was recently developed for 3-D models meshed with 20-node brick elements [25.85] and then extended for models discretized using 10-node elements [25.81]. Plotting of fringe contours from three-dimensional models requires the calculation of secondary principal stress difference at several points (x, y, z) along the light path in the 3-D FE model [25.81]. For a slice cut from c) e)

z

x

d) f)

Part C 25.5

Fig. 25.34 (a) A three-dimensional wireframe model of a crane hook. (b) FE discretization of the model. (c) Experimental isochromatics of the critical slice. (d) Isochromatics of the slice from the post processed results of FE. (e) Experimental isochromatics of a subslice. (f) Isochromatics of the subslice from the postprocessed FE results [25.81]

734

Part C

Noncontact Methods

the model, neglecting the variation of secondary principal stress direction over the light path, the fringe order is calculated as h

1 N= Fσ

(σ1s − σ2s ) dh , 0

where σ1s and σ2s represent the secondary principal stresses, h is the length of the light path in mm, Fσ is the material stress fringe value in N/mm/fringe, and N is the total fringe order. To plot isochromatics of slices, a pair of cutting planes is defined by specifying a point lying on the plane along with its normal. This enables the specification of even inclined slices for oblique incidence. The normal distance between the cutting planes determines the thickness of the slice. For ease of computations, one of the coordinates axes is considered as the viewing direction. Consider the problem of a crane hook subjected to a load as shown in Fig. 25.34. Figure 25.34 shows the wireframe model of the crane hook converted into a solid model and then meshed with 10-node quadratic

elements. The load introduces an axial pull and bending on the critical cross section of the crane hook. A photoelastic model made of conventional model material is prepared by a combination of RP and RT; the RP method employed is the FDM process. Figure 25.34c,e shows the experimental isochromatics of the slice viewed along the y-direction and the subslice viewed along the z-direction, respectively. Only for the experiment are these identified separately as a slice and a subslice whereas for the numerical simulation both are slices with appropriate cutting planes. Figure 25.34 also shows the numerically simulated fringe contours of these slices in color. The comparison is good and the typical behavior of fringes due to bending is clearly seen. In this example, as the location of critical zone could be identified quite easily, it is easier to establish an optimum slicing plan. In industrial applications one may not know the stress distribution in complex models. In such circumstances, optimum slicing plan indicating the slicing location and the viewing direction can be evolved by postprocessing the results of FEA to plot fringe contours of slices.

25.6 Interpretation of Photoelasticity Results

Part C 25.6

One of the first questions that emerges is: how does one relate the results from conducting experiments on plastics to metallic prototypes? The validity can be explained by considering the nature of the equations that are necessary for solving the stress distribution. In two-dimensional problems, the equations of equilibrium together with the boundary conditions and the compatibility conditions in terms of stress components give us a system of equations that is usually sufficient for the complete determination of the stress distribution. The compatibility conditions for plane-stress and plane-strain situations in terms of stress components are, respectively,     2 ∂2 ∂Fx ∂Fy ∂ + (σx + σ y ) = −(1 + ν) + , ∂x ∂y ∂x 2 ∂y2  2    ∂ ∂2 1 ∂Fx ∂Fy + (σx + σ y ) = − + . (1 − ν) ∂x ∂y ∂x 2 ∂y2 It is interesting to note that, in the case of constant body forces (Fx , Fy ), the equations determining the stress distribution do not contain the elastic constants of the material. Thus, the stress magnitudes and their distribution are the same for all isotropic materials. The

photoelastic model needs to be loaded carefully so that it does not experience undue large deformation in comparison to the metallic prototypes. In three-dimensional problems, the compatibility conditions are given by Beltrami and Mitchell equations [25.9]. These equations contain Poisson’s ratio and, unlike 2-D problems, the stress distribution is dependent on the Poisson’s ratio even if the body force is constant. In certain special cases, in 2-D problems also, the stress distribution is dependent on the Poisson’s ratio. When one works on multiply connected bodies, such as a plate with a hole having a resultant force acting at the boundary of a hole, the stress distribution is a function of the Poisson’s ratio. Thus in 3-D model studies or in situations where body forces change as in rotating components, the mismatch of Poisson’s ratio is a crucial factor in relating stresses in the model to those in prototypes. For roomtemperature studies, the mismatch of Poisson’s ratio between the plastics and the metallic prototypes is usually negligible and its effect is ignored. In conventional three-dimensional photoelasticity, at the stress-freezing temperature, the model’s Poisson’s ratio approaches 0.5 and its influence become significant. However, experi-

Photoelasticity

ments have shown that the mismatch of Poisson’s ratio affects only the minor principal stress significantly and the major principal stress is relatively unaffected [25.4]. When integrated and scattered light photoelasticity are applied to glass components, the study is conducted directly on prototypes and the difficulty in data interpretation does not arise. Reflection photoelasticity is intended for analyzing prototypes, and the governing equations do take care of the effect due to reinforcement and Poisson’s ratio mismatch. In dynamic photoelastic-

25.7 Stress Separation Techniques

735

ity, in addition to the usual requirement of geometrical similarity, the shape of the pulse should be similar in the model and the prototype. In impact problems, one requires that the striking body is also reproduced true to scale and that the density and elastic moduli of the striking and struck bodies have the same ratios as for the prototype. In photoplasticity, the choice of model material is a crucial task and its stress–strain behavior should closely resemble that of the metallic component being investigated.

25.7 Stress Separation Techniques Photoelasticity directly provides the information of principal stress/strain difference and the orientation of the principal stress/strain direction at the point of interest. Using these, one can find the normal stress/strain difference and in-plane shear stress/strain by invoking equations from mechanics of solids or Mohr’s circle. In a restricted sense, stress separation techniques can be used to find the individual principal stresses and, in a complete sense, they can provide all the components of the stress tensor. These methods can be either point by point or whole field. In early studies for pointby-point stress separation, a lateral extensometer was used to provide the lateral strain at the point, which provided the sum of the principal stresses. This, in conjunction with the photoelastic data, provided the individual stress components [25.9]. For whole-field stress separation, the use of interferometric techniques and electrical analogy methods for obtaining the sums of principal stresses have been proposed. All these methods require additional experimental information to be obtained by techniques other than photoelasticity. The introduction of oblique incidence technique by Drucker [25.86, 87] provided additional data for stress separation using photoelasticity itself for a point-bypoint stress separation approach. With the advent of digital computers, the use of numerical techniques to provide additional data for stress separation became quite feasible [25.88]. Various types of hybrid techniques exist [25.4]. Some give more prominence to either experimental or numerical data, while in others they are both used judiciously.

one can carry out stress separation for lines parallel to the x- or y-axis. Using the finite difference approximation, the integration of equilibrium equation can be rewritten as

25.7.1 Shear Difference Technique

The rest of the procedure is same as that of the conventional method. The usefulness of the method for stress separation in photoelastic coating technique is shown by Trebuna [25.89].

(σx ) j = (σx )i −

i

Δy

Δx ,

(25.29)

where (σx )i is the stress value at the start of the integration procedure. If the starting point is chosen on a free boundary then (σx )i is NFσ cos2 θ . h Once σx is calculated for a particular point on the grid, σ y can then be calculated. To perform the integration, one requires the data of both isochromatic and isoclinic fringe orders along the line of interest and for two more lines one above and the other below the line of interest. One of the main sources of error accumulation in the conventional shear difference method is the error in calculating the shear slope. It is well known that numerical differentiation is usually less accurate than numerical integration. Tesar suggested that the shear stress be expressed in terms of the difference of principal stresses and be differentiated as a product of two functions. (σx )i = σ1 cos2 θ =

(σx ) j = (σx )i −

j ! Δ(σ1 − σ2 ) 1 i

j !

Δy

(σ1 − σ2 ) cos 2θ

2

sin 2θΔx

Δθ Δx . Δy

Part C 25.7

−

i

In the shear difference method, one of the equations of equilibrium is integrated to separate the stresses. Thus,

j ! Δτ yx

736

Part C

Noncontact Methods

while making normal incidence along the z-direction, the slice is considered as small such that no variation in the z-direction exists. However, while using normal incidence in the y-direction, the slice is considered sufficiently thick to provide a variation of τxz . The same approximation applies to the determination of τxy . This is the inherent contradiction in three-dimensional analysis and one has to live with it for engineering analysis as it provides enormous simplicity to solve problems of practical interest.

y

z

ξ

Δz

25.7.3 Reflection Photoelasticity Δx

x

Fig. 25.35 Scheme to collect data for performing the

shear-difference technique in three dimensions. The directions of two normal and one oblique incidences are indicated

25.7.2 Three-Dimensional Photoelasticity As in two-dimensional models, the evaluation of stresses on free boundaries is still relatively simple. However, the separation of stresses inside the body requires great experience as well as understanding of the stress and strain distribution in three-dimensional bodies. For the complete determination of the stress components, a subslice has to be cut from the slice and analyzed. Let the x-axis be taken along the axis of the subslice (Fig. 25.35). For each data point, one has to use normal incidence along the y- and z-axis and one oblique incidence in the xz-plane. From the normal incidence along the z-axis and the oblique incidence in the xz-plane, (σx − σ y ), τxy , and τ yz can be obtained. From normal incidence along the y-direction, (σz − σx ) and τzx can be obtained. For stress separation one has to integrate the equilibrium equation in three dimensions [25.9]. Following (25.29), this can be written (σx ) j = (σx )i −

j ! Δτ yx i

Δy

Δx −

j ! Δτxz i

Δz

Δx .

Part C 25.7

To integrate this equation, one has to find the shear stresses for two more lines in the xy- and yz-planes, as shown in Fig. 25.35. It is worthwhile to note that,

The stress separation techniques used in reflection photoelasticity are confined to the determination of the in-plane principal strains/stress components rather than the complete strain or stress tensor at the point of interest. For flat objects, the oblique incidence method, as used in transmission photoelasticity, is possible. Standard attachments for performing oblique incidence are available. While using standard attachments and commercially obtained coating materials, the manufacturer also supplies the methodology to obtain the individual strain components. Use of a photostress separator gauge is also available. The technique is equally applicable for flat and curved objects. In this approach, one just has to paste the strain gauge at the point of interest on the birefringentcoated model and the strain gauge signal is proportional to the sum of the principal strains. Correction factors have also been developed for photostress separator gauges, for which they are used on members subjected to bending, torsion etc. In many applications involving engineering assessment of the design, a photoelastic coating is used mainly to validate the numerical technique such as finite elements for stress field evaluation or design certification. Though for flat objects the stress separation is simpler, for curved objects the evaluation of the stress field becomes quite involved. In such instances, the validation of the numerical techniques becomes much easier if the results of the numerical technique are plotted in the form of fringe contours. Using the color code [25.4], one can plot fringe contours in color for both flat and curved objects. A quick assessment of the numerical technique is possible by comparing the fringe patterns of the experiments with the fringe contours plotted.

Photoelasticity

25.9 Further Reading

737

25.8 Closure In this chapter, a brief overview of crystal optics, which is essential for understanding what causes the formation of fringes in photoelasticity, is initially presented. More clarity of these concepts can be obtained by viewing the DVD [25.5]. animations included on the A systematic but brief presentation of transmission photoelasticity is then given. Although the current emphasis on digital photoelasticity is to obtain the results autonomously, a brief note on fringe ordering is presented so that a user is sufficiently well equipped to check the results churned out by the modern methodologies. Many variants of photoelasticity are then briefly discussed wherein the focus is on appreciating the special features of the methodology rather than its details. A comprehensive overview of the recent developments of digital photoelasticity is then presented. Sustained material research has helped the development of variants of photoelasticity; for example, in 3-D

conventional photoelasticity, the material should have the capability for stress freezing, in scattered light photoelasticity the material should scatter light, in reflection photoelasticity the coating has to be contourable and also needs to be selected based on the base material, and so on. A summary of the material properties is presented in the appendix. Developments in RP have renewed designers’ interest in using photoelasticity, as models can be prepared in a matter of hours, thereby cutting the overall time required for analysis. Furthermore, the novel approach of fringe plotting from FE results has opened up an elegant approach for comparing the results of experiments with the numerical model. Thus a combined approach of RP for model making, FE for numerical analysis, and digital photoelasticity for experimental analysis would be a viable method for designers to use in the future.

25.9 Further Reading • • • • • • • •

•

• • • • • • •

phase-shifting technique, Opt. Lasers Eng. 31, 263– 278 (1999) K. Ramesh, S.K. Mangal: Data acquisition techniques in digital photoelasticity: a review, Opt. Lasers Eng. 30(1), 53–75 (1998), Errata 31(85) (1999) K. Ramesh: PHOTOSOFT H: A comprehensive photoelasticity simulation module for teaching the technique of photoelasticity, Int. J. Mech. Eng. Edu. 25(4), 306–324 (1997) J.C. Dupré, A. Lagarde: Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image processing, Exp. Mech. 37, 393–397 (1997) K. Ramesh, V. Ganapathy: Phase-shifting methodologies in photoelastic analysis – the application of Jones calculus, J. Strain Anal. Eng. Des. 31(6), 423–432 (1996) A. Asundi, M.R. Sajan: Digital dynamic photoelasticity, Opt. Lasers Eng. 20, 135–140 (1994) Y. Morimoto, Y. Morimoto Jr., T. Hayashi: Separation of isochromatics and isoclinics using Fourier transform, Exp. Tech. 18(5), 13–17 (1994) C.P. Burger: Photoelasticity. In: Handbook of Experimental Mechanics, ed. by A.S. Kobayashi (Soc.

Part C 25.9

•

K. Ramesh: Digital Photoelasticity: Advanced Techniques and Application (Springer, Berlin, Heidelberg 2000), Note: excerpts are included in this chapter with the kind permission of Springer, Berlin J.W. Dally, W.F. Riley: Experimental Stress Analysis (McGraw-Hill, New York 1991) H.K. Aben: Integrated Photoelasticity (McGrawHill, New York 1979) L.S. Srinath: Scattered Light Photoelasticity (Tata McGraw-Hill, New Delhi 1983) J.W. Dally: An introduction to dynamic photoelasticity, Exp. Mech. 20, 401–416 (1980) K. Ramesh: Digital photoelasticity – current trends and future possibilities, Proc. Int. Con. Comp. & Exp. Eng. Sci., Chennai, 2005 (Tech. Science, Forsyth 2005) pp. 2159–2164 A. Ajovalasit, S. Barone, G. Petrucci, B. Zuccarello: The influence quarter wave plates in automated photoelasticity, Opt. Lasers Eng. 38, 31–56 (2002) E.A. Patterson: Digital photoelasticity: principles, practice and potential, Strain 38, 27–39 (2002) T.Y. Chen (Ed.): Selected Papers on Photoelasticity, SPIE Milestone Series MS-158 (SPIE Opt. Eng. press, Washington 1999) S.K. Mangal, K. Ramesh: Determination of characteristic parameters in integrated photoelasticity by

738

Part C

Noncontact Methods

• • • • •

Experimental Mechanics and VCH, New York 1993) pp. 165–266 K. Ramesh: Animations to accompany the Chapter on Photoelasticity, 2008. URL: http://apm.iitm.ac.in/ smlab/kramesh/book_3.htm H. Aben, C. Guillemet: Photoelasticity of Glass (Springer, Berlin 1993) B.H. Billings (Ed.): Applications of Polarised Light, SPIE Milestone Series MS-57 (SPIE Opt. Eng., Washington 1992) J. Cernosek: Three-dimensional photoelasticity by stress freezing, Exp. Mech. 20(12), 417–426 (1980) J. Cernosek: On the photoelastic response of composites, Exp. Mech. 15, 354–357 (1975)

• • • • • •

J. Javornicky: Photoplasticity (Elsevier Scientific, New York 1974) A.J. Durelli: Applied Stress Analysis (Prentice-Hall, New Jersey 1967) A.B.J. Clark, R.J. Sanford: A comparison of static and dynamic properties of photoelastic materials, Exp. Mech. 3(6), 148–151 (1963) R.B. Heywood: Designing by Photoelasticity (Chapman Hall, London 1952) M.M. Frocht: Photoelasticity Vol. 1 (Wiley, New York 1941) M.M. Frocht: Photoelasticity Vol. 2 (Wiley, New York 1948)

25.A Appendix Table 25.5 Approximate properties of a few photoelastic materials Material

Stress fringe valueb Fσ (N/mm/fringe)

Young’s modulusc E (MPa)

Proportionalc limit (MPa)

Poisson’s ratioc ν

Polycarbonate 8 2600 3.5 0.28 Epoxy 12 3300 35 0.37 Glass 324 70 000 60 0.25 Homolite 100 26 3900 48 0.35 Homolite 911 17 1700 21 0.40 Plexiglass 140 2800 – 0.38 Polyurethane 0.2 3 0.14 0.46 Gelatin 0.1 0.3 – 0.50 a The higher the figure of merit, the more suitable the material is for photoelastic analysis b Stress fringe value corresponding to a sodium vapour source (589.3 nm) c Room-temperature properties

Figure of merita (1/mm) 325 275 216 150 100 20 15 3

Part C 25.A

Photoelasticity

25.A Appendix

739

Table 25.6 Properties of photoelastic coating materials Material

Ec (GPa)

νc

K

Strain limit (%)

Coatings suitable for high-modulus materials (metallic specimens) Polycarbonate 2.21 0.16 PS-1a 2.50 0.38 0.15 10 PS-2a 3.10 0.36 0.13 3 PS-8a 3.10 0.36 0.09 3 to 5 PL-1 liquida 2.90 0.36 0.10 3 to 5 PL-8 liquida 2.90 0.36 0.08 3 to 5 Polyester 3.86 0.04 1.5 Epoxy with anhydride 3.28 0.12 2.0 Coatings suitable for medium-modulus materials (nonmetallic specimens) PS-3a 0.21 0.42 0.02 30 PL-2 liquida 0.21 0.42 0.02 50 Coatings suitable for low-modulus materials (rubber) Polyurethane 0.004 0.008 15 PS-4a 0.004 0.50 0.009 > 50 PL-3 liquida 0.014 0.42 0.006 > 50 a

Max. usable temp. (◦ C)

150 260 200 230 200

Suitability

Flat Flat Flat Flat Contourable Contourable Flat Flat/contourable

200 200

Flat Contourable

175 150

Flat Flat Contourable

Courtesy of Measurements Group, Inc., Raleigh, NC, USA

Table 25.7 Approximate properties of some stereolithographic resins Material

Stress fringe value Fσ (N/mm/fringe) Room temp. Stress freezing temp.

Young’s modulus E (MPa) Room temp. Stress freezing temp.

Figure of merit (1/mm) Room temp.

SL5170 SL5180 SL5190 SL5530

34.10 33.62 32.22 38.58

3296 3275 3668 3681

97 97 114 95

0.7917 0.8091 0.7933

26.45 27.68 26.64

Part C 25.A

740

Part C

Noncontact Methods

References 25.1

25.2

25.3

25.4

25.5

25.6

25.7 25.8

25.9 25.10

25.11 25.12 25.13 25.14

25.15 25.16

25.17

Part C 25

25.18

D. Brewster: On the communication of the structure of doubly refracting crystals to glass, muriate of soda, flour spar and other substances by mechanical compression and dilatation, Philos. Trans. R. Soc. 106, 156–178 (1816) T.S. Majumdar, R.P. Behringer: Contact force measurements and stress-induced anisotropy in granular materials, Nature 435, 1079–1082 (2005) A.S. Voloshin, C.P. Burger: Half fringe photoelasticity – a new approach to whole field stress analysis, Exp. Mech. 23(9), 304–314 (1983) K. Ramesh: Digital Photoelasticity: Advanced Techniques and Application (Springer, Berlin, Heidelberg 2000) K. Ramesh: Animations to accompany the Chapter on Photoelasticity, 2008. URL: http://apm.iitm.ac.in /smlab/kramesh/book_3.htm K. Ramesh: Engineering Fracture mechanics, ebook (Indian Institute of Technology, Madras 2008), http://apm.iitm.ac.in/smlab/kramesh/book_4.htm A.J. Durelli, A. Shukla: Identification of Isochromatic Fringes, Exp. Mech. 23(1), 111–119 (1983) R.J. Sanford: Application of the least squares method to photoelastic analysis, Exp. Mech. 20, 192–197 (1980) J.W. Dally, W.F. Riley: Experimental Stress Analysis (McGraw-Hill, New York 1991) L.S. Srinath, K. Ramesh, V. Ramamurti: Determination of characteristic parameters in threedimensional photoelasticity, Opt. Eng. 27(3), 225– 230 (1988) H.K. Aben: Integrated Photoelasticity (McGrawHill, New York 1979) L.S. Srinath: Scattered Light Photoelasticity (Tata McGraw-Hill, New Delhi 1983) J.W. Dally: An introduction to dynamic photoelasticity, Exp. Mech. 20, 401–416 (1980) A.A. Wells, D. Post: The dynamic stress distribution surrounding a running crack – a photoelastic analysis, Proc. Soc. Exp. Stress Anal. 16(1), 69–92 (1958) G.R. Irwin: Discussion of ref. 25.14, Proc. Soc. Exp. Stress Anal. 16(1), 93–95 (1958) H.V. Rossamanith, A. Shukla: Dynamic photoelastic investigation of stress waves with running cracks, Exp. Mech. 21, 415–422 (1981) C.Y. Zhu, A. Shukla, M.H. Sadd: Prediction of dynamic contact loads in granular assemblies, ASME J. App. Mech. 58, 341–352 (1991) F. Zandman, S. Redner, J.W. Dally: Photoelastic Coatings, Soc. Experimental Stress Analysis Monograph No.3 (Soc. Experimental Stress Analysis, Westport 1977)

25.19

25.20

25.21

25.22

25.23

25.24

25.25 25.26

25.27 25.28

25.29

25.30

25.31

25.32 25.33

25.34

25.35

25.36

F. Zandman, S. Redner, E.I. Riegner: Reinforcing effect of birefringent coatings, Exp. Mech. 2(2), 55– 64 (1962) J.W. Dally, I. Alfirevich: Application of birefringent coatings to glass-fibre-reinforced plastics, Exp. Mech. 9(3), 97–102 (1969) K. Ramesh, A.K. Yadav, A.P. Vijay: Classification of crack-tip isochromatics in orthotropic composites, Eng. Fract. Mech. 53(1), 1–16 (1996) H. Pih, C.E. Knight: Photoelastic analysis of anisotropic fiber reinforced composites, J. Comp. Mater. 3(1), 94–107 (1969) B.D. Agarwal, S.K. Chaturvedi: Improved birefringent composites and assessment of photoelastic theories, Fiber Sci. Technol. 11, 399–412 (1970) R. Prabhakaran: Fabrication of birefringent anisotropic model materials, Exp. Mech. 20, 320– 321 (1980) J.W. Dally, R. Prabhakaran: Photo-orthotropic elasticity, Exp. Mech. 11, 346–356 (1971) K. Ramesh, N. Tiwari: A brief review of photoorthotropic elasticity theories, Sadhana 18, 985– 997 (1993) R. Prabhakaran: A strainoptic law for orthotropic model materials, AIAA J. 13, 723–728 (1975) B.D. Agarwal, S.K. Chaturvedi: Exact and approximate strain optic laws for orthotropic photoelastic materials, Fiber Sci. Technol. 33, 146–150 (1982) R. Prabhakaran: Photo-orthotropic elasticity: A new technique for stress analysis of composites, Opt. Eng. 21(4), 679–688 (1982) J.W. Dally, A. Mulc: Polycarbonate as a model material for three-dimensional photoplasticity, Trans. ASME J. Appl. Mech. E95, 600–605 (1973) R.B. Pipes, J.W. Dally: On the fiberreinforced birefringent composite materials, Exp. Mech. 13, 348–349 (1973) C.P. Burger: Nonlinear photomechanics, Exp. Mech. 20, 381–389 (1980) M. Ravichandran, K. Ramesh: Determination of stress field parameters for an interface crack in a bi-material by digital photoelasticity, J. Strain Anal. Eng. Des. 40(4), 327–343 (2005) K. Ramesh, S.K. Mangal: Data acquisition techniques in digital photoelasticity: A review, Opt. Lasers Eng. 30(1), 53–75 (1998), Errata 31(85) (1999) A. Ajovalasit, S. Barone, G. Petrucci: A review of automated methods for the collection and analysis of photoelastic data, J. Strain Anal. Eng. Des. 33(2), 75–91 (1998) S.L. Toh, S.H. Tang, J.D. Hovanesian: Computerised photoelastic fringe multiplication, Exp. Tech. 14(4), 21–23 (1990)

Photoelasticity

25.37

25.38

25.39

25.40

25.41 25.42

25.43

25.44

25.45

25.46

25.47

25.48

25.49

25.50

25.52

25.53

25.54

25.55

25.56

25.57

25.58

25.59

25.60

25.61

25.62

25.63

25.64

25.65

25.66

25.67

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M.J. Ekman, A.D. Nurse: Absolute determination of the isochromatic parameter by load-stepping photoelasticity, Exp. Mech. 38(3), 189–195 (1998) K. Ramesh, D.K. Tamrakar: Improved determination of retardation in digital photoelasticity by load stepping, Opt. Lasers Eng. 33, 387–400 (2000) A.D. Nurse: Full-field automated photoelasticity using a three-wavelength approach to phaseshifting, Appl. Opt. 36, 5781–5786 (1997) C. Buckberry, D. Towers: New approaches to the full-field analysis of photoelastic stress patterns, Opt. Lasers Eng. 24, 415–428 (1996) M. Ramji, V.Y. Gadre, K. Ramesh: Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity, J. Strain Anal. Eng. Des. 41(5), 333–348 (2006) G.M. Brown, J.L. Sullivan: The computer-aided holophotoelastic method, Exp. Mech. 30(2), 135– 144 (1990) G. Petrucci: Full-field automatic evaluation of an isoclinic parameter in white light, Exp. Mech. 37(4), 420–426 (1997) S. Barone, G. Burriesci, G. Petrucci: Computer aided photoelasticity by an optimum phase stepping method, Exp. Mech. 42(2), 132–139 (2002) K.R. Madhu, K. Ramesh: New boundary information encoding for effective phase unwrapping of specimens with cut outs, Strain 43, 1–4 (2007) D.C. Ghiglia, M.D. Pritt: Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley-InterScience, New York 1998) M. Ramji, E. Nithila, K. Devvrath, K. Ramesh: Assessment of autonomous phase unwrapping methodologies in digital photoelasticity, Sadhana 33(1), 27–44 (2008) M. Ramji, K. Ramesh: Isoclinic parameter unwrapping and smoothening in digital photoelasticity, Proc. Int. Conf. Opt. and Optoelectronics, Dehradun 2005 (Instruments Research & Development Establishment, Dehradun 2005) I.A. Jones, P. Wang: Complete fringe order determination in digital photoelasticity using fringe combination matching, Strain 39, 121–130 (2003) R.J. Sanford: On the range and accuracy of spectrally scanned white light photoelasticity, Proc. SEM Spring Conference on Experimental Mechanics, Los Angeles 1986 (Soc. Experimental Mechanics, Bethel 1986) pp. 901–908 K. Ramesh, S.S. Deshmukh: Automation of white light photoelasticity by phase shifting technique using colour image processing hardware, Opt. Lasers Eng. 28(1), 47–60 (1997) K. Ramesh, S.S. Deshmukh: Three fringe photoelasticity – use of colour image processing hardware to automate ordering of isochromatics, Strain 32(3), 79–86 (1996)

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25.51

T.Y. Chen, C.E. Taylor: Computerised fringe analysis in photomechanics, Exp. Mech. 29(3), 323–329 (1989) K. Ramesh, B.R. Pramod: Digital image processing of fringe patterns in photomechanics, Opt. Eng. 31(7), 1487–1498 (1992) K. Ramesh, R.K. Singh: Comparative performance evaluation of various fringe thinning algorithms in photomechanics, J. Electron. Imag. 4(1), 71–83 (1995) F.W. Hecker, B. Morche: Computer-aided Measurement of Relative Retardations in Plane Photoelasticity (Martinus Nijhoff, Dordrecht 1986) pp. 535–542 K. Creath: Phase-measurement interferometry techniques, Prog. Opt. 26, 349–393 (1998) V. Sai Prasad, K. Ramesh: Background intensity effects on various phase shifting techniques in photoelasticity, Proc. of the Conf. on Opt. and Photonics in Eng. (ABC Enterprises, New Delhi 2003) pp. 183–186 E.A. Patterson, Z.F. Wang: Towards full field automated photoelastic analysis of complex components, Strain 27(2), 49–56 (1991) A. Ajovalasit, S. Barone, G. Petrucci: A method for reducing the influence of the quarter-wave plate error in phase-shifting photoelasticity, J. Strain Anal. Eng. Des. 33(3), 207–216 (1998) ´lez-Cano: Phase measurJ.A. Quiroga, A. Gonza ing algorithm for extraction of isochromatics of photoelastic fringe patterns, Appl. Opt. 36(32), 8397–8402 (1997) D.K. Tamrakar, K. Ramesh: Simulation of errors in digital photoelasticity by Jones calculus, Strain 37(3), 105–112 (2001) W. Ji, E.A. Patterson: Simulation of error in automated photoelasticity, Exp. Mech. 38(2), 132–139 (1998) P.L. Prashant, K. Ramesh: Genesis of various optical arrangements of circular polariscope in digital photoelasticity, J. Aerosp. Sci. Technol. 58(2), 117– 132 (2006) K. Ramesh: Digital photoelasticity – current trends and future possibilities, Proc. of Int. Con. Comp. & Exp. Eng. and Sciences Chennai 2005 (Tech Science, Forsyth 2005) pp. 2159–2164 A. Ajovalasit, S. Barone, G. Petrucci, B. Zuccarello: The influence of the quarter wave plates in automated photoelasticity, Opt. Lasers Eng. 38, 31–56 (2002) V.S. Prasad, K.R. Madhu, K. Ramesh: Towards effective phase unwrapping in digital photoelasticity, Opt. Lasers Eng. 42(4), 421–436 (2004) P.L. Prashant, K.R. Madhu, K. Ramesh: New initiatives in phase unwrapping, Proc. of SPIE Third Int. Conf. on Exp. Mech., Vol. 5852 (Singapore 2004) pp. 192–197

References

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Noncontact Methods

25.69

25.70

25.71

25.72

25.73 25.74

25.75

25.76

25.77

25.78

A. Ajovalasit, S. Barone, G. Petrucci: Towards RGB photoelasticity: full-field automated photoelasticity in white light, Exp. Mech. 35(3), 193–200 (1995) K.R. Madhu, K. Ramesh: Noise removal in three fringe photoelasticity by adaptive colour difference estimation, Opt. Lasers Eng. 45, 175–182 (2007) A. Ajovalasit, G. Petrucci: Developments in RGB photoelasticity, Proc. of international Conf. on Advances in Experimental Mechanics, York 2004 (Trans Tech, Bedford 2004) pp. 205–210 E. Andrei, J. Anton, H. Aben: On data-processing in photoelastic residual stress measurement in glass, Proc. of Int. Conf. on Exp. Mech., New Delhi 2005 (Asian Commun. Exp. Mech., Japan 2005) E.A. Patterson: Digital photoelasticity: Principles, practice and potential, Strain 38, 27–39 (2002) A. Asundi, L. Tong, C.G. Boay: Phase shifting Method with a Normal Polariscope, Appl. Opt. 38, 5931–5935 (1999) E.A. Patterson, Z.F. Wang: Simultaneous Observation of Phase-stepped Images for Automated Photoelasticity, J. Strain Anal. Eng. Des. 33(1), 1–15 (1998) S. Barone, G. Burriesci, G. Petrucci: Automated photoelasticity by Phase stepping Technique, Proc. of XIV Imeko World Congress, Tampere 1997 (Finnish Society of Automation, Helsinki 1997) pp. 57–62 A. Asundi, M.R. Sajan, L. Tong: Dynamic photoelasticity by TDI imaging, Opt. Lasers Eng. 38, 3–16 (2002) J. Lesniak, S.J. Zhang, E.A. Patterson: Design and evaluation of the poleidoscope: A novel digital polariscope, Exp. Mech. 44, 128–135 (2004)

25.79

25.80

25.81

25.82

25.83

25.84

25.85

25.86

25.87 25.88

25.89

S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi: Photoelastic analysis with a single tricolour image, Opt. Lasers Eng. 29, 423–435 (1998) K. Ramesh, M.A. Kumar, S.G. Dhande: Fusion of digital photoelasticty, rapid prototyping and rapid tooling technologies, Exp. Tech. 23(2), 36–38 (1999) P.R.D. Karthick Babu, K. Ramesh: Role of finite elements in evolving the slicing plan for photoelastic analysis, Exp. Tech. 30(3), 52–56 (2006) J.D. Curtis, S.D. Hanna, E.A. Patterson, M. Taroni: On the use of stereolithography for the manufacture of photoelastic models, Exp. Mech. 43(2), 148–162 (2003) D.E. Karalekas, A. Agelopoulos: On the use of stereolithography built photoelastic models for stress analysis investigations, Mater. Des. 27(2), 100–106 (2006) K. Ramesh, A.K. Yadav, V.A. Pankhawalla: Plotting of fringe contours from finite element results, Commun. Numer. Meth. Eng. 11, 839–847 (1995) P.R.D. Karthick Babu, K. Ramesh: Development of fringe plotting scheme from 3D FE results, Commun. Numer. Meth. Eng. 22(7), 809–821 (2006) D.C. Drucker: Photoelastic separation of principal stresses by oblique incidence, ASME J. Appl. Mech. 10(3), A156–A160 (1943) M.M. Frocht: Discussion of [25.86], ASME J. Appl. Mech. 11(2), A125–A126 (1944) K. Chandrashekhara, K.A. Jacob: An experimentalnumerical hybrid technique for two-dimensional stress analysis, Strain 13, 25–31 (1977) F. Trebuna: Some problems of accelerating the measurements and evaluating the stress fields by the photostress method, Exp. Tech. 14, 36–40 (1990)

Part C 25

743

Richard J. Greene, Eann A. Patterson, Robert E. Rowlands

In this chapter an outline of the theoretical foundations for the experimental technique of thermoelastic stress analysis is presented, followed by a description of the equipment, test materials, and methods required to perform an analysis. Thermoelastic stress analysis is a technique by which maps of a linear combination of the in-plane surface stresses of a component are obtained by measuring the surface temperature changes induced by time-varying stress/strain distributions using a sensitive infrared detector. Experimental considerations relating to issues such as shielding from background radiation, edge effects, motion compensation, detector setup, calibration, and data interpretation are discussed. The potential of the technique is illustrated using a number of examples that involve isotropic as well as orthotropic materials, fracture mechanics, separation of component stresses, and vibration analysis. Applications of the method to situations involving residual stresses, elevated temperatures, and variable amplitude loading are also considered.

26.1 History and Theoretical Foundations ...... 744

26.4 Calibration ........................................... 749 26.5 Experimental Considerations ................. 26.5.1 Background Infrared Radiation Shielding .................................. 26.5.2 Edge Effects and Motion Compensation .......... 26.5.3 Specimen Preparation................. 26.5.4 Reference Signal Requirements .... 26.5.5 Infrared Image Setup .................. 26.5.6 Adiabaticity ............................... 26.5.7 Thermoelastic Data Format .......... 26.5.8 Data Processing..........................

749

750 751 751 751 752 752 752

26.6 Applications ......................................... 26.6.1 Isotropic Structural Materials ....... 26.6.2 Orthotropic Materials .................. 26.6.3 Fracture Mechanics..................... 26.6.4 Experimental Stress Separation .... 26.6.5 Residual Stress Measurement....... 26.6.6 Vibration Analysis ...................... 26.6.7 Elevated Temperature Analysis..... 26.6.8 Variable-Amplitude Loading........

753 753 753 755 756 757 757 758 759

749

26.7 Summary ............................................. 759 26.A Analytical Foundation of Thermoelastic Stress Analysis ............. 760

26.2 Equipment ........................................... 745

26.B List of Symbols ..................................... 762

26.3 Test Materials and Methods ................... 747

References .................................................. 763

Thermoelastic stress analysis (TSA) is an emerging technique that to date has been used successfully for the evaluation and validation of design concepts, fracture mechanics, damage detection, fatigue monitoring, and residual stress analysis. Maps of surface stress are obtained in real time by measuring the temperature changes induced by time-varying stress/strain distributions. Under adiabatic and reversible conditions, a cyclically loaded structure experiences in-phase temperature variations that, for isotropic materials, are

proportional to the change in the sum of the principal stresses or strains. Thermoelastic stress analysis (or thermoelasticity) uses an infrared radiometer to measure the local temperature fluctuations and relates these changes to the associated dynamic stresses by thermodynamic principles. This approach differs from dissipative methods, such as vibrothermography, which associate local temperature variations with dissipated, as opposed to stored, energy [26.1]. TSA is a full-field noncontact technique that determines stress

Part C 26

Thermoelasti 26. Thermoelastic Stress Analysis

744

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Noncontact Methods

Part C 26.1

information for actual structures in their operating environments with a sensitivity similar to that of strain gages. It is well known that the temperature of a gas decreases when the gas is expanded and increases when compressed. A similar effect occurs in solids. The temperature changes associated with elastic deformations in a solid are very small and until recently were considered to be of negligible importance. However, advances in infrared photon detector technology made it possible to measure these temperature variations efficiently and reliably. In 1982 a British company began manufacturing and marketing the first commercial system,

the SPATE 8000 [SPATE is an acronym for stress pattern analysis (by measurement of) thermal emission]. The SPATE 8000 and a subsequent improved version, the SPATE 9000, were used extensively in the USA, the UK, and Australia during the 1980s and 1990s. Recent developments in staring array technology have led to new sensors capable of acquiring data over a whole field of view instantaneously. Commercially available systems are manufactured in the USA and France and have produced rapid and continuing advances in the applications of the technique. In addition, a small number of thermoelastic analysis instruments have been developed in research laboratories [26.2, 3].

26.1 History and Theoretical Foundations The first recognition of the thermoelastic effect is traditionally attributed to Weber in 1830 [26.4]. He observed that, when a vibrating wire receives a sudden increase in tension, it experiences a delay in the change of its fundamental frequency. Weber hypothesized that this delay is due to the increased magnitude of stress causing a transient temperature change in the wire. The thermoelastic effect was given a theoretical foundation in 1853 by William Thomson, later known as Lord Kelvin [26.5]. Thomson derived a linear relationship between the temperature change of a solid and the change in the sum of the principal stresses for isotropic materials. Compton and Webster confirmed this theory experimentally in 1915 [26.6]. Theoretical considerations of thermoelasticity remained relatively dormant until the 1930s, when Zener published a series of articles on the internal friction of solids [26.7–10]. Rocca and Bever analytically investigated nonlinear effects and measured the thermoelastic response in iron and nickel near the Curie point [26.11]. During the 1950s Biot used irreversible thermodynamics to extend the theory to include anisotropic, viscoelastic, and plastic material responses [26.12–15]. In 1967 Belgen conducted the first noncontacting thermoelastic stress analysis measurements [26.16, 17]. He used an infrared radiometer to relate the stresses to the small temperature changes measured in a vibrating cantilevered beam. Mountain and Webber developed the first prototype of the commercial SPATE system for a British Ministry of Defense project in 1978 [26.1]. The theoretical basis for thermoelastic stress analysis rests on the laws of thermodynamics. Potter and Greaves derived a thermoelastic relationship for

reversible, adiabatic thermodynamic events which includes anisotropy, mean stresses, temperature-variant elasticity, and expansion coefficients [26.18]. Such behavior can be expressed as  ∂Cijkl ΔT = (εkl − αkl ΔT ) ρCε T ∂T   ∂αkl dεij , (26.1) −Cijkl αkl + ΔT ∂T where ρ is the material density, Cε the specific heat for constant strain, T the temperature, Cijkl the elasticity tensor, akl the thermal expansion coefficient tensor, and εkl the strain tensor. Special forms of this equation have been derived to include aspects such as nonzero mean stress, cyclic plasticity, residual stresses, and the temperature dependency of material properties [26.19–25]. Some of these topics are dealt with later in the chapter. Equation (26.1) is cumbersome to work with and many of its terms can often be omitted. For an isotropic material under plane stress, this relationship can be simplified to (Sect. 26.A) S∗ = K (Δσp + Δσq ) ,

(26.2)

where S∗ is the measured TSA signal, K is the thermoelastic coefficient, and σp and σq are the principal stresses. Equation (26.2) suggests that neither a state of pure shear stress nor static stresses produces any thermoelastic response for an isotropic material. Interpretation of thermoelastic stress analysis signals can therefore require some insight based on the loading and geometry of the component being analyzed. For zero mean stresses, (26.2) provides isopachic informa-

Thermoelastic Stress Analysis

(or strain), determine reliable edge stresses, smooth measured temperature data, formulate approaches for three-dimensional components, and extend the general technique to cyclic plasticity [26.24–49], orthotropic materials (Sect. 26.6) [26.32, 34, 48, 50–74], mean and residual stresses (Sect. 26.6) [26.19–22, 25, 75–79], fatigue and fracture (Sect. 26.6) [26.44, 66–72, 80–88], address nonadiabaticity [26.89–94], and to illustrate a variety of mechanical- and structural-type problems, some of which are addressed in Sect. 26.6 on applications. Several excellent review-type sources are also available [26.94–97].

26.2 Equipment A main component of a thermoelastic stress analysis system is an infrared detector capable of measuring the small temperature changes associated with the thermoelastic effect (change in stresses). The infrared detector acts as a transducer, which converts the incident radiant energy into electrical signals. The commercial SPATE system uses a photon emission detector (photodetector) instead of a thermal detector largely because of its excellent sensitivity [26.94] while the more modern systems use a focal-plane array. The SPATE system is a raster scanning camera comprised of a single infrared detector mounted in a housing with two independently positionable mirrors, whereas the focal-plane array systems are similar to charge-coupled device (CCD) video cameras in that they use a planar detector array with a compound lens to cast an image of the observed component onto the detector. Infrared photon detectors have a semiconductor component that can be characterized by the bandgap energy required to excite the semiconductor to produce an output signal. The bandgap energy is the minimum energy required to cause an excited carrier to leave the valency energy band and enter the mobile conduction band. This occurs if the incident photons from the target specimen have sufficient energy [26.1]. Photons of a maximum wavelength and less will have the required energy to excite the semiconductor (i. e., the shorter wavelengths have higher energies). The voltage output from a photodetector is proportional to the rate of incident photons. The photon emittance from a component being analyzed, known as the spectral radiant photon emittance, is an important parameter to consider when selecting a detector [26.23]. The detector will exhibit optimum sensitivity if the

maximum number of photons striking the detector has sufficient energy to excite the semiconductor. Planck’s law relates the emittance of a black body to the wavelength and the temperature of the radiating body by   2c 1 (26.3) , φλ = 4 exp(hc/λkT ) − 1 λ where φλ is the spectral radiant photon emittance, c the speed of light, λ the wavelength of light, h Planck’s constant, and k Boltzmann’s constant. At room temperature the peak photon emittance occurs at a wavelength of 12 μm. At higher temperatures the peak emittance shifts to shorter wavelengths [26.94]. An ideal photon detector would be equally responsive to all photon wavelengths having sufficient energy to excite the detector. However, as a result of absorption and other dissipative effects from the atmosphere, a photodetector is restricted to two discrete windows of operation, 3–5 μm and 8–13 μm. A detector designed for the lower range has advantages for elevated-temperature testing. However, a detector chosen for this range would not be optimally responsive to the peak spectral photon emittance wavelength at room temperature. The SPATE system uses a mercury-doped cadmium– telluride (Hg1−x Cdx Te) photovoltaic photon detector. By varying x of the semiconductor chemical composition, the required bandgap energy of the detector can be controlled. The maximum wavelength to which the detector will respond depends on the bandgap energy and therefore also on semiconductor composition. The SPATE system has approximately x = 0.2, for a bandgap energy of 0.1 eV, and it will respond to a peak photon wavelength of 12.4 μm and over a range

745

Part C 26.2

tion and therefore locates the center of Mohr’s circle. If there is nonzero proportional cyclic loading about some mean stress, TSA provides only the change in isopachic stress. The term isopachic refers to contours of constant value for the first stress invariant. Equation (26.2) is valid for bodies undergoing linear elastic, adiabatic, reversible transformation. It assumes that the stress-induced temperature changes are small and that elastic and thermal properties are constant over such temperature changes. Thermoelastic stress analysis methods have been developed to obtain individual components of stress

26.2 Equipment

746

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Part C 26.2

of 8–13 μm [26.96]. This detector therefore works well at room temperatures. The Deltatherm [26.98] and Cedip [26.99] systems employ indium antinomide (InSb) focal-plane arrays which respond in the 3–5 μm range with a peak response to photons of wavelength 5.3 μm [26.100]. These characteristics make them better suited to elevated temperature measurements and less sensitive at room temperature. Sometimes known as staring arrays, large two-dimensional focal-plane arrays are the most significant recent advance in infrared technology and enable the development of nonscanning systems [26.101]. Arrays of the order of 1024 × 1024 are available, but 256 × 320 are more commonly used in TSA systems. Optically, commercial doublemeniscus silicon–germanium coated multiple-layered lenses and aluminum-oxide coated front surface mirrors are suitable. Photon detectors require cooling to about 77 K to minimize background radiation effects and to improve the specific detectivity (the signal-to-noise ratio). Unwanted radiation can be further reduced by decreasing the detector’s angle of view. A cryogenic aperture placed directly in front of the detector will not reduce the target signal and can improve the specific detectivity by as much as an order of magnitude compared with systems without such an aperture [26.102]. Modern systems are available with either a Dewar that can be filled with liquid nitrogen or a mechanical closedcycle cooling system based on a Stirling cycle. These systems typically require 60 min and 20 min of operation, respectively, to achieve thermal stability. A Peltier thermoelectric device can be used to cool the focal-

plane array directly [26.103]. Assuming that surface emissivity does not vary with wavelength, the Stefan– Boltzmann law for photon detection expresses the total radiant photon emittance φ as [26.94] φ = eBT 3 ,

(26.4)

where e is the surface emissivity and B is the Stefan– Boltzmann constant. The detector output voltage S is linearly related to the total radiant photon emittance as S = R∗ φ + R∗ φb , where R∗ is the photodetector responsivity and φb is the incident photon rate for background emittance. Differentiating this expression with respect to time and substituting the result into (26.2) (assuming that φb does not vary with time), the relationship between the detector output voltage and the first stress invariant I for isotropic materials becomes dS −3eR∗ BαT 3 dI (26.5) = . dt ρCε dt A lock-in analyzer is employed as the signal-processing unit that extracts the thermoelastic information from the inherently noisy detector output signal. Typical output from a photon detector for high cyclic stress in a steel specimen may be a 0.020 V peak-to-peak signal buried in a large-bandwidth, 0.800 V peak-to-peak noise [26.94]. A lock-in analyzer may be described in a simplified manner (Fig. 26.1) as a series-connected signal mixer and a low-pass filter. The analog instrument mixes a reference signal at the load frequency with the detector output to extract the actual thermoelastic response [26.104]. The reference signal can originate

Image data I

Signal in

Signal channel

Correlator

DC amp. & low-pass filter

DC out

Reference channel

Reference in

Fig. 26.1 Schematic of simplified lock-in amplifier (after [26.91])

Thermoelastic Stress Analysis

erence signal, such that the x-component of the vector is dS/ dt in (26.5) and the y-component is zero for an adiabatic in-phase signal. The output-noise bandwidth of a low-pass filter is inversely proportional to the filter time constant. It is advantageous to select the filter time constant as large as possible to minimize the noise bandwidth. In the case of a single-detector system, typically a component is scanned rapidly with a small dwell time per measurement. A two-pole, low-pass filter takes 6.6 time constants to settle within 99% of a new value, so too long a time constant for a small dwell time will cause information to be dragged from previous locations. In a focal-plane array, each sensor reports the observed temperature once per image and this information, at a rate of more than 400 images per second, is used to produce the thermoelastic images so that these issues do not occur and very high-quality images can be obtained quickly. Thermoelastic stress analysis can also be conducted without a lock-in analyzer. Harwood and Cummings [26.105, 106] pioneered variable-amplitude (stationary random signal) TSA by replacing the lock-in analyzer with a fast Fourier transform (FFT) analyzer and using entirely digital signal processing. They devised techniques for measuring power spectral density and frequency-response functions, and performing modal-type analyses. It is usual for algorithms providing such functions to be included in commercial systems, however this type of analysis is only performed occasionally [26.37, 93]. The computer used in a TSA system performs three main functions. First, in single-sensor systems it provides the means to select the scan area and control the mirrors to focus the component’s photon emission onto the detector. Second, the computer samples the lock-in analyzer output signal [analog-to-digital (A/D) conversion] and stores thermoelastic data from the component. Finally, the computer is used as a video display unit to provide immediate information about the testing progress as well as to perform postprocessing and data-reduction operations.

26.3 Test Materials and Methods Thermoelastic stress analysis studies can be conducted on virtually any structure or material that is subjected to a sufficient dynamic load. The stress sensitivity for an isotropic material depends on the thermoelastic constant

K and the ambient temperature (26.2). The minimum temperature sensitivity reported for the photodetectors in both single sensor and focal plane arrays is 0.001 ◦ C. With this resolution, the available stress sensitivity for

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from a function generator, load transducer, strain gage, or any other source that is in phase with the loading frequency. Section 26.5 on experimental considerations includes some additional comments regarding reference signals. Mixing two signals of distinct frequencies, say f 1 and f 2 , produces an output signal with frequency content f 1 + f 2 and f 1 − f 2 . The output signal from the photon detector will include the actual thermoelastic response at the reference frequency ωr and other broadband frequencies due to the inherent noise. Mixing the thermoelastic response with the reference frequency will cause a portion of the resultant output to be a direct-current (DC) signal ( f r − f r = 0) and a portion which is a signal at twice the original frequency ( f r + f r = 2 f r ). Noise components are also mixed with the reference frequency, but their contribution to the DC signal is negligible. The low-pass filter of the lockin analyzer is used to recover the DC signal from the mixed output. The filter removes the portion of the signal above a cutoff frequency. Some detector noise that is close to the reference frequency will pass through the filter. The lock-in analyzer then normalizes the DC signal from the reference signal amplitude to a value proportional to the actual measured thermoelastic output. Lock-in analyzers used commonly in thermoelastic systems separate the detector output into two components: one in-phase and the other 90◦ out of phase with respect to the reference signal. The adiabatic structural response is expected to be in phase with the reference loading frequency. If heat conduction occurs in regions of high stress gradients, or if the loading frequency is insufficient for an adiabatic response, a phase shift from the reference frequency will occur and will be apparent as an out-of-phase detector signal. A quick confirmation of adiabatic response is to check for the absence of any out-of-phase detector signal. With the advent of focalplane arrays it is becoming common practice to consider the thermoelastic signal as a vector in which the magnitude relates to the measured temperature difference and the direction to the phase difference relative to the ref-

26.3 Test Materials and Methods

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Table 26.1 Common engineering materials stress sensitivity [26.107]. Note K m is defined in (26.A31) Material

K m (Pa−1 )

Δσ , T = 293 K (MPa)

Δε Corresponding strain sum

Steel Aluminum Titanium Epoxy Magnesium alloy Glass

3.5 × 10−12 8.8 × 10−12 3.5 × 10−12 6.2 × 10−11 1.4 × 10−11 3.85 × 10−12

1 0.4 1 0.055 0.22 0.91

3.5 × 10−6 4.0 × 10−6 6.5 × 10−6 11.0 × 10−6 – –

some common engineering materials has been determined, as listed in Table 26.1 [26.107]. These values compare favorably with the sensitivity available from strain gages. Moreover, the technique has the advantage of being full-field and noncontact. Thermoelastic stress analysis investigations have been performed on many different materials, components and structures. Among the materials tested are structural steels, aluminum and magnesium alloys, boron aluminum, zirconium, titanium, ceramics, copper, brass, polymethyl methacrylate (PMMA), epoxy, polyester, rubber, wood, brick, concrete, reinforced plastics, and human and animal bone. A basic thermoelastic experiment requires little preparation compared with most stress measurement techniques. Vibration isolation is unnecessary, and the method is well suited for use in industrial settings, although care needs to be taken when operating in extremely noisy environments (≥ 120 dB) [26.94]. The only component preparation required for TSA is possibly a thin flat black surface coating. A high-emissivity coating is desirable for two reasons [26.89, 90]. First, the black coating more closely approximates a black body, which enhances the photodetector sensitivity by increasing the photon flux from the structure. Second, the coating provides the

Fig. 26.2 Isopachic stress contours of an aluminum plate loaded

vertically in tension with the letters ‘UW’ cut out

uniform surface emissivity necessary for quantitative stress analysis. Most nonmetallic specimens do not require coatings, with the notable exceptions of brick and concrete. A surface coating is required for metallic materials because they tend to have both low and variable emissivity. Commercial flat black paints available in aerosol cans are well suited as surface coatings and often have emissivities of 0.9 and greater. Krylon Ultra Flat Black has been evaluated as having a surface emissivity of approximately 0.94 [26.107]. Commercial high-temperature paints are also suitable for elevatedtemperature testing. Current signal-processing capabilities allow loading frequencies between 0.5 Hz and 20 kHz. The minimum suitable frequency for a structure depends on the material properties, which affect heat transfer and the adiabatic loading requirements. The question of allowable loading frequencies for adiabatic response is addressed in Sect. 26.5 on experimental considerations. A SPATE system has a spatial resolution of 0.5 mm from a detector distance of 0.5 m and a lens attachment can be employed to improve this resolution to 0.15 mm for an 8 mm-diameter scan area. Focal-plane array systems can achieve a spatial resolution of at least 0.05 mm per pixel and, through the use of infrared camera lenses, a wide range of field of views are available at a range of distances. A sample scan is shown in Fig. 26.2. The aluminum plate is 76.2 mm wide by 25.4 mm thick with the letters ‘UW’ machined through its thickness. The plate was loaded uniaxially at 20 Hz with an 3560 N mean load (14.5 MPa) and a 534 N sinusoidal load (2.2 MPa). A thin coat of a standard commercial flat black paint was applied to enhance the surface emissivity. While the letters are highlighted for clarity, the sensitivity of the thermoelastic data is evident around the regions of stress concentration. Also, the inherent noise associated with thermoelastic measurements can be seen. Postprocessing of recorded thermoelastic data may take many forms. Typically, the raw thermoelastic data are displayed in full-field color-coded stress maps

Thermoelastic Stress Analysis

stress analysis, but only the most basic technique. More advanced techniques include the random excitation method of Harwood and Cummings mentioned earlier [26.105, 106], and residual stress testing, fracture mechanics, and high-temperature applications. Several of these will be discussed later in this chapter and are included in Sect. 26.6 on applications.

26.4 Calibration Quantitative analysis necessitates calibration of the temperature change of the thermoelastic signal S∗ in terms of stress or strain. The calibration procedure may take different forms [26.94–97, 106, 109–112]. One concept is based on the theoretical relationship between the specimen material properties and measurement system characteristics. The other approach compares the measured thermoelastic signal with either a theoretical solution or some independently determined state of stress. Although methods described herein emphasize a classical thermoelastic response, consequences of material nonlinearities and nonadiabatic analyses are included. The theoretical calibration technique relates the predicted digitally sampled photodetector output signal to the change in the appropriate linear combination of stresses. This approach evaluates the thermoelastic coefficients K , K 1 , and K 2 based on thermoelastic theory and the physics of infrared detection. As such, it involves knowledge of the mechanical and thermal properties of the material being analyzed, detector and lock-in analyzer sensitivities, and various signal attenuation correction factors. This procedure has limited usefulness because it requires an accurate knowledge of many individual factors whose values can be difficult to determine. However, this could be useful to determine if a traditional thermoelastic relationship (such as those of (26.1) and (26.2)) is appropriate or when other factors

(material nonlinearities, mean stress effects, etc.) need to be considered. Values of thermoelastic coefficients are usually determined more reliably and fairly easily by experiments on calibration specimens using the same material, painted coating, loading frequency, and ambient conditions as the test structure. Calibration specimens typically employ a geometry and loading for which the state of stress or strain is known theoretically, or independently determined by experiment, perhaps with strain gages. Examples include a beam subjected to four-point bending, a diametrically compressed disk, or uniaxial tensile coupons. Orthotropic materials usually necessitate testing two calibration specimens, with their principal material directions interchanged [26.60]. However, the low transverse strength of unidirectional composites hampers obtaining much data when loading normal to the fibers since this can be the direction of the larger of the two thermoelastic coefficients. Ju and Rowlands [26.69] circumvented this by evaluating the K from finite element modeling (FEM)-predicted stresses in a biaxial field from measured temperature data. If using a calibration specimen whose stress field is nonuniform, strain gages can be located in regions of low stress gradients, and their output (perhaps converted to stress) compared to the thermoelastic response signal S∗ . For reliability, several values should be determined and averaged to minimize the effects of scatter.

26.5 Experimental Considerations 26.5.1 Background Infrared Radiation Shielding In a typical industrial or laboratory environment, infrared radiation is emitted by the component under test

but also by all bodies surrounding a test [26.113]. However, these bodies will all be at different temperatures and have different infrared emissivities. As a result, if the target under observation is in any way reflective to incident infrared radiation, artifacts can appear in the in-

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showing isopachic contours. Various filter and statistical operations can be performed on the data to produce more coherent results [26.33, 42, 108]. The ability to separate the thermoelastically recorded stress data into individual stress components is discussed later in the chapter. Of course this section does not cover all the experimental methods associated with thermoelastic

26.5 Experimental Considerations

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frared data frames where incident radiation is recorded in the data. While it is critical that such artifacts are removed from conventional thermographic images, such reflections are only present in the thermoelastic data if they oscillate in magnitude at the loading frequency and phase of the test. In practice, there are two sources of radiation which must be eliminated, or at least minimized. The first, and usually most troublesome, is radiation from the loading frame, exciter or associated equipment whose components are experiencing a cyclic load at the same frequency as the component under test. Infrared radiation from these items will vary cyclically since they will themselves be exhibiting a thermoelastic response, and any reflection of this radiation from the component surface will appear in the recorded thermoelastic image. The second source of radiation is the component itself, which can be reflected back onto the component surface by lenses and other optics, sometimes creating narcissus artifacts where the lens system can be seen in the data image. Radiation from both of these sources can be simply and effectively reduced in most experimental cases by shielding the apparatus using conventional photographers black cloths, sheets of black paper, or any suitable material which has a low reflectivity in the infrared range. Most matt-finish paper products and cotton cloths are suitable. When combined with careful specimen surface preparation, this approach allows most reflection artifacts to be successfully removed from recorded thermoelastic data.

26.5.2 Edge Effects and Motion Compensation The photodetector of a SPATE system does not focus on a mathematical point of zero dimensions, but rather on a finite-sized spot in space. This situation can pose difficulty, particularly on the edges of a specimen. An edge effect occurs because the scanning detector sees a spot that is partly on the highly stressed specimen and partly on the stress-free background. Staring array systems record a full-field image virtually instantaneously (with exposure durations of approximately three milliseconds), but they still have a finite pixel resolution. So while this concern is less with staring than with scanning cameras, there remains an effect. The quality of an edge signal can be further reduced by the cyclic motion of the structure. The latter provides different data to the detector from the different spatial positions. The following paragraphs describe software

and mechanical motion-compensating methods. Maximum stresses often occur at an edge, and the lack of reliable thermoelastic edge data inhibits the ability to obtain such stresses accurately. If not corrected for, this aspect can cause a TSA analysis to be more qualitative than quantitative. In software motion compensation, the usual approach is to identify the two extremes of motion of a specimen under cyclic loading, and then trigger the infrared camera system to capture infrared data at these two extremes. The translation between these two data sets in the plane of the image is determined either by identifying a common point in both data manually or by using image processing techniques. The translation is used to correlate the data spatially in the plane of the image, and then the thermoelastic analysis is performed by subtracting the two spatially corrected data sets. Since motion compensation using software employs only two data sets from each cycle, it requires many more cycles to yield thermoelastic data of comparable quality to uncompensated data. This can be problematic when monitoring transient events such as rapid crack growth or damage evolution, since the surface stress distribution may change significantly over the time taken to capture the motion-compensated data. The method also requires an infrared imaging system in which the image data capture can be precisely synchronized with the loading cycle and this necessitates a snapshot rather than a rolling array approach, i. e., the reading of all pixels in the detector instantaneously rather than sequentially. In mechanical motion compensation, the detector’s view of the component under test is kept constant by the use of some form of mechanical or physical linkage between the detector and component, so that the detector becomes blind to the motion of the component. This is most usually achieved either by small motions of the infrared lens system relative to the infrared detector, or by the addition of a moveable mirror into the optical path. In both cases, the lens or mirror is oscillated at the same frequency as the loading event and, by careful adjustment of the amplitude and phase of this oscillation, results in the component motion being removed from the infrared image data. This approach appears simple; however such compensation in rarely perfect and the addition of an infrared mirror or other additional moving optical component attenuates the signal so that the quality of the captured data is degraded. The principal advantage of this method over software motion compensation is the retention of all the infrared data frames for subsequent thermoelastic analysis, thus retaining the

Thermoelastic Stress Analysis

26.5.3 Specimen Preparation Any component subjected to a change in surface stress will exhibit a change in surface temperature and hence emitted infrared radiation, regardless of its surface condition. However, the magnitude of the infrared emission from the surface of the component is greatly increased if the surface has a high emissivity in the infrared wavelength range. This usually results in the added benefit that the surface reflectivity will be low, reducing unwanted reflections of incident radiation. Ideally the surface will be a perfect black-body emitter of infrared radiation, with an emissivity coefficient of unity. Coatings are typically used on metal components and gloss-finish plastics to provide an enhanced and uniform surface emissivity. The application of a thin layer of high-carbon matt or flat black paint will achieve a surface emissivity of approximately 0.9. However, under certain conditions, coatings can introduce potential difficulties, including nonadiabaticity [26.89–92]. A coating can act as an insulating layer and thereby drag down the true thermoelastic response or cause the detector signal to lag the loading frequency. This effect becomes more pronounced with increased coating thickness and loading frequency. However, these difficulties can typically be ignored for a coating thickness of 20–30 μm (one or two layers of paint) and a loading frequency range of 5–200 Hz. Most thermoelastic stress analyses are conducted under these conditions. On a related issue, Estrada Estrada and Patterson [26.93] showed that selecting a frequently to achieve adiabaticity could result in the stress/strain history being path dependent, thereby invalidating a basic assumption of classical thermoelastic stress analysis.

26.5.4 Reference Signal Requirements Equation (26.1) relates the change in temperature to the change in strain. In practice this typically implies correlating the infrared signal with some form of signal that is representative of the applied load. This latter signal is

commonly termed the reference signal and, while it can be out of phase with the observed infrared response, it is critical that its form varies in the same manner as the loading event. Limitations are often imposed on the specific form of the reference signal by the hardware being used. Typically the signal will be between 100 mV and 10 V peak to peak and need not be symmetrical since usually an offset can be provided by the correlator. The principal requirement is that the the reference signal has a high signal-to-noise (S/N) ratio, since the correlation process can be sensitive to the presence of noise in the reference signal. Consequently, it is preferable to make use of the cleanest available source for the reference signal. Although the output from load transducers, surface strain gauges, or proximity probes in vibration tests can all be used, it is usually most successful to select the output from the signal generator used to drive the loading event. Occasionally, observation of the load cell or surface strain gauge response may suggest that the response frequency of the test component differs from the driving frequency. In such a case, a comparison should be made between the available response signals in both the loading control loop and the specimen response outputs, and the signal with the lowest noise and most accurate loading event fidelity chosen as the reference signal.

26.5.5 Infrared Image Setup The primary data in a thermoelastic stress analysis is the stream of infrared images from a staring array or a sequence of point responses from the detector in a raster-scanning system. Consequently the arrangement of the imaging system is the first critical step of the experimental setup [26.113]. A raster scanning system offers greater flexibility in the selection of the point of analysis than does a staring array camera. The spot size, or area of analysis, for each point in the scanned array is a function of the area of the point detector and the specification of the infrared lens system employed. Typically it is linearly related to the stand-off distance between the detector and the specimen. The pitch of the raster scan, or the distance between adjacent analysis points on the specimen, can then be selected independently from the spot size. This allows the operator to record anything from a rapid, widely spaced scan of the component to a detailed, closely spaced scan where each individual analysis spot is immediately adjacent to, or even overlapping, its neighbors.

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speed of data acquisition achieved in conventional, uncompensated analysis. Since both of these approaches introduce some degradation of the observed thermoelastic response, an assessment must be made as to whether the improvement in data quality from the motion compensation is considered worthwhile for a particular experimental situation.

26.5 Experimental Considerations

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In contrast, staring array devices are similar to conventional charge-coupled-devices (CCD) employed in video cameras. A planar imaging array is mounted behind a lens assembly, which typically offers a manual focus function, although not usually a zoom function. The field of view of the lens–detector assembly therefore depends on the optical magnification of the lens system, the stand-off distance between the lens and the component under observation, and the physical dimensions of the detector array. In infrared optics the attenuating effects of each element can be substantial and so infrared lenses often have fewer, simpler optical elements than similar visible-wavelength lens systems. As a result they often exhibit a greater degree of spatial distortion than their visible counterparts. This is usually seen as a modest degree of barrel distortion of the image and some degree of vignetting, i. e., loss of intensity at the corners of the recorded image. These aberrations are not significant in most thermoelastic analyses, although a barrel distortion correction is sometimes worthwhile if the precise locations of small features such as crack tips are required.

26.5.6 Adiabaticity Equations (26.1) and (26.2) assume adiabatic behavior (Sect. 26.A). It is therefore important to address what load frequency is acceptable for adiabatic material response. An adiabatic response requires that a balance be maintained between the thermal and mechanical energy, that is, the process must be reversible. The minimum loading frequency to accomplish this can depend on the thermal conductivity and stress gradients a) Signal strength Thermoelastic response Reference signal

b) y

S*

Y

θ

R θ

0

Time

0

X

x

Fig. 26.3a,b The form of the thermoelastic data recorded during a cyclic analysis: (a) the intensities of the reference signal and thermoelastic signal S∗ versus time t, and the phase relationship θ between them; and (b) the resulting thermoelastic vector, using both the polar R and θ and Cartesian X and Y representations

in the structure. Most metals can be safely considered adiabatic for frequencies above 2 Hz, but other factors such as the coating and system electronics might introduce concern below 10 Hz. Higher loading frequencies usually can improve the detail in the data since heat transfer is minimized. Experience indicates that approximately 20–25 Hz is a suitable loading frequency for aluminum and glass–epoxy and graphite–epoxy composites. Materials having lower thermal conductivities tend to necessitate increased frequency to maintain adiabaticity. This is particularly true with some nonmetallic components; for example, polyvinyl chloride (PVC) can require a loading frequency as high as 100 Hz for an adiabatic response [26.89]. Fortunately, the presence or absence of nonadiabatic response can be observed by checking the phase of the response signal with respect to the load signal. If heat conduction takes place within a component or coating, a detectable phase shift will occur between the reference and loading signals. Experience demonstrates that one can usually ignore any heat transfer between a structure and its surroundings. However, heat transfer can occur between plies in laminated composites and hence using TSA with laminated composites can raise adiabaticity concerns [26.58, 59].

26.5.7 Thermoelastic Data Format Independent of the equipment used to capture thermoelastic data, the data should always contain both magnitude and phase information from the correlation between infrared response and loading cycle. This information can be presented as a vector in either a polar or Cartesian coordinate system and both are usually available in commercial systems. The polar representation is illustrated in Fig. 26.3 where the magnitude of the thermoelastic response is the length of the vector R and the phase difference between the reference signal and observed infrared response θ is the angular coordinate. In the Cartesian representation, the vector is resolved into its x and y components X and Y , as shown in Fig. 26.3. The Cartesian representation has the advantage of presenting in a single diagram both the stress distribution and the relative tension/compression state, or mode shape in vibration testing. Examples of both representations are shown in Fig. 26.4.

26.5.8 Data Processing As discussed previously in Sect. 26.5.4, a reference signal may be out of phase with the infrared response and

Thermoelastic Stress Analysis

a large positive response along the horizontal hole centerline (relative tension) and a smaller negative response along the vertical hole centerline (relative compression); (b) Y data, showing an approximately null field; (c) R data, similar to the X data except on the hole periphery at the vertical centerline since the R data are absolute values of the X data; (d) θ data, which is mostly in phase with the tensile regions of the plate and is 180◦ out of phase with the compressive areas. The plate is loaded along the vertical axis, and the data has been filtered using a 3 × 3 median mask and normalized using the peak value in the X data

this will produce an apparent phase shift in the recorded thermoelastic data. This effect can also be generated by a global offset in the system. Such a phase shift causes a constant nonzero value to be superimposed on the entire data and can be corrected typically by applying a constant vector throughout the data map. The constant vector is deduced from data captured under conditions of zero loading amplitude. The signal-to-noise (S/N) ratio of the thermoelastic signal is not solely dependent on the magnitude of the stress change event, or the sensitivity of the infrared imaging system, but also on the quantity of information used in the correlation process. The correlation process is essentially one of noise rejection in which the thermoelastic information is extracted and other infrared and electrical noise is rejected. Consequently, the greater the quantity of infrared data which can be compared to the reference signal, the greater the success of this noise rejection procedure. Most correlation processes therefore yield better data as more infrared data flows through

a)

1

b)

Y

X

–1

–1

c)

1

d)

Y

–1

–π

them so that the signal-to-noise (S/N) ratio improves with time. Measured information inevitably contains some noise. Thermoelastic images which contain stress distributions that change slowly over the data set are good candidates for the application of filtering techniques such as a 3 × 3 median mask. On the other hand, TSA images involving large spatial gradients, or singularities such as crack or notch tips, are often distorted by filtering. This tends to reduce the maximum and minimum data points and blur sharp changes in the stress distribution. In these cases it can be preferable to accept the unfiltered data rather than apply a filter and lose information [26.38, 42].

lysis [26.114, 115]. A typical example is the support bracket for a gas turbine fuel line, shown in Fig. 26.5. The use of the technique in this type of study allows a rapid assessment of the performance of a component to be made without extensive specimen preparation, which is often attractive for industrial users.

26.6.1 Isotropic Structural Materials 26.6.2 Orthotropic Materials Many structural engineering materials can be considered to be homogeneous and isotropic in the context of macroscale stress analysis and so components made from them are ideally suited to thermoelastic stress ana-

π

R

26.6 Applications In order to illustrate the experimental topics described in the foregoing sections, a series of applications of thermoelastic stress analysis that encompass a wide range of component and loading types are presented below.

1

Recognizing that many engineering members are fabricated from fiber-reinforced composites has motivated the extension of thermoelastic stress analysis to such

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Fig. 26.4a–d Thermoelastic response of a plate in uniaxial tension with a central hole, showing: (a) X data, showing

26.6 Applications

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a)

b)

1

(σx + σy) σmax

–1

Fig. 26.5a,b Analysis of a steel fuel line support bracket from a gas turbine engine: (a) photograph of sample brackets with (from left to right) no surface preparation, light abrasion, and matt black paint; (b) thermoelastic response during vibration excitation, revealing significant stress concentrations at the changes of geometry along the bracket length. Data courtesy of Rolls-Royce Plc.

materials [26.32, 34, 44, 50–74, 79, 115, 116]. Equation (26.A38) of Sect. 26.A illustrates that the thermoelastic response of orthotropic media is no longer simply proportional to the sum of the change of the principal stresses, but rather a linear combination of the changes in the stresses. Thermoelastic applications to composites are further complicated in that stresses, stress concentration, and stress intensity factors now depend on constitutive material properties and fiber orientation (and hence the ply stacking sequence). While heat transfer between the surroundings and the external surfaces is usually negligible, that between plies of a laminate can be sufficient (depending on the stacking sequence) to negate the more simple concept that thermoelasticity measures only the surface stress of a component [26.58, 59]. Moreover, internal heat transfer effects can depend on the loading frequency as well as cause a shift between loading and temperature histories. This heat transfer between composite plies produces both difficulties and opportunities. Since the surface temperature is no longer solely dependent on the response of the surface ply, subsurface effects may be investigated by applying various loading frequencies. Wong utilized this idea to derive a potential stress separation scheme which is not applicable to isotropic materials [26.58]. The references [26.32–34, 37, 43, 44, 60] describe approaches for determining reliable TSA edge data and/or evaluating individual stress components in orthotropic composites from measured temperature information.

The literature reports several non-destructive testing (NDT) studies, including determining a damage parameter for predicting the remaining life of structural components [26.61–63]. The thermoelastic detection and characterization of damage by comparing TSA scans can successfully identify locations of crack initiation, as well as predict failure and regions of ply delamination in composite laminates. Figure 26.6 is representative of a thermoelastic analysis of an orthotropic material. Other examples include using TSA

2125 1875 1625 1375 1125 875 625 375 125 –125 –375 –625 –875 –1125 –1375 –1625 A/D

Fig. 26.6 Thermoelastic image of vertically loaded [05 /90/05 ] glass-epoxy tensile composite containing a 15◦ inclined central crack [26.72]

to decrease tensile stress concentration at a central hole in a cellulosic composite by adding auxiliary holes [26.65]. TSA is well suited for analyzing fracturetype problems in orthotropic composites [26.44,66–72]. Most of these applications to laminates have involved simple stacking sequences because of the complications cited, plus potential for delamination near crack tips, which more general ply lay-ups can present. Approaches employed here hybridize TSA with hybrid elements [26.44], J-integral [26.67], conventional and decomposed complex stress functions [26.34, 43, 68– 72], nonisoparametric p-version finite elements [26.71], and a numerical scheme to determine the location of the tip of a crack accurately [26.72]. One of the particular advantages of the thermoelastic stress analysis is the ability to yield microscale surface stress distributions (for example Fig. 26.7) in addition to macroscale responses for components. This is chiefly due to the lack of requirement for a surface coating or grating as a strain witness, and is also helped by the commercial availability of large-aperture infrared microscope lenses. The technique is especially powerful in the analysis of composite materials, where two or more materials with differing material properties are combined into a single material or structure, and where both the overall response of a component is of interest as well as the stress distribution of individual fibers or fiber bundles [26.116].

26.6.3 Fracture Mechanics Thermoelasticity has been used to determine stress intensity factors and crack-tip velocities. Fracture mechanics requires high stress sensitivity and fine spatial resolution because of the localized regions of high stress gradients. Stanley and Chan [26.81], Leaity and Smith [26.82], and Stanley and Dulieu-Smith [26.83] have published detailed papers about the methodology and limitations of thermoelasticity for fracture analysis using the SPATE system based on Westergaard’s description of the elastic normal stresses near the crack tip. Rowlands and coworkers [26.66–72] have investigated the evaluation of stress intensity factors in orthotropic composites. The combined use of thermoelasticity and photoelasticity to separate principal stress which can then be used to calculate both the stress intensity factor and the J-integral has been proposed by Sagakami et al. [26.85]. The advent of focal-plane array systems has produced significant advances in the monitoring and characterization of crack-tip stress fields due to the increase in both the spatial density of data and the speed of

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Thermoelastic Stress Analysis

(σ1 + σ2) σ0

0

Fig. 26.7 Microlevel stress field around a small central hole in a bi-

axially loaded composite specimen, showing the experimentally determined principal stress sum. Data are plotted as the normalized change in the principal stress sum Δ(σ1 + σ2 )/σ0 . Data courtesy of QinetiQ

data acquisition. To evaluate the stress intensity factor, Lesniak et al. [26.84] developed a method based on fitting the thermoelastic data to Airy stress functions using a least-squares method. In 1997, Tomlinson et al. [26.86] proposed the use of the multipoint overdeterministic method (MPODM) [26.117] combined with using Muskhelishvili’s [26.118] approach to describe the crack tip stress field. The advantage of this methodology is that the passing stress field, i. e., the stress field in the absence of the crack, is not assumed to be uniform, unlike in the Westergaard formulation. This allows cracks in the presence of stress raisers to be considered. The stress field around the crack tip is described by a pair of analytical functions, internal equilibrium and compatibility are achieved automatically, and the boundary conditions are satisfied by using conformal mapping of the crack geometry in the physical plane to a unit circle. Thermoelastic data can be collected and used to evaluate the unknown coefficients of the stress functions. For the optimization, Tomlinson et al. [26.86] used a Newton–Raphson iterative method combined with a least-squares correlation whereas Diaz et al. [26.88] employed the downhill simplex method in place of the Newton–Raphson method. However, both these solution methods require seeding with an initial estimate and so the use of a genetic algorithm allows an unseeded solution to be found. Diaz et al. [26.87] also assume the location of the crack tip to be unknown and solved for its coordinates so that the crack growth rate could be accurately

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Part C 26.6

tracked. Figure 26.8 shows a typical data set recorded from a propagating fatigue crack. The crack has grown from a machined notch shown on the left in the inset picture. The crack tip is slightly to the left of center, propagating to the right. This approach is not practical when using a SPATE system because the image is skewed by the growth of the crack during image acquisition. However Stanley and Chan showed that the degree of skew could be related to the crack growth rate [26.80]. Whereas the above analyses for determining crack-tip location assume isotropy, Ju and Rowlands [26.72] recently described a method to do this in orthotropic composites. Since thermoelastic measurements are made directly in the region surrounding the crack tip, the stress intensity factors evaluated will be indicative of those driving the crack tip forward, i. e., in the case of crack closure effects occurring, information relating to the effective stress intensity factor is obtained. This and the utilization of phase information to identify the plastic zones associated with the crack tip have been discussed by workers in the field [26.82, 88, 119]. It should be

Normalized stress 6 4 2 0 –2 60 Pixels in 40 y direction

40

20 0 0

–1

0

80 60 Pixels in x direction

1

2

20

3

4

5

Fig. 26.8 Typical thermoelastic image captured with a DT 1500

(StressPhotonics Inc., Madison, WI) from a propagating fatigue crack with permission (after Diaz et al. [26.87]). A video clip showing the development of this stress field as the crack grows in fatigue is available on the DVD-ROM of this Springer Handbook

noted that none of the above methodologies account for mean stress effects since second-order thermoelastic effects are not considered.

26.6.4 Experimental Stress Separation Thermoelastic stress analysis records information on a linear combination of the stresses (or strains), but engineering applications often necessitate knowing the magnitudes of the individual components of stress. Several techniques have been developed for separating the stresses from measured temperature data [26.26–49, 60, 111, 120–129]. Because of scatter in the recorded data, and the unreliability of measured temperature information at and near edges, some stress-separation methods also concomitantly smooth the recorded information and enhance the quality of the data at and near edges. The latter can be particularly important if the objective involves determining a stress concentration at a boundary. The stress-separation techniques employ a variety of approaches, including field equations of solid mechanics, real and complex stress functions [26.28, 29, 31, 34, 43, 44, 49, 111], hybrid elements [26.44], finite element- and/or boundary elementtype concepts [26.41, 45], nonlinear effects [26.47], least-squares, and finite differences [26.42]. TSA information is the only measured data used in virtually all of these cited studies. On the other hand, TSA and photoelastic data can be combined [26.126, 129] to yield the principal stresses. Barone and Patterson [26.130] analyzed the stress using thermoelasticity in a polycarbonate birefringent coating bonded to a component of interest, thus demonstrating that such a coating could act as a strain witness in thermoelasticity as well as photoelasticity [26.131]. By combining these approaches and evaluating simultaneously both the sum (TSA) and difference (photoelastic isochromatics) of the principal stresses, Greene and Patterson evaluated the two individual principal stresses [26.120]. Most of the developed stress-separation methods assume isotropic material response, while others are valid for orthotropic behavior [26.32, 34, 43, 60]. Those formulated for orthotropy are also applicable under isotropy. Thermoelastic stress analyses usually assume adiabatic, reversible thermodynamics, and therefore employ cyclic mechanical loading. Weldman et al. [26.128] utilized the frequency-dependent depth effect to formulate a nonadiabatic TSA theory for evaluating individual strains in laminated composites, but no experimental results are presented. Most TSA stress-separation schemes developed to date emphasize linear-elastic

Thermoelastic Stress Analysis

26.6.5 Residual Stress Measurement The nondestructive determination of residual stresses is an active and emerging topic of thermoelastic research [26.22, 25, 75, 76, 79]. A number of techniques have been proposed but none have yet found wide acceptance. An early technique, known as thermal evaluation for residual stress analysis (TERSA) [26.75], used a laser to heat a statically loaded component. A mechanical chopper in the photon signal path provides a pseudodynamic signal. Alternative approaches are to measure the change in yield strain [26.76] or the change in the coefficient of thermal expansion [26.25] of a body arising from the plastic deformation accompanying the introduction of residual stresses. A more common technique (elaborated upon below) is based on the second-order mean stress effects found in (26.1) [26.22]. In this method it is assumed that a residual stress field will produce a shift in the mean stress compared to the mean induced by the applied load. The mean stress is determined thermoelastically. For isotropic materials, (26.1) can be rewritten as [26.20]     T˙ ν ∂E 1 ∂ν − I I˙ ρCε = − α + T E ∂T E 2 ∂T   (1 + ν) ∂E 1 ∂ν − σij σ˙ ij . (26.6) + E ∂T E 2 ∂T In this expression the elastic constants are not assumed to be constant but are functions of temperature.

In the uniaxial case (σp = I , σq = σr = 0 and σ˙ p = I˙, σ˙ q = σ˙ r = 0) this becomes   T˙ ν ∂E ρCε = − α + 2 (26.7) σp σ˙p , T E ∂T and under an applied stress of the form σp = σ¯p + Δσp sin ωt

(26.8)

expression (26.7) can be integrated to give [26.21]   ΔT˙ 1 ∂E ρCε = − α+ 2 σ¯ p Δσp sin ωt T E ∂T 1 ∂E (26.9) (Δσp )2 cos 2ωt , + 4E 2 ∂T assuming that any steady-state component of ΔT will be eliminated by heat transfer to the surroundings. Most photon detectors have a linear response to changes in the radiant photon emittance described by expressions (26.4) and (26.5). In normal operation TSA systems only provide the harmonic components of the measured signal at the frequency of the reference signal and hence the measurement will be   4eR∗ BT04 1 ∂E (26.10) α− 2 σ¯ p Δσp . S=− ρCε E ∂T Thus by measurement of both first and second harmonic of the infrared signal it has been suggested that the mean stress and hence residual stress can be evaluated [26.21, 22]. The validity of these expressions has been demonstrated through correlation with experimental results [26.19, 77], however the routine practical application of thermoelastic stress analysis to the determination of residual stress has yet to occur and only a few exemplars have been published [26.78, 79].

26.6.6 Vibration Analysis The effectiveness of thermoelastic stress analysis in the study of constant-amplitude sinusoidal loading events, makes it particularly suited to vibration testing [26.132–134]. The technique is doubly attractive in that it can readily yield modal shapes in addition to full-field stress data. The data from a gas turbine blade vibration analysis is presented in Fig. 26.9 as an example. The blade of Fig. 26.9 was excited in its first four modes of vibration 1. first flap, the simple oscillation back and forth of the tip of the blade relative to the root;

757

Part C 26.6

plane stress. There are few three-dimensional (3-D) stress-separation formulations and fewer experimental results [26.30, 36, 40, 48, 127]. The idea of bonded TSA gages is developed in [26.35]. This concept involves adhering longitudinal/uniaxial isotropic strips or rods to the surface of the structure of interest. Such TSA stress gages are potentially applicable to components made of orthotropic as well as isotropic material. Many of the separation schemes combine experimental, analytical, and computational features. They can be somewhat computationally intensive, so one must be attentive to numerical stability, robustness, and reliability. Future separation considerations could include nonlinear and/or inelastic constitutive response, laminated composites, loaded boundaries, three-dimensional situations, further advantageously synergy of temperature data with analytical and numerical tools and/or other measured information, and applications to more engineering-type situations.

26.6 Applications

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Part C 26.6

2. second flap, where the tip motion is reversed relative to part of the blade and a nodal point exists part way up the blade; 3. first torsion, where each corner of the blade tip oscillates in opposite directions, twisting the blade along its vertical axis; 4. third flap, where the tip is moving in the same direction as in image Fig. 26.9a (first flap) but two nodal points are present along the length of the blade.

optimized to make best use of the small thermoelastic signal available. It can take many tens of minutes to begin to extract data at the higher frequencies. Additional complications can arise since the temperature–strain relationship may not be path independent, which invalidates the basic assumption of classical thermoelastic stress analysis and requires more sophisticated interpretation [26.93].

26.6.7 Elevated Temperature Analysis There is both a lower and upper limit at which thermoelastic data can be successfully collected. The lower limit is associated with maintaining adiabatic conditions and has been discussed in Sect. 26.5.6. The upper limit is controlled by the integration time of the infrared detector, which must be reduced (i. e., the shutter speed must be increased) in order to obtain a snapshot but this reduces the signal-to-noise (S/N) ratio until eventually reliable data capture is not viable. The manufacturers of the SPATE hardware suggest that analyses up to 20 kHz are possible, however there is no published work to support this claim. Successful analyses up to a few kHz are certainly achievable using staring array detector systems, as demonstrated by the example in Fig. 26.10. Recording data at these frequencies is nontrivial and experimental conditions must be a)

b)

c)

High-temperature thermoelasticity is another promising area of research. The noncontact nature of the technique, and the ability to perform through a furnace window if necessary, are important here. Enke [26.135] has shown that thermoelasticity is able to maintain its stress sensitivity at elevated temperatures, while Lesniak and Bartel [26.136] have designed a special furnace for making thermoelastic measurements. Resistive heating of metallic components has also been used since it avoids obstruction of the field of view, which is common with most irradiative methods [26.137]. Theoretically, there is no upper temperature limit to the technique, and measurements have been successfully made at temperatures in excess of 1000 ◦ C. Calciumfluoride-coated windows, which are clear from the d) 1

(σ1 + σ2) σ0

–1

Fig. 26.9a–d Thermoelastic response of a gas turbine blade, normalized by the peak thermoelastic response: (a) first flap at 68 Hz; (b) second flap at 237 Hz; (c) first torsion at 433 Hz; and (d) third flap at 614 Hz. Data courtesy of Rolls-Royce

Plc.

Thermoelastic Stress Analysis

b)

Fig. 26.10a,b Thermoelastic re1

(σ1 + σ2) σ0

sponse of a Trent 500 high-frequency compressor blade HP1: (a) first flap 701 Hz; (b) second flap 3400 Hz, which contains systematic noise in addition to the thermoelastic data

–1

visible through the far infrared, are suitable up to the 1000 ◦ C range. The SPATE system is designed for use at room temperature in ambient conditions by taking advantage of the peak spectral radiant emittance of structures at room temperature in the wavelength range 8–14 μm. Most focal-plane array systems operate in the 3–5 μm range, which is advantageous for measurements at elevated temperatures. Fortunately, the SPATE detector will respond to this window if a notch filter is inserted into the optical path. Enke [26.135] suggests some inexpensive laboratory materials that may perform the task sufficiently well. Equation (26.4) shows that photon emittance is highly temperature dependent. At elevated temperatures the emittance can saturate the photodetector. Photodetector saturation implies that the detector no longer returns to a neutral state upon removal of the incident radiation [26.1]. This problem can be alleviated by adding an attenuating filter to the system to reduce the incident radiation to an level according to   acceptable 1 (26.11) , D = log T∗

where D is the optical density of the filter and T ∗ the transmittance. An alternative method is to insert a physical aperture, typically a thin anodized metallic circular plate with a small central hole. Not all temperatures are equally desirable for thermoelastic testing. Certain temperature ranges are advantageous because the maximum photon flux falls within an acceptable atmospheric transmission window. If possible, testing should have an ambient condition around 170 ◦ C, 460–540 ◦ C or 620–950 ◦ C [26.23].

26.6.8 Variable-Amplitude Loading Although much thermoelastic testing is performed under constant-amplitude, and often sinusoidal, loading conditions, there is no reason why high-quality data cannot be captured during loading events where the load is irregular, random or transient. Examples might include the randomized loading profiles used to test automotive chassis, loading events where a number of different cyclic loading signals of different frequencies are combined, and impact events such as automotive door slam or aircraft landing-gear loading.

26.7 Summary Thermoelastic stress analysis is a relatively straightforward experimental technique for surface stress measurement, particularly in terms of the simple surface preparation required. It yields the change in the sum of the principal surface stresses during a loading event, and is particularly well suited to the study of components undergoing cyclic fatigue or vibration testing, although any component undergoing time-varying loading is suitable for analysis. Thermoelastic stress analysis has achieved considerable popularity during the last quarter century,

and its use has been accelerated by the availability of focal-plane array cameras in the last decade. Potential new uses for thermoelastic stress analysis continue to emerge as the technique develops. Fundamental research in this area continues to produce significant advancements as software, hardware, and signal-processing techniques are refined. Thermoelasticity offers several advantages compared with other methods. The technique provides full-field, noncontact stress information with a resolution similar to a strain gage and at high spatial resolution. Virtually any type

759

Part C 26.7

a)

26.7 Summary

760

Part C

Noncontact Methods

Part C 26.A

of material component or structure can be analyzed over a broad frequency range with little setup and specimen preparation. However, the fact that the output signal involves a linear combination of the stresses, not individual stress components, necessitates some postprocessing to separate the stresses. Applications to

cases involving random, rather than cyclic, loading are also more difficult. Thermoelasticity is maturing and current research is addressing the method’s challenges and producing advancements quickly. Continuing efforts are enabling this concept to become a practical tool for quantitative engineering applications.

26.A Analytical Foundation of Thermoelastic Stress Analysis The analytical foundations of TSA have been discussed previously [26.18, 20–22, 96] and are developed from classical mechanics [26.138–146]. The first law of thermodynamics states that for a closed system undergoing a reversible process dU = ΔW + ΔQ ,

(26.A1)

where dU is the increase in internal energy, ΔW is the work done on the system, and ΔQ is the heat transferred from the surroundings to the system. Since the variation of internal energy is process independent, its infinitesimal change is mathematically denoted by the exact differential d whereas heat and work are processdependent quantities and their changes are signified by Δ (the symbol d denotes exact differentials and Δ other variations). It is convenient to rewrite (26.A1) as du = Δw + Δq

(26.A2)

with lower-case letters u, w, and q representing the corresponding quantities per unit volume. Mechanical work done by external loading on the system is absorbed as strain energy stored within the deformed body. Based on a unit volume, the work done can be expressed as the strain energy density dw = σij dεij

i, j = 1, 2, 3 ,

(26.A3)

where σij and εij are components of the stress and strain tensors, respectively. Thermodynamics defines entropy per unit volume s as dq = T ds

(26.A4)

with T being the absolute temperature [T (K) = T (◦ C) + 273.16]. The quantity s in (26.A4) is regarded as a thermodynamic parameter (a state variable). Combining (26.A2–26.A4) produces du = du(εij , s) = σij dεij + T ds

i, j = 1, 2, 3 , (26.A5)

where εij is another thermodynamic state variable which defines u in a path (process)-independent manner. This means that internal energy u is a function of these two variables but not the process which leads to changes in ε and s. With elastic solids, one prefers a different set of state variables, i. e., σij and T , which necessitates introducing Gibbs free energy as a thermodynamic potential. Gibbs free energy per unit volume g is defined as g = u − Ts − σij εij ,

(26.A6)

which upon differentiating becomes dg = du − T ds − s dT − σij dεij − εij dσij . (26.A7) Combining (26.A5) and (26.A7) gives dg = −s dT − εij dσij .

(26.A8)

Equation (26.A8) shows that the Gibbs free energy g depends on temperature and stress so one can write g = g(T, σij ) , whose differential form is ∂g ∂g dσij . dT + dg = ∂T ∂σij

(26.A9)

(26.A10)

Comparing (26.A8) and (26.A10) with respect to temperature and stress, respectively, yields ∂g (26.A11) = −s , ∂T ∂g = −εij . (26.A12) ∂σij Since entropy is a thermodynamic property, it can be described as a function of two independent (state) variables, in this case temperature and stress. Therefore, s = s(T, σij ) from which one obtains ∂s ∂s ds = dσij . dT + ∂T ∂σij

(26.A13)

(26.A14)

Thermoelastic Stress Analysis

 −

∂s ∂s dσij dT + ∂T ∂σij

 =

∂2 g ∂2 g dT + dσij . ∂T∂σij ∂T 2 (26.A15)

Using (26.A12) and (26.A14) to simplify (26.A15) leads to − ds =

∂εij ∂2 g dT − dσij . ∂T ∂T 2

∂εij (26.A24) = αδij . ∂T For isotropy, substituting (26.A24) into (26.A22) results in dT (26.A25) = −αδij dσij = −α dσkk . ρCσ T Integrating (26.A25) between two equilibrium states, 0 and 1, T1

(26.A16)

Thermodynamics in general defines specific heat (per unit mass) at constant stress or pressure Cσ as   ∂q ρCσ = , (26.A17) ∂T σ where ρ is the density. Note that the density is not constant since Δρ/ρ = −εii and at least for isotropic materials this first strain invariant (change) is proportional to the first stress invariant (change), which TSA measures. Substituting (26.A4) into (26.A17), and noting (26.A13), produces     ∂s(T, σij ) T∂s =T . (26.A18) ρCσ = ∂T σ ∂T σ Partially differentiating (26.A11) with respect to T , i. e., ∂s ∂2 g =− 2 ∂T ∂T

or for isotropic media as

(26.A19)

ρCσ gives

ρCσ = −T

. (26.A20) ∂T 2 Combining (26.A4), (26.A16), and (26.A20) produces dq dT ∂εij = ρCσ + dσij . T T ∂T

(26.A21)

For an adiabatic process, dq = 0, so under these conditions the above simplifies to ∂εij dT =− dσij . ρCσ T ∂T

(26.A22)

Moreover, assuming thermal expansion coefficients, αij (α), are defined for anisotropic media as ∂εij = αij . ∂T

(26.A23)

(26.A26)

  = −α (σkk )1 − (σkk )0

(26.A27)

(σkk )0



ρCσ ln or

(σ kk )1

dσkk

T0

T1 T0





ΔT ρCσ ln 1 + T0

 = −αΔσkk ;

k= p∼r

= α(Δσp + Δσq + Δσr ) , (26.A28)

where σkk is the first stress invariant, σp , σq , and σr are the principal stresses, and ΔT is the temperature change between the two equilibrium states associated with the stress changes. Expanding the natural logarithm term of (26.A28) in terms of a power series yields

    ΔT 1 ΔT 2 1 ΔT 3 − + − ... ρCσ T0 2 T0 3 T0 = −αΔσkk .

and substituting (26.A19) into (26.A18) results in ∂2 g

dT = −α T

(26.A29)

During a TSA experiment ΔT associated with the cyclic stresses is very small (of the order of 0.001 ◦ C) compared with the ambient temperature T0 so the higher-order terms of the above power series can be neglected. Therefore, ρCσ

ΔT = −αΔσkk T0

(26.A30)

or α Δσkk ; ρCσ k = p ∼ r ; or ΔT = −T0 K m Δσkk

ΔT = −T0

(26.A31)

Under plane-stress isotropy, this can be written as S∗ = K (Δσp + Δσq ) ,

(26.A32)

761

Part C 26.A

Equations (26.A9) and (26.A13) state that both the Gibbs free energy g and entropy s are functions of temperature and stress. Combining (26.A11) and (26.A14) therefore gives

26.A Analytical Foundation of Thermoelastic Stress Analysis

762

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Part C 26.B

where S∗ is the TSA system output signal and K a thermomechanical coefficient. For proportional loading, (26.A32) becomes S∗ = K Δ(σp + σq )

(26.A33)

and (σp + σq ) is the first stress invariant, i. e., the isopachic stress. For general anisotropic media, substituting (26.A23) into (26.A22) gives ρCσ

dT = −αij dσij T

i, j = 1, 2, 3 .

(26.A34)

Following the same procedures as with (26.A26– 26.A30) for isotropy but replacing α and σkk by αij and σij , respectively, results in ρCσ

ΔT = − αij Δσij T0 = − (α11 Δσ11 + 2α12 Δσ12 + 2α13 Δσ13 + α22 Δσ22 + 2α23 Δσ23 + α33 Δσ33 ) (26.A35)

or, in the directions of material symmetry of an orthotropic material, ΔT = −(α11 Δσ11 + α22 Δσ22 + α33 Δσ33 ) , ρCσ T0 = −αii Δσii i = 1, 2, 3 (principal material directions) . (26.A36) Under plane-stress orthotropy, T0 ΔT = − (α11 Δσ11 + α22 Δσ22 ) ρCσ or S∗ = K 1 Δσ11 + K 2 Δσ22 ,

(26.A37)

(26.A38)

where σ11 and σ22 of (26.A37) and (26.A38) are the stresses in the directions of material symmetry, and K 1 and K 2 are thermomechanical coefficients. Some authors write the expressions in terms of the specific heat at constant strain/volume Cε rather than at constant stress Cσ . References [26.96, 147] derive the relationship between Cσ and Cε . Moreover, Wong et al. extended the classical analysis to account for the temperature dependency of material properties [26.20–22].

26.B List of Symbols B c Cε Cσ Cijkl D e E f g h I, I˙ k K, K 1 , K 2 Q q R R∗ S

Stefan–Boltzmann constant Speed of light Specific heat at constant deformation or volume Specific heat at constant stress or pressure Elasticity tensor Optical density of filter Emissivity Modulus of elasticity Frequency Gibbs free energy Planck’s constant First stress invariant σx + σ y + σz and its rate with respect to time Boltzmann’s constant Thermomechanical coefficients Heat transferred Heat transferred per unit volume Vector representing the magnitude of thermoelastic signal Photodetector responsivity Photodetector output voltage

S∗ s T, T0 , T˙ T∗ t U u W w x, y X, Y αi, j Δ δij θ εij λ

Measured TSA signal Entropy Absolute temperature, initial temperature, temperature change with respect to time Transmittance of a filter Time Internal energy Internal energy per unit volume Work Work per unit volume Cartesian coordinates x- and y-direction components of the thermoelastic response R Thermal expansion coefficient tensor Denotes a change Kronecker delta Phase difference between reference signal and observed infrared response Strain tensor Wavelength of light

Thermoelastic Stress Analysis

Poisson’s ratio Density Denotes under constant stress as a subscript σp , σq , σr ; σ˙ p , σ˙ q , σ˙ r Principal stresses and their rates with respect to time Mean and amplitude of uniaxial σ¯ p , Δσp stress

σ11 , σ22 , σ33 σm φ φb φλ ω

Stresses in directions of material symmetry Stress sum = σx + σ y Total radiant photon emittance Incident photon rate for background emittance Spectral radiant photon emittance Frequency

References 26.1

26.2

26.3

26.4

26.5 26.6

26.7

26.8

26.9

26.10

26.11

26.12 26.13 26.14

26.15 26.16

D.S. Mountain, J.M.B. Webber: Stress pattern analysis by thermal emission (SPATE), Proc. Soc. Photo Opt. Inst Eng., Vol. 164 (1978) pp. 189–196 T.G. Ryall, A.K. Wong: Design of a focal-plane array thermographic system for stress analysis, Exp. Mech. 35, 144–147 (1995) A.K. Wong, T.G. Ryall: Performance of the FAST system for stress analysis, Exp. Mech. 35, 148–152 (1995) W. Weber: Über die spezifische Wärme fester Körper insbesondere der Metalle, Ann. Phys. Chem. 96, 177–213 (1830) W. Thomson (Lord Kelvin): On dynamical theory of heat, Trans. R. Soc. Edinburgh 20, 261–283 (1853) K.T. Compton, D.B. Webster: Temperature changes accompanying the adiabatic compression of steel: verification of W. Thomson’s theory to a very high accuracy, Phys. Rev. 5, 159–166 (1915) C. Zener: Internal friction in solids. I. Theory of internal friction in reeds, Phys. Rev. 52, 230–235 (1937) C. Zener: Internal friction in solids. II. General theory of thermoelastic internal friction, Phys. Rev. 53, 90–99 (1938) C. Zener: Internal friction in solids. IV. Relation between cold work and internal friction, Phys. Rev. 53, 582–586 (1938) C. Zener: Internal friction in solids. V. General theory of microscopic eddy currents, Phys. Rev. 53, 1010–1013 (1938) R. Rocca, M. Bever: The thermoelastic effect in iron and nickel (as a function of temperature), Trans. MME 188, 327–333 (1950) M.A. Biot: On anisotropic viscoelasticity, J. Appl. Phys. 25, 1385–1391 (1954) M.A. Biot: Plasticity and consolidation in a porous anisotropic solid, J. Appl. Phys. 26, 182–185 (1955) M.A. Biot: Irreversible thermodynamics with application to viscoelasticity, Phys. Rev. 97, 1463–1469 (1955) M.A. Biot: Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27, 240–253 (1956) M.H. Belgen: Structural stress measurements with an infrared radiometer, ISA Trans. 6, 49–53 (1967)

26.17

26.18

26.19

26.20

26.21

26.22

26.23

26.24

26.25

26.26

26.27

26.28

26.29

26.30

M.H. Belgen: Infrared radiometric stress instrumentation application range study, NASA Rep. CR-I067 (1968) R.T. Potter, L.J. Greaves: The application of thermoelastic stress analysis techniques to fibre composites, Proc. SPIE, Vol. 817 (1987) pp. 134–146 S. Machin, J.G. Sparrow, M.G. Stinson: Mean stress dependence of the thermoelastic constant, Strain 23, 27–30 (1987) K. Wong, R. Jones, J.G. Sparrow: Thermoelastic constant or thermoelastic parameter, J. Phys. Chem. Solids 48(8), 749–753 (1987) K. Wong, J.G. Sparrow, S.A. Dunn: On the revised theory of the thermoelasticity, J. Phys. Chem. Solids 49(4), 395–400 (1988) K. Wong, S.A. Dunn, J.G. Sparrow: Residual stress measurement by means of the thermoelastic effect, Nature 332, 613–615 (1988) N.F. Enke: Thermographic Stress Analysis of Isotropic Materials. Ph.D. Thesis (University of Wisconsin-Madison, Madison 1989) N.F. Enke, B.I. Sandor: Cyclic plasticity analysis by differential infrared thermography, Proc. VI Int. Cong. Exp. Mech. (1988) pp. 836–842 S. Quinn, J.M. Dulieu-Barton, J.M. Langlands: Progress in thermoelastic residual stress measurement, Strain 40, 127–133 (2004) T.G. Ryall, A.K. Wong: Determining stress components from thermoelastic data – a theoretical study, Mech. Mater. 7, 205–214 (1988) P. Stanley: Stress separation from SPATE data for a rotationally symmetrical pressure vessel, Proc. SPIE, Vol. 108(4) (1989) pp. 72–83 Y.M. Huang, H. Abdel Mohsen, R.E. Rowlands: Determination of individual stresses thermoelastically, Exp. Mech. 30(1), 88–94 (1990) Y.M. Huang, R.E. Rowlands, J.R. Lesniak: Simultaneous stress separation, smoothing of measured thermoelastic information, and enhanced boundary data, Exp. Mech. 30, 398–403 (1990) W. Weldman, T.G. Ryall, R. Jones: On the determination of stress components in 3-D from thermoelastic data, Compos. Struct. 36, 553–557 (1990)

763

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ν ρ σ

References

764

Part C

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Part C 26

26.31

26.32

26.33

26.34

26.35

26.36

26.37

26.38

26.39

26.40

26.41

26.42

26.43 26.44

26.45

26.46

26.47

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Photoacousti

27. Photoacoustic Characterization of Materials

The basic principles of photoacoustic generation of ultrasonic waves and applications to materials characterization of solid structures are discussed in this chapter. Photoacoustic techniques are a subset of ultrasonic methods wherein stress waves are used to obtain information about structural and material properties. In photoacoustic techniques, the ultrasound is typically generated using lasers, thereby enabling noncontact nondestructive characterization of the material properties of structures. Photoacoustic techniques have found application over a wide range of length scales ranging from macrostructures to nanometer-sized thin films and coatings. In this chapter, the basics of photoacoustics primarily as they relate to nondestructive characterization of solid materials are discussed. In Sect. 27.1, the basics of stress waves in solids is outlined. In Sect. 27.2, the process of photoacoustic generation is described. The major techniques of optical detection of ultrasound are then described in Sect. 27.3. The final section of this chapter is then devoted to some representative recent applications of photoacoustic characterization of materials. The objective here is to describe the basic principles involved, and to provide illustrative applications which take specific advantage of some of the unique features of the technique.

Photoacoustics (also known as optoacoustics, laser ultrasonics, etc.) deals with the optical generation and detection of stress waves in a solid, liquid or gaseous medium. Typically, the technique uses modulated laser irradiation to generate high-frequency stress waves (ultrasonic waves) by either ablating the medium or through rapid thermal expansion. The resulting stress

27.1 Elastic Wave Propagation in Solids ......... 27.1.1 Plane Waves in Unbounded Media . 27.1.2 Elastic Waves on Surfaces .............. 27.1.3 Guided Elastic Waves in Layered Media ......................... 27.1.4 Material Parameters Characterizable Using Elastic Waves

770 771 772 774 776

27.2 Photoacoustic Generation ..................... 27.2.1 Photoacoustic Generation: Some Experimental Results............ 27.2.2 Photoacoustic Generation: Models . 27.2.3 Practical Considerations: Lasers for Photoacoustic Generation ........

777

27.3 Optical Detection of Ultrasound ............. 27.3.1 Ultrasonic Modulation of Light ....... 27.3.2 Optical Interferometry................... 27.3.3 Practical Considerations: Systems for Optical Detection of Ultrasound .

783 783 785

777 780 783

789

27.4 Applications of Photoacoustics............... 789 27.4.1 Photoacoustic Methods for Nondestructive Imaging of Structures ................................ 789 27.4.2 Photoacoustic Methods for Materials Characterization ........ 793 27.5 Closing Remarks ................................... 798 References .................................................. 798

wave packets are also typically measured using optical probes. Photoacoustics therefore provides a noncontact way of carrying out ultrasonic interrogation of a medium to provide information about its properties. Photoacoustics can be used for nondestructive imaging of structures in order to reveal flaws in the structure, as well as to obtain the material properties of the structure.

Part C 27

Sridhar Krishnaswamy

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Part C

Noncontact Methods

Photoacoustic measurement systems are particularly attractive for nondestructive structural and materials characterization of solids because:

• • Part C 27.1

• • • •

they are noncontact; they can be nondestructive if the optical power is kept sufficiently small; they can be used for in situ measurements in an industrial setting; they are couplant independent (unlike contact acoustic techniques), providing absolute measurements of ultrasonic wave displacements; they have a very small footprint and so can be operated on curved complex surfaces; they are broadband systems providing information from the kHz to the GHz range, enabling the probing of macrostructures to very thin films.

Over the past two decades, photoacoustic methods have evolved from being primarily laboratory research tools, which worked best on highly polished optically reflective specimens, to being incorporated in a wide range of industries for process monitoring applications. Photoacoustic methods are currently being used for imaging of flaws in real composite aerospace structures. They have been used for in situ process control in steel mills to measure the thickness of rolled sheets on the fly. Photoacoustic metrology tools are also widely used in the semiconductor industry for making wafer thickness measurements among other things. For a more comprehensive discussion of the principles of photoacoustic metrology, the reader is referred to the books by Scruby and Drain [27.1] and Gusev and Karabutov [27.2], as well as to several excellent review articles on laser generation of ultrasound [27.3], and optical detection of ultrasound [27.4–6].

27.1 Elastic Wave Propagation in Solids There are many excellent books on stress waves in solids [27.7–9]. Here, only a selective review of elastic waves in solids is given. The field equations of linear elastodynamics are

•

the equations of motion ∇ · σ = ρu¨

•

σij, j = ρu¨ i ,

(27.1)

the constitutive relations σ = C·ε

•

→

→

σij = Cijkl εkl ,

(27.2)

σ I = (σ11 σ22 σ33 σ23 =σ32 σ31 =σ13 σ12 =σ21 )T

and the strain–displacement relations  1 ε= ∇u + (∇u)T 2

→

view of these symmetries, there are at most 21 independent elastic stiffness constants for the most anisotropic material. With increasing levels of material symmetry, the number of independent elastic stiffness constants decreases, with only three for cubic crystals, and only two for isotropic materials. For simplicity, a contracted notation is often preferred. The six independent components of the stress and strain tensors are stacked up as six-dimensional column vectors: εI = (ε11 ε22 ε33 2ε23 =2ε32 2ε31 =2ε13 2ε12 =2ε21 )T .

1 εkl = (u k,l + u l,k ) , 2 (27.3)

where σ is the stress tensor field, ε is the infinitesimal strain tensor field, C is the elasticity tensor, u is the displacement vector field, and ρ is the material density. Superposed dots imply time differentiation. From angular momentum balance considerations of nonpolar media, it can be shown that the stress tensor σ is symmetric, i.e., σij = σ ji . From its definition, the strain tensor ε is symmetric as well: εij = ε ji . It follows therefore, that the elasticity tensor has the following minor symmetries: Cijkl = C jikl and Cijkl = Cijlk . From thermodynamic considerations, the elasticity tensor also has the following major symmetry: Cijkl = Cklij . In

(27.4)

Thus, ij = 11 → I = 1; ij = 22 → I = 2; ij = 33 → I = 3; ij = 23 or 32 → I = 4; ij = 31 or 13 → I = 5; ij = 12 or 21 → I = 6. Capital subscripts will be used whenever the contracted notation is used. The constitutive relations in contracted notation then become ⎞⎛ ⎞ ⎛ ⎞ ⎛ σ1 c11 c12 c13 c14 c15 c16 ε1 ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎜σ2 ⎟ ⎜c12 c22 c23 c24 c25 c26 ⎟ ⎜ε2 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎜σ3 ⎟ ⎜c13 c23 c33 c34 c35 c36 ⎟ ⎜ε3 ⎟ ⎟⎜ ⎟ ⎜ ⎟=⎜ ⎜σ ⎟ ⎜c c c c c c ⎟ ⎜ε ⎟ ⎜ 4 ⎟ ⎜ 14 24 34 44 45 46 ⎟ ⎜ 4 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎝σ5 ⎠ ⎝c15 c25 c35 c45 c55 c56 ⎠ ⎝ε5 ⎠ σ6 c16 c26 c36 c46 c56 c66 ε6 (27.5) → σ I = C I J εJ .

Photoacoustic Characterization of Materials

As will be seen later in this chapter, photoacoustic measurements can be useful in determining the anisotropic elastic stiffness tensor.

27.1.1 Plane Waves in Unbounded Media

u(r, t) = U exp[ik( · r − vt)] ,

(27.6)

where U is the displacement amplitude vector,  is a unit vector along the propagation direction of the wave, r = x1 eˆ 1 + x2 eˆ 2 + x3 eˆ 3 is the position vector, k = 2π/λ is the wavenumber, λ is the wavelength, and v is the phase velocity of the wave. The angular frequency of the harmonic wave is related to the wavenumber and velocity through ω = kv = 2πv/λ. The eˆ i are unit vectors√along the 1, 2, and 3 directions, and the symbol i = −1. Substituting the above into the field equations (27.1)–(27.3), results in the so-called Christoffel equation 

(27.7) Γik − δik ρv2 Ui = 0 , where Γik = Cijkl j l

(27.8)

is called the Christoffel matrix. The existence of plane waves propagating along any direction in a general anisotropic unbounded media follows directly from the existence of real solutions to the eigenvalue problem represented by (27.7). The eigenvalues are obtained by solving the secular equation det(Γik − δik ρv2 ) = 0 .

(27.9)

It can be readily seen that the Christoffel matrix is symmetric, and under some nonrestrictive conditions on the elastic stiffness tensor, one can show that it is also positive definite. From the spectral theorem for positivedefinite symmetric matrices, it follows that there are three positive, real eigenvalues for Γij . This implies that the plane-wave phase velocities v (which are just the square root of these eigenvalues divided by the density) are guaranteed to be real and so will represent propagating modes. It should also be noted that, as the eigenvalues are independent of the frequency (and there are no boundary conditions to be satisfied here), the plane waves in an unbounded anisotropic media are

nondispersive (i. e., the phase velocity is independent of the frequency). Again from the spectral theorem, corresponding to each of the real eigenvalues there is at least one real eigenvector and, furthermore, one can always find three orthogonal eigenvectors, say, U (i) . Therefore, in any homogeneous anisotropic material, one can always propagate three types of plane harmonic waves along any chosen propagation direction . In general, these three waves will have different phase velocities v(i) , and the corresponding particle displacement vectors U (i) will be mutually orthogonal. Each of these modes is called a normal mode of propagation. The direction of the displacement vector is called the polarization direction of the wave. Note, however, that the particle displacement vector need not be parallel or perpendicular to the propagation direction in general. If the polarization direction of a wave is parallel to the propagation direction, the wave is called a pure longitudinal wave. Waves with polarization direction normal to the propagation direction are pure shear waves. If the polarization directions are neither parallel nor perpendicular to the propagation direction, the waves are neither pure longitudinal nor pure shear. In such cases, the mode whose polarization makes the smallest angle to the propagation direction is called a quasi-longitudinal wave, and the other two are called quasi-shear waves. Plane Waves in Unbounded Isotropic Media For an isotropic material, as all directions are equivalent, consider a convenient propagation direction such as  = eˆ 1 so that the Christoffel matrix readily simplifies to ⎞ ⎛ c11 0 0 ⎟ ⎜ (27.10)  = ⎝ 0 c14 0 ⎠ . 0 0 c14

The secular or characteristic equation then becomes





 c11 − ρv2 c44 − ρv2 c44 − ρv2 = 0 , (27.11) whose roots are v(1) ≡ vL =



c11 ρ

(27.12)

with the corresponding polarization parallel to the propagation direction, which therefore represents a pure longitudinal wave; and two degenerate roots c44 (27.13) v(2) = v(3) ≡ vT = ρ

771

Part C 27.1

Consider a homogenous unbounded linear elastic anisotropic medium. Seek plane harmonic waves in such a medium given by the following displacement field:

27.1 Elastic Wave Propagation in Solids

772

Part C

Noncontact Methods

with the corresponding polarization along any direction on the plane perpendicular to the propagation direction, which therefore represent pure shear waves.

Part C 27.1

Plane Waves in Unbounded Anisotropic Media For a general anisotropic material, it is not easy to simplify the secular equation analytically for arbitrary propagation directions, even though it may be possible to obtain analytically tractable expressions for special cases of propagation along certain material symmetry directions. In general, however, one seeks numerical solutions for the anisotropic problem. It is customary to represent the inverse of the phase velocity along any given propagation direction  by means of slowness surfaces in so-called k-space with axes k i /ω (which is the reciprocal of velocity, hence slowness). Any direction in this space represents the propagation direction, and the distance to the slowness surface gives the reciprocal of the phase velocity of the associated mode in this direction. Figure 27.1 shows the slowness curves for cubic Si material. Note that for an isotropic material the slowness surfaces are just spheres, the inner one representing the longitudinal mode, and the two shear slowness surfaces being degenerate. Group Velocity The group velocity (which is the velocity with which a non-monochromatic wave packet of finite frequency

content propagates in a general dispersive medium) is given by (g)

Vj =

∂ω , ∂k j

(27.14)

where k j = k j and ω(k) in general. It can be shown that the group velocity (which is also the direction of energy flow) is always perpendicular to the slowness surface. It should be noted that it is usually the group velocity that is measured in experiments.

27.1.2 Elastic Waves on Surfaces Surface waves are waves that propagate along the surface of a body, and which typically decay in amplitude very rapidly perpendicular to the surface. Consider a half-space of an anisotropic but homogeneous medium (Fig. 27.2). In this case, in addition to the field equations of motion (27.1)–(27.3), the top surface (x3 = 0) is assumed to be traction free: σi3 = Ci3kl εkl = Ci3kl u k,l = 0 on x3 = 0 for i = 1, 2, 3 .

(27.15)

Seek so-called inhomogeneous plane wave solutions of the form u(r, t) = U exp[ik 3 x3 ] exp[ik( 1 x1 + 2 x2 − vt)] , (27.16)

× 10 2

kl2/ω

–4

[001]

1.5 1 Pure shear [010] ρ 1/2 Polarized: ⎛ ⎛ ⎝c44 ⎝

0.5

[100]

0

kl1/ω

–0.5

Quasi-longitudinal

–1 Quasi-shear

–1.5 –2 –2

–1.5

–1

– 0.5

0

Fig. 27.1 Slowness curves for silicon

0.5

1

1.5

2 × 10– 4

where it is required that the displacements decay with depth (x3 -direction) and the propagation vector be restricted to the x1 –x2 plane, i. e.,  = 1 eˆ 1 + 2 eˆ 2 . Such waves are called Rayleigh waves. Note that the 3 term is not to be thought of as the x3 component of the propagation vector (which is confined to be on the x1 –x2 plane), but rather a term that characterizes the decay of the wave amplitudes with depth. In fact, both 3 and the wave velocity v are to be determined from the solution to the boundary-value problem. If 3 = 0 or is pure real, then the wave does not decay with depth, and is not a Rayleigh wave. If 3 is complex, then in order to have finite displacements at x3 → ∞, we note that the imaginary part of 3 must be positive. This is the so-called radiation condition. Furthermore, to have a propagating wave mode, the velocity v must be positive and real. Here again the equations of motion obviously formally reduce to the same Christoffel equation (27.7), where the Christoffel matrix is again given by Γik = Cijkl j l . The solution to the eigenvalue problem

Photoacoustic Characterization of Materials

3   

αn U (n) exp ik (n) 3 x3

u(r, t) =

n=1

 × exp[ik( 1 x1 + 2 x2 − vt)] ,

(27.17)

where αn are weighting constants. Using the above displacement field in the traction-free boundary conditions (27.15) results in a set of three homogenous equations for the weighting constants αn : σi3 =[ik]

3   

C3 jkl αn Uk(n) l(n) exp ik (n) 3 x3

Swapping out the elastic stiffness in favor of the longitudinal and transverse velocities vL and vT one obtains 2    2 2 vT 3 + vT2 − v2 vL2 23 + vL2 − v2 = 0 , (27.22) whose six roots are ⎧ 

2 1/2 ⎪ ⎪ v ⎪ ±i 1 − , ⎪ vT ⎪ ⎪ ⎪ ⎨ 

2 1/2

3 = ±i 1 − vv , T ⎪ ⎪ ⎪   ⎪

2 1/2 ⎪ ⎪ ⎪ . ⎩±i 1 − vvL

=0 ,

(27.18)

where (n) 1:2 = 1:2 is used for simplicity of notation. The above can be cast as dmn αn = 0 ,

(27.19)

where the elements of the 3 × 3 d matrix are: dmn = C3mkl Uk(n) l(n) (no sum over (n) intended here). Nontrivial solutions to the above are obtained if det(dmn ) = 0 .

(27.20)

Rayleigh Waves on Isotropic Media The isotropic surface wave problem is analytically tractable. Since all directions are equivalent, pick  = eˆ 1 as the propagation direction. The corresponding secular equation is 2  1 1 (C11 − C12 ) 23 + (C11 − C12 ) − ρv2 2 2   2 × C11 3 + C11 − ρv2 = 0 . (27.21)

(27.23)

Since it is required that the displacements decay with depth, and since the velocities have to be real, the three admissible roots for the Rayleigh wave velocity v ≤ vT < vL are only those corresponding to the negative exponential. The corresponding eigenvectors are ⎛ ⎞ 0 ⎜ ⎟ U (1) = ⎝1⎠ , 0 ⎛ ⎞ 

2 1/2  vT  v ⎜i v 1 − vT ⎟ ⎜ ⎟ U (2) = ⎜ ⎟, ⎝ ⎠ 0 ⎛

n=1

 × exp[ik( 1 x1 + 2 x2 − vt)]

773

vT v vT v

⎞

⎜ ⎟ ⎜ ⎟ 0 U (3) = ⎜ ⎟   1/2

2 ⎝ v  ⎠ −i vL 1 − vvL

(27.24)

and the boundary conditions (27.19) become ⎛ ⎞ 0 (U3(2)+U1(2) (2) (U3(3)+U1(3) (3) 3 ) 3 ) ⎜ (1) ⎟ ⎝ 3 ⎠ 0 0 (2) (3) (3) (3) +c U ) (c U

+c U ) 0 (c11 U3(2) (2) 12 1 11 3 3 12 1 ⎛ ⎞ ⎛ ⎞3 α1 0 ⎜ ⎟ ⎜ ⎟ × ⎝α2 ⎠ = ⎝0⎠ . (27.25) α3 0 The determinant of the matrix above vanishes for two cases. One case corresponds to a shear wave in the bulk of the material with no decay with depth and this is not a surface wave. The other case leads to the characteristic Rayleigh wave equation    vT2 3 =0, β − 8(β − 1) β − 2 1 − 2 vL β=

v2 . vT2

(27.26)

Part C 27.1

again formally yields the same secular equation (27.9), except that in this case this leads to a sixth-order polynomial equation in both 3 and v, both of which are as yet undetermined. It is best to think of this as a sixth-order polynomial equation in 3 with velocity as a parameter. In general, there will be six roots for 3 (three pairs of complex conjugates). Of these, only three are admissible so that the waves decay with depth according to the radiation condition. Denote the admissible values by

(n) 3 , n = 1, 2, 3 and the corresponding eigenvectors as U (n) . That is, there are three possible surface plane harmonic wave solutions, and the most general solution for the displacement field is a linear combination of these three

27.1 Elastic Wave Propagation in Solids

774

Part C

Noncontact Methods

This can be solved numerically for any given isotropic material. Let the velocity corresponding to the solution to (27.26) be called vR . The corresponding Rayleigh wave displacement field is given by

Part C 27.1

u(r, t) = U0 exp[ik(x 1 − vR t)] ⎡ ⎤ ⎤ ⎛ ⎡  1 1  1 1 ⎞  2 2 2 4 2 4 2 2 vR vR vR vR ⎜exp⎣k 1− 2 x 3⎦− 1− 2 1− 2 exp⎣k 1− 2 x 3⎦⎟ ⎜ ⎟ vL vL vL vT ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ ⎧ ⎫ ⎤ ⎡ ⎜ ⎟  1 ⎪ ⎪ ⎜ ⎟, 2 ⎪ ⎡ ×⎜ vR 2 ⎥ ⎪ ⎢ ⎪ ⎪ ⎤ ⎟ ⎪ ⎪ x exp k 1− ⎦ ⎣ 3 1⎪ 1 ⎪     ⎜ ⎟ 2 ⎨ ⎬ v 2 2 2 2 ⎜ ⎟ T vR vR ⎜ −i 1− ⎟ exp⎣k 1− 2 x 3⎦ −  2 1 1 ⎜ ⎟    vL ⎪ vL ⎪ 2 2 ⎪ ⎪ 4 4 ⎝ ⎠ vR vR ⎪ ⎪ ⎪ ⎪ 1− 2 1− 2 ⎪ ⎪ ⎩ ⎭ v v L

L

(27.27)

where U0 is the amplitude. It should be noted that, for Rayleigh waves in isotropic media, the vectorial displacement is entirely in the plane contained by the propagation vector and the normal to the surface (the so-called sagittal plane). The x3 decay of the two nonzero components is not equal. The u 3 component is phase delayed with respect to the u 1 component by 90◦ , and they are not of equal magnitude. Therefore, the particle displacement is elliptical in the sagittal plane. Also note that the velocity is independent of the frequency, and therefore the waves are nondispersive.

may be larger than the slow (quasi-)transverse velocity for the bulk material. Along neighboring directions to these isolated directions, a pseudo-Rayleigh wave solution exists which actually does not quite satisfy the displacement decay condition at infinite depth. (These pseudo-Rayleigh waves have energy vectors with nonzero component perpendicular to the surface.) In practice, these waves can actually be observed experimentally if the source-to-receiver distance is not too great.

27.1.3 Guided Elastic Waves in Layered Media Elastic waves can also be guided in structures of finite geometry. Guided waves in rods, beams, etc. are described in the literature. Of particular importance is the case of guided waves in layered media such as composite materials and coatings [27.11]. The simplest case is that of guided waves in plates. Consider an infinitely wide plate of thickness h made of an anisotropic but homogeneous medium (Fig. 27.3). Here, in addition to the field equations of elastodynamics, there are two traction-free boundary condition to be satisfied on the two end planes x3 = 0, h. Once again seek solutions of the form u(r, t) = U exp[ik 3 x3 ] exp[ik( 1 x1 + 2 x2 − vt)] , (27.28)

Rayleigh Waves on Anisotropic Crystals For a general anisotropic medium, the solution to the Rayleigh wave problem is quite involved and is typically only obtainable numerically. It has been shown [27.10] that Rayleigh waves exist for every direction in a general anisotropic medium, but their velocity depends on the propagation direction. In some cases, the decay term 3 can be complex and not just pure imaginary, which means that the decay with x3 can be damped oscillatory. Along certain isolated directions, the Rayleigh surface wave phase velocity

which represents an inhomogeneous plane wave propagating parallel to the plane of the plate with an amplitude variation along the thickness direction given by the 3 term. Again, as for the surface wave case, 3 is at this point undetermined along with the wave velocity v, but the depth decay condition is no longer valid. As in the Rayleigh wave problem, solving the Christoffel equation will lead to six roots for 3 (three pairs of complex conjugates), but now all of these are admissible. Let the eigenvalues be denoted by (n) 3 , n = 1, 2, . . . , 6,

x1

x2

x1

x2 x3

l x3

h

Fig. 27.2 Surface waves on a half-space

Fig. 27.3 Lamb waves in thin plates

Photoacoustic Characterization of Materials

and let U (n) be the corresponding eigenvectors. There are therefore six plane-wave solutions to the Christoffel equation, each of which is called a partial wave. The displacement field is therefore a linear combination of the six partial wave solutions

Ωsym (ω,v,vL ,vT ) = (k2−β 2 )2 cos(αh/2) sin(βh/2) + 4k2 αβ sin(αh/2) cos(βh/2) =0 ,

n=1

(27.29)

where

σi3 = (ik)

6   

Ci3kl αn Uk(n) l(n) exp −ik (n) 3 x3

n=1  × exp [ik ( 1 x1 + 2 x2 − vt)] .

(27.30)

Using the above in the traction-free boundary conditions leads to a set of six homogenous equations: dα = 0 ,

(27.31)

where the d matrix is now 6 × 6 and is given by dmn

  (1,2,3) dmn = (4,5,6) dmn

(n)

h

,

12 000 10 000

(27.32) 8000

where for convenience of notation we define (n) 1:2

= 1:2 . Nontrivial solutions for the weighting constants are obtained if det(dmn ) = 0 .

The first of the two dispersion relations corresponds to symmetric modes, and the second gives rise to antisymmetric modes. These dispersion relations are quite problematic because α, β can be real or complex depending on whether the phase velocity is less than or greater than the bulk longitudinal or bulk shear wave speeds in the material. Numerical solutions to the dispersion relations indicate that Lamb waves are highly dispersive and multimodal. Figure 27.4 shows Lamb wave dispersion for an isotropic aluminum plate. Lamb wave phase velocity (m/s)

(1,2,3) = C3mkl Uk(n) l(n) dmn (no sum over n intended here) (4,5,6) dmn = C3mkl Uk(n) l(n) eik 3

1/2 v2 α = ik 1 − 2 , vL 1/2  v2 β = ik 1 − 2 . vT 

(27.33)

This provides the dispersion relation for the guided waves. Note that, unlike for bulk waves and surface waves, now the solution depends on the wavenumber k. Therefore, we can now expect dispersive solutions where the velocity will depend on frequency. Guided Waves in Isotropic Plates For an isotropic plate, one can show that the above leads to so-called Lamb waves. Lamb wave dispersion parti-

6000

S0 4000 2000 0

A0 0

2

4

6

8

10

f h (MHz mm)

Fig. 27.4 Lamb-wave dispersion curve for an aluminum plate. The horizontal axis is the frequency–thickness product and the vertical axis is the phase velocity. The fundamental antisymmetric mode A0 , the fundamental symmetric mode S0 , and higher-order modes are shown

Part C 27.1

+ 4k2 αβ cos(αh/2) sin(βh/2) (27.34) =0 ,

where αn are weighting constants. The corresponding stress components σi3 can then be readily expressed as

775

tions into two separate dispersion relations [27.7]

Ωasym (ω,v,vL ,vT ) = (k2−β 2 )2 sin(αh/2) cos(βh/2)

6   

x αn U (n) exp ik (n) u(r, t) = 3 3

 × exp [ik ( 1 x1 + 2 x2 − vt)] ,

27.1 Elastic Wave Propagation in Solids

776

Part C

Noncontact Methods

Part C 27.1

Guided Waves in Multilayered Structures Consider the N-layer structure on a substrate shown in Figure 27.5. Acoustic waves may be coupled into the structure either via air or a coupling liquid on the top surface. We are interested in the guided plane acoustic waves that can be supported in such a system with propagation in the plane of the structure. The six partial wave solutions obtained in Sect. 27.1.3 are valid for a single layer, and these can be used to assemble the solution to the multilayer problem. The solution approach is to determine the partial wave solutions for each layer, and thereby the general solution in each layer as a linear combination of these partial wave solutions. We then impose traction and displacement boundary and continuity conditions at the interfaces. This is most effectively done by means of the transfer matrix formulation [27.12, 13]. For this purpose, a state vector is created consisting of the three displacements and the three tractions that are necessary to ensure continuity conditions:

S = (u 1 u 2 u 3 σ33 σ32 σ31 )T .

(27.35)

This vector must be continuous across each interface. For any single layer, following (27.29, 30), we have the state vector in terms of the weighting constants as follows: S = D(x3 ) (α1 α2 α3 α4 α5 α6 )T ,

(27.36)

where the common term exp[ik( 1 x1 + 2 x2 − vt)] is omitted for simplicity, and the matrix D(x3 ) is determined either analytically or numerically given the material properties and frequency and wavenumber. Note that the above state vector can be used for a substrate (treated as a half-space) by recognizing that in the R (θ)

θ Coupling water/air

Top

Layer 1

h1 h2

Layer 2 .. .

Plane wave hn

Layer N

Substrate: half-space x3

Fig. 27.5 Waves in layered structures

T (θ)

x1

Phase velocity (m/s) 8000

6000

Longitudinal velocity of Ti

4000

Shear velocity of Ti 2000

0

2

4

6

8

10

f h (MHz mm)

Fig. 27.6 Guided wave dispersion curves for an aluminum layer on a titanium half-space (Al/Ti)

substrate layer the radiation condition is valid, which means that for this last layer only three partial waves are admissible. For a single layer of finite thickness, the state vector at the two sides are related through direct elimination of the weighting constants vector: [S(−) (x3 = 0)] = [D(x3 = 0)][D(x3 = h)]−1 [S(+) (x3 = h)] ≡ [T1 ][S(+) (x3 = h)] , (27.37) where T1 is called the transfer matrix of the layer. For an N-layer system, it readily follows from continuity conditions that the state vector at the top and bottom surfaces are related through [S(−) ] = [T1 ][T2 ][T3 ] . . . [T N ][S(+) ] ≡ [T][S(+) ] , (27.38)

where T is called the global transfer matrix of the N-layer system. Given the boundary conditions for the top and bottom surfaces, or the radiation condition if the bottom layer is a substrate (half-space), the above provides the appropriate dispersion relations for the various guided-wave modes possible. This can be solved numerically for a given system. For illustrative purposes, Fig. 27.6 shows the guided-wave dispersion curves for an aluminum layer on a titanium substrate.

27.1.4 Material Parameters Characterizable Using Elastic Waves Since elastic wave propagation in a solid depends on the material properties and the geometry of the struc-

Photoacoustic Characterization of Materials

measurements can often be made independent of the support conditions of the structure. An example of this will be seen in section Sect. 27.4.2 on photoacoustic characterization of ultrathin films. It should be noted that techniques based on interrogation using linear elastic waves can only access stiffness properties and cannot in general reveal information about the strength of a material. If large-amplitude stress waves can be generated, material nonlinearity can be probed. However, we will not address this issue here since photoacoustic generation is generally kept in the so-called thermoelastic regime where the stress waves are typically of small amplitude.

27.2 Photoacoustic Generation Photoacoustics arguably traces its history back to Alexander Graham Bell, but the field really got its impetus in the early 1960s starting with the demonstration of laser generation of ultrasound by White [27.14]. Since then, lasers have been used to generate ultrasound in solids, liquids, and gases for a number of applications. A comprehensive review of laser generation of ultrasound is given in Hutchins [27.3] and Scruby and Drain [27.1]. Here we will restrict attention to the generation of ultrasound in solids using pulsed lasers. The basic mechanisms involved in laser generation of ultrasound in a solid are easy to outline. A pulsed laser beam impinges on a material and is partially or entirely absorbed by it. The optical power that is absorbed by the material is converted to heat, leading to rapid localized temperature increase. This results in rapid thermal expansion of a local region, which in turn leads to generation of ultrasound into the medium. If the optical power is kept low enough that the material does not melt and ablate, the generation regime is called thermoelastic (Fig. 27.7a). If the optical power is high enough to lead to melting of the material and plasma formation, once again ultrasound is generated, but in this case via momentum transfer due to material ejection (Fig. 27.7b). The ablative regime of generation is typically not acceptable for nondestructive characterization of materials, but is useful in some process monitoring applications especially since it produces strong bulk wave generation normal to the surface. In some cases where a strong ultrasonic signal is needed but ablation is unacceptable, a sacrificial layer (typically a coating or a fluid) is used either unconstrained on the surface of the test medium or constrained between the

medium and an optically transparent plate. The sacrificial layer is then ablated by the laser, again leading to strong ultrasound generation in the medium due to momentum transfer.

27.2.1 Photoacoustic Generation: Some Experimental Results Before delving into the theory of photoacoustic generation, it is illustrative to look at some of the kinds of ultrasonic signals that can be generated using photoacoustic generation. Photoacoustic Longitudinal and Shear Wave Generation Using a Point-Focused Laser Source Figure 27.8 shows the surface normal displacement at the epicentral location generated by a point-focused thermoelastic source and measured using a homodyne interferometer (described in Sect. 27.3.2). The arrival a)

b) Thermoelastic expansion

Thermoelastic expansion and momentum transfer due to material ejection

x3

x1 Ablation Generating laser

Generating laser

Fig. 27.7a,b Laser generation of ultrasound in (a) the thermoelastic regime and (b) the ablative regime

777

Part C 27.2

ture, elastic waves can be used to measure some of these parameters. For instance, by measuring the bulk wave speeds of an anisotropic medium in different directions, it is possible to obtain the elasticity tensor knowing the density of the material. Using guided acoustic waves in multilayered structures, it may be possible to infer both material stiffnesses as well as layer thicknesses. The primary advantage of using elastic waves over performing standard load–deflection type of measurements arises from the fact that these measurements can be made locally. That is, while load–deflection measurements require accurate knowledge of the boundary or support conditions of the entire structure, elastic wave

27.2 Photoacoustic Generation

778

Part C

Noncontact Methods

0 Epicentral displacement

–8

–10

Part C 27.2

–12

–14

Thermoelastic generation

Longitudinal wave arrival

–16 0.8

1

1.2

1.4

1.6

1.8

2

2.2

Time (μs)

Fig. 27.8 Experimentally measured epicentral surface normal displacement

of a longitudinal wave is seen distinctly, followed by a slowly increasing wash, until the shear wave arrives. a)

The directivity (i. e., the wave amplitude at different angles to the surface normal) of the longitudinal and shear waves generated by a thermoelastic source have been calculated theoretically and measured experimentally by a number of researchers [27.1]. The longitudinal and shear wave directivity using a shear dipole model (described later) for the thermoelastic laser source have been obtained using a mass–spring lattice elastodynamic calculation [27.15] and are shown in Fig. 27.9. It is seen that, for thermoelastic generation, longitudinal waves are most efficiently generated in directions that are at an angle to the surface normal. Epicentral longitudinal waves are much weaker. This is consistent with the results of other models as well as with those from experiments [27.16]. Photoacoustic Rayleigh Wave Generation Using a Point- and Line-Focused Laser Source Thermoelastic generation of Rayleigh waves has been extensively studied (see Scruby and Drain [27.1]). a) Displacement (arb. units)

90 120

60

150

30

180

0

b) Displacement (arb. units) Laser source

b)

Rayleigh wave

90 120

60

150

30

180

0

Laser source

Fig. 27.9a,b Calculated directivity pattern of thermoelastic laser generated ultrasound: (a) longitudinal waves and (b) shear waves

0

0.2

0.4

0.6

0.8

1

1.2

1.4 1.6 Time (μs)

Fig. 27.10a,b Rayleigh wave generation using a pointfocused laser source. (a) Experimental results for surface normal displacements. (b) Theoretical calculations of the surface normal (solid) and horizontal (dotted) displacements

Photoacoustic Characterization of Materials

a) Displacement (arb. units)

27.2 Photoacoustic Generation

779

Displacement (arb. units)

5 0

0

5

10

15

20

25

–5

–15

Rayleigh wave 0

2

4

6

b) Displacement (arb. units)

8 Time (μs)

Photoacoustic Guided-Wave Generation in Multilayered Structures Thermoelastic generation of ultrasound on multilayered structures has also been demonstrated, motivated by potential applications to the characterization of coatings. In these cases, again the various guided modes are gen-

0.1 0 –0.1 Rayleigh wave

–0.2 –0.3

0

2

a thin aluminum plate. The surface normal displacements at some distance from the generating source shows the dispersive nature of Lamb waves

4

a) Displacement (arb. units) 6

8 Time (μs)

Fig. 27.11a,b Surface normal displacement showing a monopolar Rayleigh wave from a thermoelastic line source: (a) theoretical and (b) experimental

Shown in Fig. 27.10a are measurements of the far-field surface normal displacement made using a homodyne interferometer (discussed in Sect. 27.3.2). Figure 27.10b shows the theoretically calculated surface displacements in the far field of a thermoelastic point source, using a shear-dipole model [27.15]. A strong bipolar Rayleigh wave is seen to be generated by the thermoelastic source. If the laser source is a line source rather than a point-focused source, the Rayleigh wave becomes a monopolar pulse, as shown both theoretically and experimentally in Fig. 27.11 [27.15]. Photoacoustic Lamb Wave Generation Thermoelastic Lamb wave generation in thin plates has been experimentally studied by Dewhurst et al. [27.17], and has been modeled by Spicer et al. [27.18]. Figure 27.12 shows the Lamb waves that are generated by a single thermoelastic line source on a thin aluminum plate. The detection system used was a broadband twowave mixing interferometer (Sect. 27.3.2). The figure shows the dispersive nature of Lamb wave propagation.

0

200

400

600

800 Time (ns)

600

800 Time (ns)

b) Displacement (arb. units)

0

200

400

Fig. 27.13a,b Thermoelastic generation on a 2.2 μm Ti

thin-film coating on an aluminum substrate. Surface normal displacement at a distance of 1.5 mm (a) measured experimentally and (b) predicted by the theoretical model

Part C 27.2

Fig. 27.12 Thermoelastically generated Lamb waves in

–10

–20

30 Time (μs)

780

Part C

Noncontact Methods

Part C 27.2

erally dispersive. Murray et al. [27.19] have performed comparisons of theoretical and experimental signals in layered structures on a substrate. High-frequency ultrasonic guided waves were generated with a miniature Nd:YAG laser with a 1 ns pulse width and 3 μJ per pulse, focused down to a point source of about 40 μm diameter. Figure 27.13a shows the ultrasonic surface displacements measured using a broadband stabilized Michelson interferometer (described in Sect. 27.3.2) at a distance of 1.5 mm from the laser source. The specimen was a 2.2 μm Ti coating on aluminum. Also shown in Fig. 27.13b are the theoretically expected signals using a thermoelastic model for the two-layer case. It is seen that the thermoelastic model adequately captures the experimental behavior. From these examples, it is clear that photoacoustic generation of bulk and guided waves is technically feasible using pulsed laser sources, and theoretical models for the process are well established. In the next section, the basic theoretical models for photoacoustic generation are outlined.

27.2.2 Photoacoustic Generation: Models It is important to characterize the ultrasound generated by laser heating of a material in order to determine the amplitude, frequency content, and directivity of the ultrasound generated. In general, models are also useful in extracting material property information from the measured data. There are two modes of photoacoustic generation in solids: ablative and thermoelastic. If the material ablates, the ultrasound that results from momentum transfer can be modeled as arising from a normal impulsive force applied to the surface. Analytical solutions to this problem can be obtained by appropriate temporal and spatial convolution of the elastodynamic solution for a point impulsive force on a half-space [27.7]. Recently, a complete model of ultrasound generation in the ablative regime has been given by Murray et al. [27.20]. As ablative generation is typically not used in nondestructive evaluation, we will not explore this further here. We will only consider thermoelastic generation. The basic problem of thermoelastic generation of ultrasound can be decomposed into three subproblems: (i) electromagnetic energy absorption by the medium (ii) the consequent thermal diffusion problem with heat sources (due to the electromagnetic energy absorbed)

(iii) the resulting elastodynamic problem with volumetric sources (due to thermal expansion) There are two important cases that are relevant to materials characterization: (i) bulk-wave photoacoustics where typically onedimensional bulk waves of GHz frequency range are launched using femtosecond pulsed laser sources to characterize thin films and coatings (ii) guided-wave photoacoustics where nanosecond focused laser sources are used to launch kHz to MHz ultrasonic waves Guided-Wave Photoacoustic Generation For simplicity a fully decoupled linear analysis for homogeneous, isotropic materials is considered. The optical energy that is absorbed depends on the wavelength of the laser light and the properties of the absorbing material. The optical intensity variation with depth inside an absorbing medium that is illuminated by a light beam at normal incidence is given by an exponential decay relation

I (x1 , x2 , x3 , t) = I0 (x1 , x2 , t) exp(−γ x3 ) ,

(27.39)

where I0 (x1 , x2 , t)is the incident intensity distribution at the surface (which is a function of the laser parameters) and γ is an absorption coefficient characteristic of the material for the given wavelength of light. The optical energy absorbed by the material leads to a distributed heat source in the material given by q(x1 , x2 , x3 , t) = q0 (x1 , x2 , t)γ exp(−γ x3 ) , (27.40)

where q0 is proportional to I0 and has the same spatial and temporal characteristics as the incident laser source. The corresponding thermal problem is then solved for the given thermal source distribution using the equations of heat conduction ∂T (27.41) = κ∇ 2 T + q , ρC ∂t where T is the temperature, ρ is the material density, C is the heat capacity at constant volume, and κ is the thermal diffusivity. The necessary thermal boundary conditions arise from the fact that there is no heat flux across the surface, and initially the medium is at uniform temperature. The temperature distribution can be calculated for a given laser source and material by solving the heat conduction equation (27.41). For most metals, heat diffusion can be significant and needs to be taken into

Photoacoustic Characterization of Materials

u = ∇φ + ∇ × ψ ,

(27.42)

where φ and ψ are the scalar and vector potentials, respectively. The equations of motion including the volumetric expansion source then become 1 φ¨ = φT , vL2 1 ∇ 2 ψ − 2 ψ¨ = 0 , vT ∇2φ −

(27.43)

where vL and vT are the longitudinal and shear wave speeds of the material, respectively, and a superposed dot indicates time differentiation. The temperature rise from the laser energy absorbed leads to a volumetric expansion source given by φT =

3λ + 2μ αT T , λ + 2μ

(27.44)

where αT is the coefficient of linear thermal expansion, and λ and μ are the Lamé elastic constants of the material. The above wave equations need to be solved along with the boundary conditions that the surface tractions vanish. Given the laser source parameters, the resulting temperature and elastodynamic fields are obtained from the above system of equations using transform techniques [27.21, 22]. The heat conduction and the elastodynamic equations are transformed using onesided Laplace transform in time, and either a Fourier (for a line source) or Hankel (for a point source) transform in space. Closed-form solutions can be obtained analytically in the transformed domain, and numerically inverted back into the physical domain [27.21–23]. For most metallic materials, the absorption coefficient is high enough that the optical energy does not penetrate very much into the material. The optical penetration depth defined as 1/γ is on the order of a few nanometers for most metals over the optical wavelengths typically used for laser generation of ultrasound. If, in addition, the time scale of interest is such that significant thermal diffusion does not occur,

and therefore the volumetric thermoelastic expansion source is confined to the surface region, it has been shown that the volumetric sources can be replaced by an equivalent traction boundary condition on the surface. Scruby et al. [27.24] have argued that the relevant elastodynamic problem is that of shear dipoles acting on the surface of the body. Their argument was based on the consideration that a point expansion source in the interior of a solid can be modeled as three mutually orthogonal dipoles [27.7], and this degenerates into a pair of orthogonal dipoles as the expansion source moves to a free surface (Fig. 27.7a). This approach was given a rigorous basis in the form of the surface center of expansion (SCOE) model proposed by Rose [27.25], which predicts all the major features that have been observed in thermoelastic generation. Specifically, if the optical penetration depth is very small (i. e., γ → ∞), and the laser beam is assumed to be focused into an infinitely long line along the x1 -direction, and with a delta-function temporal dependence, the resulting heat source simplifies to q(x1 , x2 , x3 , t) = Q 0 δ(x1 )δ(x3 )δ(t) ,

(27.45)

where Q 0 is the strength of the heat source proportional to the laser energy input. If thermal diffusion is also neglected, the resulting simplified problem can be explicitly solved [27.21, 23]. The corresponding in-plane stresses are given by [27.23] σ31 = Dδ (x1 )H(t) ,

(27.46)

where a prime indicates differentiation with respect to the argument, and H(t) is the Heaviside step function. The above indicates that the shear dipole model of Scruby et al. [27.24] is indeed valid in this limit of no thermal diffusion and no optical penetration. The dipole magnitude D is given by [27.23] D=

αT Q 0 2μ (3λ + 2μ) . λ + 2μ ρC

(27.47)

Solutions to the more general case where the temporal and spatial characteristics are more typical of real laser pulses can be readily obtained from the above solution using a convolution over space and time. If optical penetration depth is significant or if thermal diffusion is important, solutions to the complete system of equations will have to be obtained numerically [27.22, 23]. The shear-dipole model can also be extended to multilayered structures by using the transfer matrix formulation described in Sect. 27.1.3. Murray et al. [27.19] have shown results for guided-wave photoacoustic generation in a two-layer system consisting of a 2.2 μm Ti

781

Part C 27.2

account by solving the full heat conduction equation. For insulators, heat conduction may be neglected and the resulting adiabatic temperature rise is readily obtained by setting κ → 0 in (27.41). Next the elastodynamic problem is considered. For isotropic materials, the equations of elastodynamics can be cast in terms of the scalar and vector displacement potentials [27.7]. The elastic displacement field u can be decomposed into

27.2 Photoacoustic Generation

782

Part C

Noncontact Methods

coating on an aluminum substrate. Typical results are shown in Fig. 27.13b.

Part C 27.2

Bulk-Wave Photoacoustic Generation For photoacoustic characterization of very thin films and coatings, an alternate approach is to use very highfrequency (very short-wavelength) bulk longitudinal waves that are launched into the depth of the specimen. This technique has come to be known as picosecond ultrasonics [27.26]. This necessitates the use of unfocused femtosecond laser sources that impinge on the specimen surface. If the diameter of the heating region (typically tens of μm) is much larger than the film thickness (typically nanometers), and the optical skin depth is much less than the film thickness, a one-dimensional thermoelastic model can be used [27.27, 28]. The temperature field T therefore only varies with the depth direction x, and the only nonzero displacement component is the one along the x-direction, denoted by u. To solve the complete thermoelastic problem in multilayer structures, a photothermoelastic transfer matrix approach is adopted, as shown in Fig. 27.14. Following Miklos et al. [27.29] we apply the classical partially coupled thermoelastic differential equations for the heat transfer and wave propagation in homogeneous isotropic or cubic media as follows:

κ

∂T ∂2 T − ρC = −q ∂t ∂x 2 ∂2u ∂2u ∂T c 2 −ρ 2 = λ , ∂x ∂x ∂t

be expressed as λ = cαT , where αT is the thermal expansion coefficient, and q is the absorbed energy due to laser irradiation (27.40). The linear coupled thermoelastic equations (27.48) hold for every thin layer in the multilayer structure, with total thickness h, as shown in Fig. 27.14. The boundary conditions are assumed to be no heat transfer or tractions across the surfaces at x = 0 and x = h. It is also assumed that all thin layers are in perfect contact, which ensures continuity of displacement, temperature, heat flux J = −κ∂T /∂x, and elastic stress σ = c∂u/∂x − λT at every interface. The coupled problem given by (27.48) can be readily solved using the transfer-matrix formulation a) Reflection coefficient change (arb. units) 1.2 1

0.6

where ρ is the density, κ is the thermal conductivity, C is the specific heat per unit volume, c is the effective elastic stiffness, λ is the thermal stress tensor, which can

Supported film

0.4 Unsupported film

0.2 0

(27.48)

Acoustic echoes

0.8

0

100

200

300 400 Delay time (ps)

b) Elastic strain (10– 4) Al

2

10 ps 30 ps 50 ps 70 ps

1 y Pump laser

Layer 1

Layer 2

···

Layer n-1

Layer n

0

Si3N4

–1 Heat flux and elastic wave

d1 0

d2 x1

0

dn–1 x2

dn xn–1

xn (h) x

Fig. 27.14 Geometry of the photothermoelastic model for multilayer structures

100

200

300

400

500 600 Depth (nm)

Fig. 27.15 (a) Simulated transient optical reflectivity change due to femtosecond laser heating for both supported and unsupported 300 nm aluminum and 300 nm silicon nitride double-layer thin films; (b) corresponding thermoelastic strain pulse shape and propagation in the unsupported double-layer thin film

Photoacoustic Characterization of Materials

properties of the materials determine the thermoreflectivity signal.

27.2.3 Practical Considerations: Lasers for Photoacoustic Generation Several different types of lasers are commercially available. The material in which the ultrasound is to be generated, and the desired frequency content of the ultrasound, dictate the type of laser to use. The generating laser wavelength, its energy, its pulse duration, and its repetition rate are all parameters that can be selected based on applications. The repetition rate is important primarily for speed of testing. The pulse duration along with other parameters such as the spatial extent of the generating volume/area dictate the frequency content of the ultrasound generated. A modulated continuous-wave laser is possibly adequate for low-frequency (order of tens of kHz) generation. Typically, laser pulses on the order of 10 ns are used for the generation of ultrasound in the 100 kHz–10 Mhz frequency range. For even higher frequencies, pulse widths on the order of 100 ps (resulting in ultrasound in the order of 100 MHz frequency range) or even femtosecond (for GHz range) laser systems may be necessary. The optical power required depends on the material to be tested and whether laser damage is acceptable or not. Lasers with optical power ranging from nanojoules to microjoules to several hundred millijoules have all been used to produce ultrasound in structures. Finally, the choice of laser wavelength primarily depends on the material absorption. Laser wavelengths ranging from the ultraviolet to the infrared and higher have been used for laser generation of ultrasound.

27.3 Optical Detection of Ultrasound Optical detection of ultrasound is attractive because it is noncontact, with high detection bandwidth (unlike resonant piezoelectric transducers), and it can provide absolute measurement of the ultrasonic signal. In this section, the ways in which ultrasonic signal information can be encoded onto a light beam are first described, followed by a discussion of two methods of demodulating the encoded information using optical interferometry. For a more complete review of optical detection of ultrasound, the reader is referred to a number of excellent review articles on the subject [27.4–6].

783

27.3.1 Ultrasonic Modulation of Light To monitor ultrasound optically, a light beam should be made to interact with the object undergoing such motion. Interaction of acoustic waves and light waves in transparent media has a long history (see, for instance, [27.32]), and will not be reviewed here. Attention will be confined here to opaque solids that either reflect or scatter light. In this case, the light beam can only be used to monitor the surface motion associated with the ultrasound.

Part C 27.3

as described in Sect. 27.1.3 but now in the Laplacetransformed (over time) domain. As an illustrative example, bulk-wave photoacoustic generation in a 300 nm aluminum and 300 nm silicon nitride double-layer film with and without silicon substrate are shown in Fig. 27.15. Figure 27.15a shows the simulation of the transient thermoelastic strain that propagates in the unsupported film. The laser pulse duration used in the calculation is a 100 fs Hanning function [27.29]. The elastic strain generated on the near surface propagates toward the interface with the velocity of longitudinal waves in aluminum and arrives at the Al and Si3 N4 interface at about 50 ps. Part of the energy is transmitted into the silicon nitride layer and the rest is reflected back toward the front surface, as shown in Fig. 27.15a. Also shown in Fig. 27.15b is the transient change in the surface reflectivity due to temperature and strain changes induced by the laser. It should be pointed out that, in picosecond ultrasonics, the transient reflectivity is typically monitored in a snapshot manner using a femtosecond laser pulse that is progressively time delayed with respect to the photoacoustic generation pulse (see [27.30, 31], for instance). This is necessitated by the very high temporal resolutions needed to monitor the transient photothermoacoustic phenomena that arise from femtosecond laser irradiation. The initial transient rise and subsequently exponential-like decay process are due to the thermoreflectivity changes. The superimposed multiple spikes are due to the acoustic waves that are reflected from the Al/Si3 N4 interface and the Si3 N4 /silicon or air interface. The elastic properties and density of each layer affect the acoustic wave arrival time as well as the signal amplitude, while the thermal

27.3 Optical Detection of Ultrasound

784

Part C

Noncontact Methods

Typically laser beams are used as the optical source, and these can provide monochromatic, linearly polarized, plane light beams. The electric field of such beams can be expressed as E = a exp[i(ωopt t − φ)] ,

(27.49)

Part C 27.3

where E is the electric field of amplitude a, frequency ωopt , and phase φ. It is important to note that extant photodetectors cannot directly track the optical phase (the optical frequency ωopt is just too high), and as such only the optical intensity (proportional to P = E E ∗ = a2 , where ∗ represents the complex conjugate) can be directly measured. There are a number of ways in which ultrasound can affect the light beam. These can be broadly classified into intensity-modulated techniques and phase/frequency-modulated techniques. Intensity Modulation Induced by Ultrasound The intensity of the reflected light beam can change due to ultrasound-induced changes in the refractive index of the medium, and this can be monitored directly using a photodetector. Though these changes are typically very small for most materials, this method has been used successfully in picosecond ultrasonics [27.30] to measure the properties of thin films [27.31] and nanostructures [27.33]. Another intensity-based technique utilizes the surface tilt associated with ultrasonic motion [27.34]. The probe light beam is tightly focused onto an optically reflective object surface. A partial aperture (usually called a knife edge in this context) is placed behind a recollimating lens located in the path of the reflected beam prior to being focused onto a photodetector. The reflected light beam will undergo a slight tilt due to the ultrasonic displacement. This in turn will cause varying portions of light to be blocked by the knife edge, resulting in an intensity change at the photodetector. A third class of intensity-based techniques is applicable to continuous or tone-burst surface acoustic wave (SAW) packets of known frequency and velocity. In this case, the ultrasonic surface displacement acts like a surface diffraction grating, and an incident plane light beam will undergo diffraction in the presence of the SAW wave packet. A photodetector placed in the direction of either of the two expected diffracted first-order beams can be used to monitor the SAW wave packet [27.35]. Recently, diffraction detection has been used to measure the mechanical properties of thin films [27.36]. In general, intensitymodulated techniques are typically less sensitive than phase/frequency-modulated techniques. As such, their

use in nondestructive characterization has been limited. The reader is referred to the review papers [27.4, 5] for further information on intensity-modulation techniques. Phase or Frequency Modulation Induced by Ultrasound Ultrasonic motion on the surface of a body also affects the phase or frequency of the reflected or scattered light. For simplicity, consider an object surface illuminated at normal incidence by a light beam, as shown in Fig. 27.16. Let the surface normal displacement at the point of measurement be u(t) due to ultrasonic motion, where t is time. We shall assume that the surface tilt is not so large that the reflected optical beam is tilted significantly away. Therefore the object surface displacement just changes the phase of the light by causing a change in the path length (equal to twice the ultrasonic normal displacement) that the light has to travel. In the presence of ultrasonic displacement, the electric field can therefore be expressed as

E s = as exp[i(ωopt t − 2kopt u(t) − ϕs )] ,

(27.50)

where kopt = 2π/λopt is the optical wavenumber, λopt is the optical wavelength, and ϕs is the optical phase (from some common reference point) in the absence of ultrasound. For time-varying phase modulation such as that caused by an ultrasonic wave packet, it is also possible to view the optical interaction with the surface motion as an instantaneous Doppler shift in optical frequency. To see this, note that (27.50) can be equivalently written in terms of the surface velocity V (t) = du/ dt as follows: E s = as exp[i(ω˜ opt t − ϕs )] ,

(27.51)

u (x1, x2, t) Incident light

Reflected light

Fig. 27.16 Phase modulation of light due to ultrasonic dis-

placement

Photoacoustic Characterization of Materials

where the instantaneous optical frequency is now given by 

't ω˜ opt t =

ωopt

2V 1− c

 dt ,

(27.52)

0

27.3.2 Optical Interferometry The phase of a single optical beam cannot be measured directly since the optical frequency is too high to be monitored directly by any extant photodetector. Therefore, a demodulation scheme has to be used to retrieve phase-encoded information. There are a number of optical interferometers that perform this demodulation (see [27.37] for a general discussion on optical interferometry). Here we will only consider two systems that have found wide extensive application in photoacoustic metrology. Another common device that has found extensive applications in photoacoustic metrology is the confocal Fabry–Pérot interferometer. Due to space limitations, this will not be discussed here, and the reader is referred to the review paper by Dewhurst and Shan [27.6]. Reference-Beam Interferometers The simplest optical interferometer is the two-beam Michelson setup shown in Fig. 27.17. The output from a laser is split into two at a beam splitter and one of the beams is sent to the test object, and the other is sent to a reference mirror. Upon reflection, the two beams are recombined parallel to each other and made to interfere at the photodetector. The electric fields at the

photodetector plane can be written as E r = ar exp[i(ωopt t − kopt L r )] , E s = as exp[i(ωopt t − kopt (L s + 2u(t)))] ,

(27.53) (27.54)

where (i = r, s) refer to the reference and signal beams, respectively. Here E i are the electric fields of the two beams of amplitudes ai and optical frequency ωopt . The phases φi = kopt L i are due to the different path lengths L i that the two beams travel from a point of common phase (say at the point just prior to the two beams splitting at the beam splitter). Here, it is convenient to consider the phase term for the signal beam as being comprised of a static part kopt L s due to the static path length, and a time-varying part due to the time-varying ultrasonic displacement u(t). The total electric field at the photodetector plane is then the sum of the fields of the two beams, and the resulting intensity is therefore obtained as PD = Ptot {1 + M cos[kopt (L r − L s ) − 2kopt u(t)]} , (27.55)

Pi = ai2

where are the optical intensities (directly proportional to the power in Watts) of the two beams individually. In the last expression above, we have defined the total optical power Ptot = Pr + Ps . The factor √ 2 Pr Ps M= Ptot is known as the modulation depth of the interference and ranges between 0 (when one of the beams is not present) to 1 (when the two beams are of equal intensity). If the phase change due to the signal of interest 2kopt u(t)  1 – as is the case for typical ultrasonic displacements – the best sensitivity is obtained by ensuring I

Photodetector

Laser Signal beam

Reference beam

Fig. 27.17 Two-beam homodyne (Michelson) interferome-

ter

785

Δφ

Fig. 27.18 Two-beam interferometer output intensity as

a function of phase change. The largest variation in output intensity for small phase changes occurs at quadrature

Part C 27.3

where c is the speed of light. The surface velocity associated with the ultrasonic motion therefore leads to a frequency shift of the optical beam.

27.3 Optical Detection of Ultrasound

786

Part C

Noncontact Methods

Part C 27.3

that the static phase difference is maintained at quadrature, i. e., at kopt (L r − L s ) = π2 (Fig. 27.18). This can be achieved by choosing the reference and signal beam path lengths appropriately. The two-beam Michelson interferometer that is maintained at quadrature therefore provides an output optical power at the photodetector given by PD = Ptot [1 + M(2kopt u(t))] ,

kopt u  1 . (27.56)

This shows that the output of a Michelson interferometer that is operated at quadrature is proportional to the ultrasonic displacement. In reality, even the static optical path is not quite static because of low-frequency ambient vibration that can move the various optical components or the object around. If the signal of interest is high frequency (say several kHz or higher) – which is the case for ultrasonic signals – it is possible to use an active stabilization system using a moving mirror (typically mounted on a piezoelectric stack) on the reference leg such that the static (or, more appropriately, low-frequency) phase difference is always actively kept constant by means of a feedback controller. The piezoelectric mirror can also be used to calibrate the full fringe interferometric output by intentionally inducing an optical phase change in the reference leg that is larger than 2π. This will provide both the total power Ptot and the modulation depth M. Therefore, an absolute measurement of the ultrasonic displacement can be obtained from (27.56). It is important to characterize the signal-to-noise ratio (SNR), or equivalently the minimum detectable displacement of the optical interferometer. There are several possible noise sources in an optical detection system. These include noise from the laser source, in the photodetector, in the electronics, and noise in the a)

Speckled object Photodetector Planar reference

b)

Speckled object Photodetector Wavefront-matched reference

Fig. 27.19a,b Ultrasound detection on rough surfaces. (a) Interference of speckled signal and planar reference beams is nonoptimal. (b) Wavefront-matched interference

optical path due to ambient vibrations and thermal currents. Most of these noise sources can be stabilized against or minimized by careful design and isolation, leaving only quantum or shot noise arising from random fluctuations in the photocurrent. Shot noise increases with increasing optical power, and it therefore basically sets the absolute limit of detection for optical measurement systems. It can be shown that the SNR of a shot-noise-limited Michelson interferometer operating at quadrature is given by [27.5] ( ηPtot (27.57) , SNR = kopt MU hνopt B where η is the detector quantum efficiency, νopt is the optical (circular) frequency in Hertz, h is Planck’s constant, B is the detection bandwidth, and U is the ultrasonic displacement. The minimum detectable ultrasonic signal can then be readily determined from (27.57) based on the somewhat arbitrary criterion that a signal is detectable if it is equal to the noise magnitude, i. e., if SNR = 1. For a detection bandwidth of 1 Hz, detector efficiency of 0.5, modulation depth M of 0.8, and total collected optical power of 1 W from a green laser (514 nm), the minimum detectable sensitivity is on the order of 10−17 m. Of all possible configurations, the two-beam homodyne interferometer provides the best shot-noise detection sensitivity as long as the object beam is specularly reflective. If the object surface is rough, the scattered object beam will in general be a speckled beam. In this case, the performance of the Michelson interferometer will be several orders of magnitude poorer due to two factors. First, the total optical power Ptot collected will be lower than from a mirror surface. Secondly, the mixing of a nonplanar object beam (one where the optical phase varies randomly across the beam) with a planar reference beam is not efficient (Fig. 27.19a), and indeed could be counterproductive with the worst-case situation leading to complete signal cancellation occurring. Therefore, interferometers such as the Michelson, which use a planar reference beam, are best used in the laboratory on optically mirror-like surfaces. For optically scattering surfaces, self-referential interferometers such as the adaptive interferometers based on two-wave mixing in photorefractive crystals are preferred. Dynamic Holographic Interferometers This class of interferometers is based on dynamic holographic recording typically in photorefractive media.

Photoacoustic Characterization of Materials

(a) creation of optical intensity gratings due to coherent interference of the interacting beams, leading to (b) nonuniform photoexcitation of electric charges in the PRC, which then diffuse/drift to create (c) a space-charge field within the PRC, which in turn creates (d) a refractive index grating via the electro-optical effect, and which causes (e) diffraction of the interacting beams A net consequence of this is that at the output of the PRC we have not only a portion of the transmitted probe beam, but also a part of the pump beam which is diffracted into the direction of the probe beam. The pump beam diffracted into the signal beam direction has the same wavefront structure as the transmitted signal beam. Since the PRC process has a certain response time (depending on the material, the applied electric field, and the total incident optical intensity), it turns out that it is unable to adapt to sufficiently high-frequency modulations in the signal beam. The PRC can only adapt to changes in the incident beams that are slower than the response time. This makes two-wave mixing in-

λ/2

PRC BC

λ/2

terferometers especially useful for ultrasound detection. High-frequency ultrasound-induced phase modulations are essentially not seen by the PRC, and therefore the diffracted pump beam will have the same wavefront structure as the unmodulated signal beam. The transmitted signal beam, however, obviously will contain the ultrasound-induced phase modulation. By interfering the diffracted (but unmodulated) pump beam with the transmitted (modulated) signal beam (both otherwise with the same wavefront structure) one obtains a highly efficient interferometer. Furthermore, any lowfrequency modulation in the interfering beams (such as those caused by noise from ambient vibration, or slow motion of the object) will be compensated for by the PRC as it adapts and creates a new hologram. Two-wave mixing interferometers therefore do not need any additional active stabilization against ambient noise. Several different types of photorefractive two-wave mixing interferometers have been described in the literature [27.40–42]. Here we will describe the isotropic diffraction configuration [27.44] shown in Fig. 27.20. For simplicity, optical activity and birefringence effects in the PRC will be neglected. Let the signal beam obtained from the scatter from the test object be s-polarized. As shown in Fig. 27.20, a half-wave plate (HWP) is used to rotate the incident signal beam polarization by 45◦ leading to both s- and p-polarized phase-modulated components of equal intensity given by a (27.58) E s0 = √ exp{i[ωopt t − φ(t)]} . 2 A photorefractive grating is created by the interference of the s-polarized component of the signal beam with the s-polarized pump beam. The diffracted pump beam (also s-polarized for this configuration of the PRC) upon exiting the crystal is then given by [27.44] a E r = √ exp(iωopt t) exp(−αL/2) 2 × {[exp(γ L) − 1] + exp[−iφ(t)]} , (27.59) BS

Signal beam PD 2 Pump beam

PD 1

Fig. 27.20 Configuration of isotropic

diffraction setup (λ/2: half-wave plate, PRC: photorefractive crystal, BC: Berek compensator, BS: beam splitter; PD: photodetector)

787

Part C 27.3

One approach is to planarize the speckled object beam by using optical phase conjugation [27.38, 39]. The planarized object beam can then be effectively interfered with a planar reference beam in a two-beam homodyne or heterodyne interferometer. Alternatively, a reference beam with the same speckle structure as the static object beam can be holographically reconstructed for interference with the object beam containing the ultrasonic information (Fig. 27.19b) [27.40–42]. This is most readily achieved by using the process of two-wave mixing in photorefractive media [27.43]. Two-wave mixing is essentially a dynamic holographic process in which two coherent optical beams (pump/reference and probe/signal beams) interact within a photorefractive crystal (PRC). The process of TWM can be briefly summarized as

27.3 Optical Detection of Ultrasound

788

Part C

Noncontact Methods

a)

Detection

beam splitter (PBS) oriented at 45◦ to the s- and pdirections, giving rise to two sets of optical beams that interfere at the two photodetectors. The intensities recorded at the two photodetectors are then given by (for φ(t)  π/2) Ptot −αL 2γr L + 2 e2γr L {cos(γi L − φL ) e e PD1 = 4 +φ(t) sin(γi L − φL ) + sin(γi L)}  + 2φ(t) sin φL + 1 Ptot −αL 2γr L PD2 = − 2 e2γr L {cos(γi L − φL ) e e 4 +φ(t) sin(γi L − φL ) − sin(γi L)}  − 2φ(t) sin φL + 1 , (27.61)

Source

25 m

3 1

Part C 27.3

R

R

30 mm

70 mm

b) 0.32

R

0.24 0.16

S1

0.08

P1

P3

0 –0.08 –0.16

RR

–0.24 10

20

30

40

50

60 Time (μm)

Fig. 27.21a,b Photoacoustically generated waves detected on an un-

polished aluminum block using a two-wave mixing interferometer. (a) Specimen configuration. (b) Optically detected ultrasonic signal. The vertical axis is displacement in nanometers (R: direct Rayleigh wave; RR: once reflected Rayleigh wave; P1: once reflected longitudinal wave; P3: three-times reflected longitudinal wave; S1: once reflected shear wave)

where γ = γr + iγi is the complex photorefractive gain, α is the intensity absorption coefficient of the crystal, and L is the length of the crystal. The p-polarized component of the signal beam is transmitted by the PRC undisturbed (except for absorption) and may be written as a E s = √ exp(−αL/2) exp{i[ωopt t − φ(t)]} . 2 (27.60)

Upon exiting the PRC, the diffracted pump and the transmitted signal beams are now orthogonally polarized. A Berek’s wave plate is interposed so as to introduce an additional phase shift of φL in the transmitted signal beam to put the interference at quadrature. The two beams are then passed through a polarizing

where Ptot = a2 is proportional to the optical power collected in the scattered object beam. In the case of a pure real photorefractive gain, quadrature is obtained by setting φL = π/2. In this case, upon electronically subtracting the two photodetector signals using a differential amplifier, the output signal is 

(27.62) S = Ptot e−αL eγ L − 1 ϕ(t) . Since the phase modulation φ(t) = 2kopt u(t) , the output signal is directly proportional to the ultrasonic displacement. The signal-to-noise ratio of the two-wave mixing interferometer in the isotropic configuration for real photorefractive gain can be shown to be [27.44] ( ηPtot − αL eγ L − 1 e 2  SNR = 2kopt U 1/2 . hvopt e2γ L + 1 (27.63)

It is clear that, the lower the absorption and the higher the photorefractive gain, the better the SNR. Minimum detectable sensitivities on the order of 10−15 m have been reported on optically scattering surfaces using dynamic holographic interferometers [27.42]. As an illustration, Fig. 27.21 shows the ultrasonic surface displacements monitored using a two-wave mixing interferometer on an unpolished aluminum block using a 500 mW laser source. The generation was using a photoacoustic point source using a 15 mJ Nd:YAG pulsed laser.

Photoacoustic Characterization of Materials

27.3.3 Practical Considerations: Systems for Optical Detection of Ultrasound

put together by an experienced engineer. Of greater interest are confocal Fabry–Pérot systems and the dynamic holographic interferometers, which work well on unpolished surfaces not only in the laboratory but also in industrial settings. These detection systems are commercially available from a number of sources.

27.4 Applications of Photoacoustics Photoacoustic methods have found wide-ranging applications in both industry and academic research. Here we will consider some illustrative applications in nondestructive flaw identification and materials characterization.

27.4.1 Photoacoustic Methods for Nondestructive Imaging of Structures Photoacoustic techniques have been used for nondestructive flaw detection in metallic and composite structures. Here we review a few representative example applications in flaw imaging using bulk waves, surface acoustic waves, and Lamb waves. Flaw Imaging Using Bulk Waves As discussed earlier, thermoelastic generation of bulk waves in the epicentral direction is generally weak in materials for which the optical penetration depth and thermal diffusion effects are small. Therefore, laser ultrasonic techniques using bulk waves have been used primarily for imaging of defects in composite structures (where the penetration depth is large), or on structures

that are coated with a sacrificial film that enhances epicentral generation. Recently, Zhou et al. [27.45] have developed efficient photoacoustic generation layers which they have used in conjunction with an ultrasonic imaging camera to image the interior of aluminum and composite structures (Fig. 27.22). Lockheed Martin has recently installed a large-scale laser ultrasonic facility for inspecting polymer-matrix composite structures in aircraft such as the joint strike fighter [27.46]. In this system, a pulsed CO2 laser was used to thermoelastically generate bulk waves into the composite part. A coaxial long-pulse Nd:YAG detection laser demodulated by a confocal Fabry–Pérot was used to monitor the back reflections of the bulk waves. The system has been demonstrated on prototype F-22 inlet ducts. Yawn et al. [27.46] estimate that the inspection time using the noncontact laser system is about 70 min as opposed to about 24 h for a conventional ultrasonic squirter system. Flaw Imaging Using Surface Acoustic Waves Photoacoustic systems can also be used to generate and detect surface acoustic waves on specimens such as the aircraft wheel shown in Fig. 27.23a,b. Here the high de-

Fig. 27.22 Imaging of subsurface features in aluminum structures using photoacoustically generated bulk waves. The subsurface features were 12.7 mm inside an aluminum block, and the feature size is on the order of 6.35 mm

789

Part C 27.4

Reference-beam interferometers, discussed in Sect. 27.3.2, are primarily laboratory tools which work well typically only on highly polished reflective specimens. These systems can be very easily

27.4 Applications of Photoacoustics

790

Part C

Noncontact Methods

a)

b) EDM notch 15 mm

11 mm

Part C 27.4

Laser generation Fiber-optic interferometer probe

c) Surface displacement (nm)

d) Surface displacement (nm)

0.15

0.15 Direct rayleigh

0.1

0.11 0.07

Reflection from crack

0.05 0.03 0

–0.05

–0.01

0

2

4

6

8

–0.05

10 Time (μs)

0

2

4

6

8

10 Time (μs)

e) Reflection coefficient 0.5 0.4 0.3 0.2 0.1 0 –0.1 –5

Actual crack position

0

5

10

15

20 25 Position (mm)

Fig. 27.23a–e Photoacoustic surface acoustic wave imaging. (a) Aircraft wheel part containing cracks. (b) Schematic of the scanning setup. Signal from locations (c) without a crack and (d) with a crack indicated by presence of reflections. (e) Reflection coefficient measured at different scanning locations of the wheel

gree of double curvature of the wheel makes the use of contact transducers difficult. Huang et al. [27.47] used

laser generation along with dual-probe heterodyne interferometer detection. The presence of cracks along

Photoacoustic Characterization of Materials

Tomographic Imaging Using Lamb Waves Tomographic imaging of plate structures using Lamb waves is often desired when the test area is not directly accessible and so must be probed from outside the area. Computer algorithms are used to reconstruct variations of a physical quantity (such as ultrasound attenuation) within a cross-sectional area from its integrated projection in all directions across that area. Photoacoustic tomographic systems using attenuation of ultrasound for tomographic reconstruction need to take into account the high degree of variability in the generated ultrasound arising from variation in the thermal absorption at different locations on the plate. A schematic of the setup is shown in Fig. 27.24a. Narrow-band Lamb waves were generated using an array of ther-

moelastic sources. Figure 27.24b shows a cross-section of the simulated corrosion defect produced in epoxybonded aluminum plates. The specimen is composed of two aluminum plates of thickness 0.65 mm and an approximately 13 μm-thick epoxy film. Corrosion was simulated by partially removing the surface of the bottom plate and inserting a fine nickel powder in the cavity prior to bonding. Figure 27.24c,d show typical narrow-band Lamb waves detected using the dual-probe interferometer in the presence and absence of the inc) Amplitude (mV) 0.2

0.15 0.1 0.05 0 –0.05 –0.1 –0.15 –0.2

0

5

10

15 Time (μs)

5

10

15 Time (μs)

d) Amplitude (mV) 0.2

0.15

a)

0.1

YAG Laser

0.05 Photodiode

0

Dual-probe fiber-optic interferometer

–0.05 –0.1

Defects Specimen

Oscilloscope

b)

Computer display

–0.15

GPIB

Line source Rotation and translation stages

5.5 mm 0.65 mm 0.65 mm

Aluminum plates

0.22 mm

Ni powder

Bonded together by epoxy film

Fig. 27.24a–e Photoacoustic Lamb-wave tomography. (a) Setup. (b) Cross-section of bonded thin plates con-

taining an inclusion. Typical Lamb-wave signals after band-pass filtering (c) without defect, (d) with the defect between the two detecting points. (e) Superposed image of the tomographic image (solid line) and a conventional ultrasonic C-scan image

–0.2

e)

0

791

Part C 27.4

the doubly curved location is indicated by the presence of reflected ultrasound signals in some locations but not in others (Fig. 27.23c,d). A reflection coefficient, calculated as the ratio of the reflected signal to the original signal power, is plotted in Fig. 27.23e and provides a measure of the length of the crack. Such pitch-catch approaches to detecting cracks using laser ultrasonics are feasible on cracks that are large enough to provide a sufficiently strong reflected signal.

27.4 Applications of Photoacoustics

792

Part C

Noncontact Methods

Part C 27.4

clusion. By tomographically scanning the plate, Nagata et al. [27.48] were able to create a tomographic image of the specimen, as shown in Fig. 27.24e. Also shown superposed in Fig. 27.24e is an ultrasonic C-scan image of the same sample obtained using a commercial scanning acoustic microscope. The size and the shape of the tomographically reconstructed image is consistent with that of the C-scan. a)

Scanning Laser Source Imaging of Surface-Breaking Flaws In the applications described thus far, photoacoustic methods have been used in a conventional pitch-catch ultrasonic inspection mode, except that lasers were used to generate and detect the ultrasound. For detecting very small cracks, the pitch-catch technique requires that the crack reflect a significant fraction of the in-

Laser Receiver source x2

Scanning

Crack

x1 x3

b) Interferometer signal (mV) 4

4

4

3

3

3

2

2

2

1

1

1

0

0

0

–1

–1

–1

–2

0

0.5

1

1.5

2

–2

2.5 3 Time (μs)

0.5

0

1

1.5

2

2.5 3 Time (μs)

–2

0

0.5

1

1.5

2

2.5 3 Time (μs)

c) Ultrasonic amplitude (mV) 6 II

5

Crack

4 I 3 III

2 1 0

0

1

2

3

4 5 6 SLS position (mm)

Fig. 27.25a–c The scanning laser source (SLS) technique. (a) Schematic of the SLS technique. (b) Ultrasonic surface normal displacement as a function of time (in μs) at three locations of the scanning laser source – left: far from the defect; center: close to the defect; and right: behind the defect. (c) Typical characteristic signature of ultrasonic peak-to-peak amplitude versus SLS location as the source is scanned over a surface-breaking defect

Photoacoustic Characterization of Materials

793

tering of the generated signal by the defect (zone III in Fig. 27.25c). The variation in signal amplitude is due to two mechanisms: (a) near-field scattering by the defect (b) changes in the conditions of generation of ultrasound when the SLS is in the vicinity of the defect As such, the SLS technique is not very sensitive to flaw orientation [27.50]. In addition to the amplitude signature shown above, spectral variations in the detected ultrasonic signal also show characteristic features [27.50]. Both amplitude and spectral variations can form the basis for an inspection procedure using the SLS technique. The SLS technique has been applied to detect small electric discharge machined notches in titanium engine disk blade attachment slots [27.49].

27.4.2 Photoacoustic Methods for Materials Characterization Photoacoustic metrology has been used to characterize the properties of materials ranging from the macroto the microscale. Here we describe applications to thin-film and coating characterization and to the determination of material anisotropy. Characterization of Material Anisotropy Photoacoustic methods have been used by a number of researchers to investigate the anisotropy of materials [27.51–55]. Both point- and line-focused laser Specimen

y Circular array receiver

r θ Point source

x

Fig. 27.26 Point source generation and multiplexed array detection of surface acoustic waves in anisotropic materials

Part C 27.4

cident wave, and furthermore that the generating and receiving locations be in line and normal to the crack. Recently, Kromine et al. [27.49, 50] have developed a scanning laser source (SLS) technique for detecting very small surface-breaking cracks that are arbitrarily oriented with respect to the generating and detecting directions. The SLS technique has no counterpart in conventional ultrasonic inspection methodologies as it relies on near-field scattering and variations in thermoelastic generation of ultrasound in the presence and absence of defects. In the SLS technique, the ultrasound generation source, which is a point- or line-focused high-power laser beam, is swept across the test specimen surface and passes over surface-breaking flaws (Fig. 27.25a). The generated ultrasonic wave packet is detected using an optical interferometer or a conventional contact piezoelectric transducer either at a fixed location on the specimen or at a fixed distance between the source and receiver. The ultrasonic signal that arrives at the Rayleigh wave speed is monitored as the SLS is scanned. Kromine et al. [27.50] and Sohn et al. [27.15] have shown that the amplitude and frequency of the measured ultrasonic signal have specific variations when the laser source approaches, passes over, and moves behind the defect. Kromine et al. [27.50] have experimentally verified the SLS technique on an aluminum specimen with a surface-breaking fatigue crack of 4 mm length and 50 μm width. A broadband heterodyne interferometer with 1–15 MHz bandwidth was used as the ultrasonic detector. The SLS was formed by focusing a pulsed Nd:YAG laser beam (pulse duration 10 ns, energy 3 mJ). The detected ultrasonic signal at three locations of the SLS position is presented in Fig. 27.25b. The Rayleigh wave amplitude as a function of the SLS position is shown in Fig. 27.25c. Several revealing aspects of the Rayleigh wave amplitude signature should be noted. In the absence of a defect or when the source is far ahead of the defect, the amplitude of the generated ultrasonic direct signal is constant (zone I in Fig. 27.25c). The Rayleigh wave signal is of sufficient amplitude above the noise floor to be easily measured by the laser detector (Fig. 27.25b), but the reflection is within the noise floor. As the source approaches the defect, the amplitude of the detected signal significantly increases (zone II in Fig. 27.25c). This increase is readily detectable even with a low sensitivity laser interferometer as compared to weak echoes from the flaw (Fig. 27.25b). As the source moves behind the defect, the amplitude drops lower than in zone I due to scat-

27.4 Applications of Photoacoustics

794

Part C

Noncontact Methods

Part C 27.4

sources have been used to generate the ultrasound. The ultrasound generated by a point laser source is typically detected by a point receiver and group velocity information is obtained at different angles. Doyle and Scala [27.56] have used a line-focused laser to determine the elastic constants of composite materials using the measured phase velocities of surface acoustic waves. Huang and Achenbach [27.57] used a line source and a dual-probe Michelson interferometer to provide accurate measurements of time of flight of SAWs on silicon. Zhou et al. [27.58] have recently used a multiplexed two-wave mixing interferometer with eight detection channels to provide group velocity slowness images. Figure 27.26 shows the configuration of the optical beams for anisotropic material characterization using a) θ = 90° θ = 80° θ = 70° θ = 60° θ = 50° θ = 40° θ = 30° θ = 20° θ = 10° θ = 0°

0.8

1

1.2

b) × 10– 4

1.4

1.6

1.8 Time (μs)

120

Slowness (ms/mm) 0.3

90

2.1

SAWs. In this setup, a pulsed Nd:YAG laser (pulse energy of approximately 1 mJ) is focused by a lens system onto the sample surface to generate the SAWs. The eight optical probe beams are obtained using a circular diffraction grating and focused onto the sample surface by a lens system to fall on a circle of radius r centered about the generation spot. The whole array of eight points was rotated every 2◦ to obtain the material anisotropy over the entire 360◦ range. Since a point source and point receiver configuration is used, the surface wave group velocity is obtained in this case. Figure 27.27a shows the time-domain SAW signals on (001) silicon from 0◦ to 90◦ . The group velocity slowness in each direction is obtained from the timedomain data through a cross-correlation technique and is shown in Fig. 27.27b where the filled circles are the experimental values. Also shown in Fig. 27.27b is the theoretical group velocity slowness calculated using nominal material values. The discontinuities that appear in both the experimental and the theoretical curves are due to the presence of pseudo-surface waves. Zhou et al. [27.58] have also obtained group velocity slowness curves on the (0001) surface of a block of quartz. The time-domain traces and the corresponding group velocity slowness curves are shown in Fig. 27.28. The multiple pulses observed are due to a combination of the presence of SAWs and pseudo-SAWs as well as the energy folding that occurs in anisotropic materials such as (0001) quartz. The group velocity slowness curves obtained experimentally can be further processed to ob-

60

Experiment Theory

0.2

2 150

30

0.1

1.9 0 1.8 180

0

–0.1 1.9 210

330

–0.2

2 –0.3 2.1

240

300 270

Fig. 27.27 (a) SAW signals detected in different directions on z-cut silicon. (b) Slowness curve for z-cut silicon

–0.3 –0.2 –0.1

0

0.1 0.2 0.3 Slowness (ms/mm)

Fig. 27.28 Group velocity slowness of z-cut quartz

Photoacoustic Characterization of Materials

tain the anisotropic material constants as described by Castagnede et al. [27.59].

PZT driver

Attenuator 780 nm femtosecond laser

λ/2

532 nm CW laser

Aperture

PBS

λ/4

Stabilizer

PZT BS

Sample Optical fibers Oscilloscope

BPD

Fig. 27.29 Guided-wave photoacoustic setup (BS: beam splitter, PBS: polarized beam splitter, BPD: balanced photodetector, λ/4: quarter-wave plate, λ/2: half-wave plate, PZT: piezoelectric mounted mirror)

of hundreds of nanometers, followed by a Cr adhesion metallic layer with thickness about 100 nm right on the steel substrate. The properties of the interpolated layer are taken to be the average of those of the DLC and the metallic layers. The transfer matrix method described in Sect. 27.1.3, was used to obtain the theoretical guided-wave dispersion curves. To derive the mechanical properties, an inverse problem has to be solved to calculate the parameters from the measured velocity dispersion curve v( f ). A nonlinear regression method is used to minimize the least-square error function y=

N 2 1 theo v − vmeas , N

795

(27.64)

i=1

where vmeas are the measured velocities, and vtheo are the theoretically calculated velocities, which are functions of the mechanical properties of each layer. A simplex method of least-square curve fitting is useful for fitting a function of more than one variable [27.64]. The thicknesses of the coatings were separately measured and used in the calculation. The reliability of the results mainly depends on the accuracy of the experiments, the choice of initial parameters, and the number of fitted variables. The Young’s modulus, Poisson’s ratio, and density of the DLC coating layer were set as variables to be determined in the iteration. Repetitive fitting showed a variation of up to 5% for the fitted values of Young’s moduli and densities.

Part C 27.4

Characterization of the Mechanical Properties of Coatings The small footprint and noncontact nature of photoacoustic methods make them especially useful for characterizing coatings. Several optical techniques have been devised and implemented. A pump-probe technique has been used in which very high-frequency (GHz) acoustic waves are generated that propagate perpendicular to the film and reflect off of the film/substrate interface [27.26, 31]. This bulk wave technique requires an ultrafast laser source, and material attenuation of high-frequency ultrasound limits the useful measurement range to reasonably thin films. For thicker films, guided-wave ultrasonic techniques are more practical. The impulsive stimulated thermal scattering (ISTS) [27.60] technique and the phase velocity scanning (PVS) [27.61] technique both use a spatially periodic irradiance pattern to generate single-frequency surface acoustic wave (SAW) tone bursts which are detected through probe-beam diffraction, interferometry, or contact transducers. Broadband techniques [27.19, 62] can also be used where SAWs are generated with a simple pulsed laser point or line source which are then detected with an interferometer after some propagation distance along the film. Ultrahard coatings such as diamond, diamond-like carbon (DLC), cubic BN, etc., are of great interest due to their unique mechanical, thermal, and electrical properties. In particular, the high hardness and stiffness of diamond-like thin films make them excellent coating materials for tribological applications. Unfortunately, the coating properties are highly sensitive to the processing parameters, and photoacoustic methods provide a way to measure the properties of these coatings nondestructively. Figure 27.29 shows the guided-wave photoacoustic setup used for characterizing multilayer Cr-DLC specimens. A pulsed laser was line-focused to a line width of 10 μm on the surface of the specimen to generate broadband guided acoustic waves. The acoustic waves were monitored by a stabilized balanced Michelson interferometer. By monitoring the guided waves at multiple source to receiver locations (Fig. 27.30a), the dispersion curves for these waves were obtained, as shown in Fig. 27.30b. To interpret the measurements, the multilayer DLC specimen was modeled as a three-layer system [27.63]. Below the top layer of DLC, there is an interpolated transition layer of Cr and DLC with various thickness

27.4 Applications of Photoacoustics

796

Part C

Noncontact Methods

Part C 27.4

The experimentally measured dispersion curves for several Cr-DLC coatings are shown in Fig. 27.30b as dots. The solid lines in Fig. 27.30b are the theoretically fitted dispersion curves as described above for the inverse problem. The good agreement indicates that material properties can be extracted with a high degree of confidence. Note that significant variations in DLC a) Signal amplitude (arb. units) 0.2 0.1 0 –0.1 –0.2 –0.3

3

3.2

3.4

3.6

3.8

0.1

0

–0.1

–0.2 5

5.2

5.4

5.6

5.8

6

Time (μs)

b) Phase velocity (m/s) 3060 (A) Cr-DLC specimens #1 #2 #3 #4 #5 #6

3040

3020

properties can be obtained, arising from fabrication process variations [27.63]. Photoacoustic characterization of such coatings is therefore very useful in process control applications to assure quality. Characterization of the Mechanical Properties of Thin Films Photoacoustic techniques can also be used to characterize the properties of free-standing nanometer-sized thin films [27.36, 65]. Such thin films are widely used in micro-electromechanical systems (MEMS) devices such as radiofrequency (RF) switches, pressure sensors, and micromirrors. Described below are guided-wave and bulk-wave photoacoustic methods for characterizing free-standing thin films. Two-layer thin films of Al/Si3 N4 were fabricated on a standard Si wafer using standard microfabrication processes [27.65]. The film thicknesses were in the range of hundreds of nanometers. Several windows were etched in the wafer to provide unsupported membranes of Al/Si3 N4 which were only edge supported (Fig. 27.31). In a first set of experiments, bulk-wave photoacoustic measurements were made. A standard pump-probe optical setup operated at 780 nm using a Ti:sapphire femtosecond laser (100 fs pulse duration, 80 MHz repetition rate) was used in this work. Both the substrate-supported and unsupported region were measured as, indicated in Fig. 27.31. Figure 27.32 shows the normalized measured pump-probe signals for the thin films on the silicon substrate supported region. As shown in Fig. 27.32, the time of flight of the first ultrasonic echo reflected from the first Al/Si3 N4 interface is marked as τ1 and that of the echo from the Si3 N4 /Si interface is marked as τ2 . To deduce Pump pulses Probe pulses

3000

Al Si3N4

2980

GW

0 d1 d2

BW

Si 2960 0

50

100

Frequency (MHz)

Fig. 27.30a,b Guided-wave photoacoustic measurement of Cr-DLC coatings. (a) Broadband SAW signals detected at two source–receiver locations. (b) SAW dispersion

curves

x

150

Michelson interferometer

Fig. 27.31 Bulk-wave and guided-wave photoacoustic

characterization of free-standing thin films: schematic diagram of the compact optical setup for both bulk-wave and guided-wave detection

Photoacoustic Characterization of Materials

0.2

2

Short source –receiver distance

0.1

τ1

τ2 0 5 4

1

3

0.5

2 1 0

50

100

150

200

250 300 350 Delay time (ps)

Fig. 27.32 The experimental transient thermoreflectivity

Part C 27.4

Sample 6

0

797

a) Relative amplitude (arb. units)

Reflection coefficient change (arb. units)

1.5

27.4 Applications of Photoacoustics

– 0.1 – 0.2 – 0.3 Long source – receiver distance

– 0.4 0.2

0.8

1

1.2 1.4 Time (μs)

1.6 Specimen #1 Specimen #2 Specimen #3 Specimen #4

1.4

where h is the thickness, ρ is the density of the film, and σ is the in-plane stress in the film (typically caused by residual stress). The flexural rigidity D is related to the Young’s modulus E, Poisson’s ratio ν, and the geometry of the film.

0.6

b) (phase velocity υa0)2 (km2/s2)

signals for double-layer thin films

the elastic moduli E accurately, the theoretical simulated transient thermoelastic signals (Sect. 27.2.2) with various moduli were calculated and compared with the experimental signals, and the moduli that give the smallest error for the time of flight of acoustic echoes were determined iteratively. In general, the measured Young’s modulus of the aluminum layer falls in the range 47–65 GPa. The experimentally determined Young’s moduli of silicon nitride films range from about 220 to 280 GPa and are in good agreement with the Young’s modulus of low-pressure chemical vapor deposition (LPCVD)-fabricated silicon nitride reported in the literature. The same set of specimens was also tested using guided-wave photoacoustics. Only the lowest-order antisymmetric Lamb-wave mode was efficiently generated and detected in such ultrathin films. For ultrathin films such as these, for small wavenumbers, a simple expression for the acoustic phase velocity of A0 mode in terms of the wavenumber k can be derived [27.65]: ( D 2 σ ω (27.65) k + , v A0 = = k ρh ρ

0.4

1.2 1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4 (k h)2

Fig. 27.33a,b Guided-wave photoacoustic characterization of two-layer ultrathin films. (a) Measured signals at two source-to-receiver locations for thin-film sample #1, and (b) measured dispersion curves (dots) and linear fitted curves (lines) for samples #1–#4. The average measured flexural rigidities are 4.82, 5.82, 3.64, 1.76 × 10−9 Nm; and the residual stresses are 235, 299, 334, 242 MPa, respectively

Figure 27.33a shows typical measured time traces of the A0 mode at two source-to-receiver positions. Figure 27.33b shows the experimentally determined A0 mode dispersion curves for four specimens. From the figure, it is clear that (27.65) represents the measurements well, thereby enabling direct determination of the residual stresses in the film [27.65].

798

Part C

Noncontact Methods

27.5 Closing Remarks

Part C 27

In this chapter, only a selective review of photoacoustic metrology as applied to mechanical characterization of solids has been provided. Applications of photoacoustic methods of course extend well beyond the ones discussed here. Photoacoustic spectroscopy is a wellestablished set of methods for the characterization of the composition of condensed and gaseous matter [27.66]. Recent developments in biomedical photoacoustics promise new methods of characterization

and imaging of tumors and blood vessels [27.67, 68]. Photoacoustic methods are also being used to characterize nanostructures such as superlattices [27.69] and nanoparticles [27.70]. The photoacoustic phenomenon, which Alexander Graham Bell originally investigated, possibly with applications to telephony in mind, has since yielded several powerful metrology tools for both the laboratory researcher and for industrial applications.

References 27.1

27.2 27.3

27.4

27.5

27.6

27.7 27.8 27.9 27.10

27.11

27.12

27.13

27.14

27.15

C.B. Scruby, L.E. Drain: Laser Ultrasonics: Techniques and Application (Adam Hilder, New York 1990) V.E. Gusev, A.A. Karabutov: Laser Optoacoustics (American Institute of Physics, New York 1993) D.A. Hutchins: Ultrasonic Generation by Pulsed Lasers. In: Physical Acoustics, Vol. XVIII, ed. by W.P. Mason, R.N. Thurston (Academic, New York 1988) pp. 21–123 J.P. Monchalin: Optical detection of ultrasound, IEEE Trans. Ultrason. Ferroelectr. Frequ. Control, 33(5), 485–499 (1986) J.W. Wagner: Optical Detection of Ultrasound. In: Physical Acoustics, Vol. XIX, ed. by R.N. Thurston, A.D. Pierce (Academic, New York 1990) R.J. Dewhurst, Q. Shan: Optical remote measurement of ultrasound, Meas. Sci. Technol. 10, R139–R168 (1999) J.D. Achenbach: Wave Propagation in Elastic Solids (North-Holland/Elsevier, Amsterdam 1973) B.A. Auld: Acoustic Fields and Waves in Solids (Krieger, Malabar 1990) D. Royer, E. Dieulesaint: Elastic Waves in Solids (Springer, New York 1996) G.W. Farnell: Properties of Elastic Surface Waves. In: Physical Acoustics, Vol. VI, ed. by W.P. Mason, R.N. Thurston (Academic, New York 1970) G.W. Farnell, E.L. Adler: Elastic Wave Propagation in Thin Layers. In: Physical Acoustics, Vol. IX, ed. by W.P. Mason, R.N. Thurston (Academic, New York 1974) W.T. Thomson: Transmission of Elastic Waves through a Stratified Solid Medium, J. Appl. Phys. 21, 89–93 (1950) N.A. Haskell: The dispersion of surface waves on multilayered media, Bull. Seism. Soc. Am. 43(1), 17–34 (1953) R.M. White: Generation of elastic waves by transient surface heating, J. Appl. Phys. 24(12), 3559–3567 (1963) Y. Sohn, S. Krishnaswamy: Interaction of a scanning laser-generated ultrasonic line source with

27.16

27.17

27.18

27.19

27.20

27.21

27.22

27.23

27.24

27.25

27.26

27.27

a surface-breaking flaw, J. Acoust. Soc. Am. 115(1), 172–181 (2004) A.M. Aindow, D.A. Hutchins, R.J. Dewhurst, S.B. Palmer: Laser-generated ultrasonic pulses at free metal-surfaces, J. Acoust. Soc. Am. 70, 449– 455 (1981) R.J. Dewhurst, C. Edwards, A.D.W. McKie, S.B. Palmer: Estimation of the thickness of thin metal sheet using laser generated ultrasound, Appl. Phys. Lett. 51, 1066–1068 (1987) J. Spicer, A.D.W. McKie, J.W. Wagner: Quantitative theory for laser ultrasonic waves in a thin plate, Appl. Phys. Lett. 57(18), 1882–1884 (1990) T.W. Murray, S. Krishnaswamy, J.D. Achenbach: Laser generation of ultrasound in films and coatings, Appl. Phys. Lett. 74(23), 3561–3563 (1999) T.W. Murray, J.W. Wagner: Laser generation of acoustic waves in the ablative regime, J. Appl. Phys. 85(4), 2031–2040 (1999) F.A. McDonald: Practical quantitative theory of photoacoustic pulse generation, Appl. Phys. Lett. 54(16), 1504–1506 (1989) J. Spicer: Laser Ultrasonics in Finite Structures: Comprehensive Modeling with Supporting Experiments. Ph.D. Thesis (The Johns Hopkins University, Baltimore 1991) I. Arias: Modeling of the Detection of Surface Breaking Cracks by Laser Ultrasonics. Ph.D. Thesis (Northwestern University, Evanston 2003) C.B. Scruby, R.J. Dewhurst, D. Hutchins, S. Palmer: Quantitative studies of thermally-generated elastic wave in laser irradiated solids, J. Appl. Phys. 51, 6210–6216 (1980) L. Rose: Point-source representation for laser generated ultrasound, J. Acoust. Soc. Am. 75(3), 723–732 (1984) C. Thomsen, H.T. Grahn, H.J. Maris, J. Tauc: Surface generation detection of phonons by picosecond light-pulses, Phys. Rev. B 34, 4129–4138 (1986) C.J.K. Richardson, M.J. Ehrlich, J.W. Wagner: Interferometric detection of ultrafast thermo-elastic

Photoacoustic Characterization of Materials

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27.29

27.31

27.32

27.33

27.34

27.35

27.36

27.37 27.38

27.39

27.40

27.41

27.42

27.43

27.44

27.45

27.46

27.47

27.48

27.49

27.50

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27.54

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P. Delaye, A. Blouin, D. Drolet, L.A. Montmorillon, G. Roosen, J.P. Monchalin: Detection of ultrasonic motion of a scattering surface by photorefractive InP:Fe under an applied dc field, J. Opt. Soc. Am. B 14(7), 1723–1734 (1997) Y. Zhou, G. Petculescu, I.M. Komsky, S. Krishnaswamy: A high-resolution, real-time ultrasonic imaging system for NDI applications, SPIE, Vol. 6177 (2006) K.R. Yawn, T.E. Drake, M.A. Osterkamp, S.Y. Chuang, D. Kaiser, C. Marquardt, B. Filkins, P. Lorraine, K. Martin, J. Miller: Large-Scale Laser Ultrasonic Facility for Aerospace Applications. In: Review of Progress in Quantitative Nondestructive Evaluation, Vol. 18, ed. by D.O. Thompson, D.E. Chimenti (Kluwer Academic/Plenum, New York 1999) pp. 387–393 J. Huang, Y. Nagata, S. Krishnaswamy, J.D. Achenbach: Laser based ultrasonics for flaw detection, IEEE Ultrasonic Symposium, ed. by M. Levy, S.C. Schneider (IEEE, New York 1994) pp. 1205– 1209 Y. Nagata, J. Huang, J.D. Achenbach, S. Krishnaswamy: Computed Tomography Using LaserBased Ultrasonics. In: Review of Progress in Quantitative Nondestructive Evaluation, Vol. 14, ed. by D.O. Thompson, D.E. Chimenti (Plenum, New York 1995) A. Kromine, P. Fomitchov, S. Krishnaswamy, J.D. Achenbach: Scanning Laser Source Technique and its Applications to Turbine Disk Inspection. In: Review of Progress in Quantitative Nondestructive Evaluation, Vol. 18A, ed. by D.O. Thompson, D.E. Chimenti (Plenum, New York 1998) pp. 381– 386 A.K. Kromine, P.A. Fomitchov, S. Krishnaswamy, J.D. Achenbach: Laser ultrasonic detection of surface breaking discontinuities: Scanning laser source technique, Mater. Eval. 58(2), 173–177 (2000) A.A. Maznev, A. Akthakul, K.A. Nelson: Surface acoustic modes in thin films on anisotropic substrates, J. Appl. Phys. 86(5), 2818–2824 (1999) D.C. Hurley, V.K. Tewary, A.J. Richards: Surface acoustic wave methods to determine the anisotropic elastic properties of thin films, Meas. Sci. Technol. 12(9), 1486–1494 (2001) B. Audoin, C. Bescond, M. Deschamps: Measurement of stiffness coefficients of anistropic materials from pointlike generation and detection of acoustic waves, J. Appl. Phys. 80(7), 3760–3771 (1996) A. Neubrand, P. Hess: Laser generation and detection of surface acoustic waves: elastic properties of surface layers, J. Appl. Phys. 71(1), 227–238 (1992) A.G. Every, W. Sachse: Determination of the elastic constants of anisotropic solids from acoustic-wave group-velocity measurements, Phys. Rev. B 42, 8196–8205 (1990)

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transients in thin films: theory with supporting experiment, J. Opt. Soc. Am. B 16, 1007–1015 (1999) Z. Bozoki, A. Miklos, D. Bicanic: Photothermoelastic transfer matrix, Appl. Phys. Lett. 64, 1362–1364 (1994) A. Miklos, Z. Bozoki, A. Lorincz: Picosecond transient reflectance of thin metal films, J. Appl. Phys. 66, 2968–2972 (1989) H.T. Grahn, H.J. Maris, J. Tauc: Picosecond ultrasonics, IEEE J. Quantum Electron. 25(12), 2562–2569 (1989) C. Thomsen, H.T. Grahn, H.J. Maris, J. Tauc: Ultrasonic experiments with picosecond time resolution, J. Phys. Colloques. C10 46, 765–772 (1985) P.A. Fleury: Light scattering as a probe of phonons and other excitations. In: Physical Acoustics, Vol. VI, ed. by W.P. Mason, R.N. Thurston (Academic, New York 1970) G.A. Antonelli, H.J. Maris, S.G. Malhotra, M.E. Harper: Picosecond ultrasonics study of the vibrational modes of a nanostructure, J. Appl. Phys. 91(5), 3261–3267 (2002) R. Adler, A. Korpel, P. Desmares: An instrument for making surface waves visible, IEEE Trans. Sonics. Ulrason. SU-15, 157–160 (1967) E.G.H. Lean, C.C. Tseng, C.G. Powell: Optical probing of acoustic surface-wave harmonic generation, Appl. Phys. Lett. 16(1), 32–35 (1970) J.A. Rogers, G.R. Bogart, R.E. Miller: Noncontact quantitative spatial mapping of stress and flexural rigidity in thin membranes using a picosecond transient grating photoacoustic technique, J. Acoust. Soc. Am. 109, 547–553 (2001) D. Malacara: Optical Shop Testing (Wiley, New York 1992) M. Paul, B. Betz, W. Arnold: Interferometric detection of ultrasound from rough surfaces using optical phase conjugation, Appl. Phys. Lett. 50, 1569–1571 (1987) P. Delaye, A. Blouin, D. Drolet, J.P. Monchalin: Heterodyne-detection of ultrasound from rough surfaces using a double-phase conjugate mirror, Appl. Phys. Lett. 67, 3251–3253 (1995) R.K. Ing, J.P. Monchalin: Broadband optical detection of ultrasound by two-wave mixing in a photorefractive crystal, Appl. Phys. Lett. 59, 3233–3235 (1991) P. Delaye, L.A. de Montmorillon, G. Roosen: Transmission of time modulated optical signals through an absorbing photorefractive crystal, Opt. Commun. 118, 154–164 (1995) B.F. Pouet, R.K. Ing, S. Krishnaswamy, D. Royer: Heterodyne interferometer with two-wave mixing in photorefractive crystals for ultrasound detection on rough srufaces, Appl. Phys. Lett. 69, 3782–2784 (1996) P. Yeh: Introduction to Photorefractive Nonlinear Optics (Wiley, New York 1993)

References

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P.A. Doyle, C.M. Scala: Toward laser-ultrasonic characterization of an orthotropic isotropic interface, J. Acoust. Soc. Am. 93(3), 1385–1392 (1993) J. Huang, J.D. Achenbach: Measurement of material anisotropy by dual-probe laser interferometer, Res. Nondestr. Eval. 5, 225–235 (1994) Y. Zhou, T.W. Murray, S. Krishnaswamy: Photoacoustic imaging of surface wave slowness using multiplexed two-wave mixing interferometry, IEEE UFFC 49(8), 1118–1123 (2002) B. Castagnede, K.Y. Kim, W. Sachse, M.O. Thompson: Determination of the elastic constants of anistropic materials using laser-generated ultrasonic signals, J. Appl. Phys. 70(1), 150–157 (1991) A.R. Duggal, J.A. Rogers, K.A. Nelson: Real-time optical characterization of surface acoustic modes of polyimide thin-film coatings, J. Appl. Phys. 72(7), 2823–2839 (1992) A. Harata, H. Nishimura, T. Sawada: Laser-induced surface acoustic-waves and photothermal surface gratings generated by crossing two pulsed laserbeams, Appl. Phys. Lett. 57(2), 132–134 (1990) D. Schneider, T. Schwartz, A.S. Bradfordm, Q. Shan, R.J. Dewhurst: Controlling the quality of thin films by surface acoustic waves, Ultrasonics 35, 345–356 (1997) F. Zhang, S. Krishnaswamy, D. Fei, D.A. Rebinsky, B. Feng: Ultrasonic characterization of mechanical

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properties of diamond-like carbon hard coatings, Thin Solid Films 503, 250–258 (2006) J.D. Achenbach, J.O. Kim, Y.C. Lee: Acoustic microscopy measurement of elastic-constants and mass density. In: Advances in Acoustic Microscopy, ed. by A. Briggs (Plenum, New York 1995) C.M. Hernandez, T.W. Murray, S. Krishnaswamy: Photoacoustic characterization of the mechanical properties of thin films, Appl. Phys. Lett. 80(4), 691–693 (2002) A. Rosencwaig: Photoacoustics and photoacoustic spectroscopy (Wiley, New York 1980) R.G.M. Kolkman, E. Hondebrink, W. Steenburgen, F.F.M. de Mul: In vivo photoacoustic imaging of blood vessels using an extreme-narrow aperture sensor, IEEE J. Sel. Top. Quantum Electron. 9(2), 343–346 (2003) V.G. Andreev, A.A. Karabutov, A.A. Oraevsky: Detection of ultrawide-band ultrasound pulses in optoacoustic tomography, IEEE Trans. Ultrason, Ferroelectr. Frequ. Control 50(10), 1183–1390 (2003) N.W. Pu: Ultrafast excitation and detection of acoustic phonon modes in superlattices, Phys. Rev. B 72, 115428 (2005) M. Nisoli, S. de Silvestri, A. Cavalleri, A.M. Malvezzi, A. Stella, G. Lanzani, P. Cheyssac, R. Kofman: Coherent acoustic oscillation in metallic nanoparticles generated with femtosecond optical pulses, Phys. Rev. B 55, R13424 (1997 )

801

X-Ray Stress 28. X-Ray Stress Analysis

Jonathan D. Almer, Robert A. Winholtz

28.1 Relevant Properties of X-Rays ................ 28.1.1 X-Ray Diffraction........................ 28.1.2 X-Ray Attenuation...................... 28.1.3 Fluorescence .............................

802 802 802 803

28.2 Methodology ........................................ 28.2.1 Measurement Geometry .............. 28.2.2 Biaxial Analysis .......................... 28.2.3 Triaxial Analysis ......................... 28.2.4 Determination of Diffraction Peak Positions ........

804 805 805 805

28.3 Micromechanics of Multiphase Materials. 28.3.1 Macrostresses and Microstresses... 28.3.2 Equilibrium Conditions ............... 28.3.3 Diffraction Elastic Constants......... 28.4 Instrumentation ................................... 28.4.1 Conventional X-Ray Diffractometers .......................... 28.4.2 Special-Purpose Stress Diffractometers .......................... 28.4.3 X-Ray Detectors ......................... 28.4.4 Synchrotron and Neutron Facilities 28.5 Experimental Uncertainties ................... 28.5.1 Random Errors ........................... 28.5.2 Systematic Errors ........................ 28.5.3 Sample-Related Issues................ 28.6 Case Studies ......................................... 28.6.1 Biaxial Stress ............................. 28.6.2 Triaxial Stress ............................ 28.6.3 Oscillatory Data Not Applicable to the Classic Model.................... 28.6.4 Synchrotron Example: Nondestructive, Depth-Resolved Stress........................................ 28.6.5 Emerging Techniques and Studies

807 807 808 808 809 809 810 810 810 810 811 811 812 813 813 814 815

815 816

806

28.7 Summary ............................................. 817 28.8 Further Reading ................................... 818 References .................................................. 818

X-rays are an important tool for measuring stresses, particularly residual stresses, in crystalline materials. x-ray stress measurements are used to help solve material failure problems, check quality control, verify computational results, and for fundamental materials research. X-ray diffraction can be used to precisely determine the distance between planes of atoms in crystalline materials through the measurement of peak positions. These positions can be used to determine elastic strains

in each crystalline phase of the material. These strains can then be converted to stresses using appropriate elastic constants. Plastic deformation can be detected through changes in diffraction peak widths rather than peak shifts. Laboratory-based x-ray sources typically penetrate only a few tens of microns into common materials, yielding stresses averaged over the near-surface region. Deeper depths can be accessed using destructive layer-removal methods, with appropriate corrections. Alternatively, more highly penetrating neutrons or high-

Part C 28

X-ray diffraction is a powerful non-destructive technique capable of measuring elastic strain in all crystalline phases of a material, which can be converted to stress using appropriate elastic constants. With laboratory sources typical penetration depths are on the micron-level, and deeper depths can be evaluated using (destructive) layer removal methods or higher-energy x-rays (esp. from synchrotron sources) or neutrons. Diffraction also provides complementary information on crystallographic texture and plastic deformation. Potential sources of errors in stress measurements are outlined. Finally, some case studies and emerging techniques and studies in this field are highlighted.

802

Part C

Noncontact Methods

energy x-rays [28.1, 2]. can be used to examine the interior of components nondestructively. The subject of stress measurement with x-ray diffraction has been treated in great depth by Noyan and Cohen [28.3] and more recently by Hauk [28.4]. In addition, related reports have been presented by the Society for Automotive Engineers [28.5], the Cen-

tre Technique de Industries Méchaniques [28.6], the American Society for Metals [28.7], and most recently as a United Kingdom good practice guide [28.8]. Here we provide an updated account of the subject, including case studies from both laboratory- and synchrotron-based experiments, together with a summary of emerging studies in the field.

28.1 Relevant Properties of X-Rays Part C 28.1

In this section, several properties of x-rays relevant to diffraction stress analysis are summarized.

28.1.1 X-Ray Diffraction X-rays provide a means of examining the atomic-scale structure of materials because their wavelengths are similar to the size of atoms. The measurement of lattice spacings in crystalline materials with x-rays utilizes diffraction, the constructive interference of x-ray waves scattered by the electrons in the material. If the crystal is oriented properly with respect to the incident x-ray beam, constructive interference can be established in particular directions, giving rise to a diffracted x-ray beam. Constructive interference arises when the path length for successive planes in the crystal equals an integral number of wavelengths. This condition is summarized with Bragg’s law nλ = 2d sin θ .

(28.1)

Here, λ is the wavelength of the x-rays, d is the interplanar spacing in the crystal, θ is the diffraction angle, and n is the number of wavelengths in the path difference between successive planes. In practice, n is eliminated from (28.1) by setting it equal to 1 and considering the diffraction peaks arising with n > 1 to be occurring from higher-order lattice planes, i. e., diffraction from (100) planes with n = 4 are considered as arising from the (400) planes with n = 1. The interplanar spacing or d-spacing in the crystal is measured along a direction that bisects the incident and diffracted x-ray beams. A vector in this direction with length 2 sin θ/λ is termed the diffraction vector, as shown in Fig. 28.1. Equation (28.1) can be used in two ways to determine d-spacings in materials. First, a nominally monochromatic x-ray beam can be used and the diffraction angle θ precisely measured. Second, a white or polychromatic x-ray beam can be incident on the specimen and the wavelengths scattered at a particular angle

measured. This is usually done with a solid-state detector that can precisely measure the energy of detected photons using the wavelength–energy relationship for electromagnetic radiation hc (28.2) , E where h is Planck’s constant, c is the speed of light, and E is the energy of the photons. Substituting numerical values for the constants, if λ is in Angstroms and E is in keV, the wavelength energy relation becomes λ=

λ=

12.39842 . E

(28.3)

28.1.2 X-Ray Attenuation As x-rays pass through matter they are attenuated. This is an important consideration in measuring stresses with x-rays. For typically used reflection geometries, the attenuation defines the depth being sampled in a stress measurement while, for transmission geometries, it will define the thickness of a specimen that can be examined. An x-ray beam of intensity I0 passing through a thickness t of a material will be attenuated to an intensity I given by I = I0 exp(−μl t) ,

(28.4)

where μl is the linear attenuation coefficient for the material. The linear attenuation coefficient can be computed for a sample having N elements using μl = ρ

N 

wi (μm )i ,

(28.5)

i=1

where ρ is the sample density and wi and μm,i are, respectively, the weight fraction and mass attenuation coefficient of element i. The μm are inherent properties

X-Ray Stress Analysis

a)

28.1 Relevant Properties of X-Rays

803

b) X3 Ψ

X3' σ

Diffraction vector X1

θ

Ψ

X2

c)

Ψ

Diffraction vector

Fig. 28.1 (a) Coordinate systems used in diffraction stress measurement, showing the specimen coordinates (X i ) and laboratory coordinates (X i ) and the manner in which the specimen is rotated to orient the diffraction vector along the laboratory X 3 axis; definition of tilt angle ψ for (b) Ω-gonio-metry and (c) Ψ -goniometry, with the in-plane stress direction indicated

of a given element and commonly tabulated. Between absorption edges (which lead to fluorescence, see below) these values vary with wavelength λ and elemental atomic number Z as μm ∝ Z 4 λ3 ,

(28.6)

which when combined with (28.2) indicates that absorption decreases considerably with x-ray energy. Table 28.1 lists μl values for selected materials and common laboratory x-ray energies. Also tabulated is the inverse value τ1/e , which represents the x-ray path length at which I = I0 /e ≈ 0.37I0 , and which is seen to be in the micron range. The actual penetration depth into the sample depends on the diffraction geometry used, and this is covered further in Sect. 28.2.1.

28.1.3 Fluorescence When the incident x-ray energy is higher than the (element-specific) energy required to remove an electron (e.g., K-edge electron) from its shell, the electron

will be ejected. This ejected electron is called a photoelectron and the emitted characteristic radiation is called fluorescent radiation. For strain measurements, fluorescence is unwanted as it increases x-ray background levels, thereby decreasing the signal-to-background ratio and reducing accuracy in peak position determination. The fluorescence signal can be eliminated by using x-ray energies below the fluorescence energies of the elements comprising the sample, though this is not always practical. For tube sources, containing both continuous (bremsstrahlung) radiation and characteristic (Kβ and Kα ) radiation, the fraction of higher-energy (fluorescence-forming) x-rays can be reduced relative to the signal-forming (typically Kα ) x-rays through the use of an incident-beam energy filter in the form of a metal or oxide foil or monochromator. Alternatively, fluorescence signals can be reduced by the use of a diffracted-beam energy filter (foil or monochromator) and/or through use of an energy-discriminating (e.g., solid-state) detector.

Part C 28.1

σ

804

Part C

Noncontact Methods

Table 28.1 X-ray penetration depths, common reflection/radiation combinations for stress measurements, and elastic properties for selected materials Material (structure)

μ (1/mm) and (τ(1/e), μm) Mo–Kα Cu–Kα λ = 0.71 Å λ = 1.54 Å

Cr–Kα λ = 2.29Å

14.3

132

397

(69.9)

(7.6)

(2.5)

Al (fcc)

Elastic aniso- a tropy

1.2

310

2547

904

(3.3)

(0.4)

(1.1)

Ferrite (bcc)

2.3

Part C 28.2

311

2633

934

(3.2)

(0.4)

(1.1)

Austenite (fcc)

3.3

hkl b

2θ (deg)b

111

53.8

200

49.9

311 (Cr) 200

139.0

52.1 141.7

211 (Cr)

156.0

178.7

732 (Mo) 111

154.5

163.2 209.8

200

124.4

420 (Cu) 111 422

439

1291

(2.4)

(2.3)

(0.8)

Nickel (fcc)

2.5

107

919

2718

(9.4)

(1.1)

(0.4)

-

b c d

155.4

200

52.3

163.6

145.7

152.1

164.0 90.9 83.0

139.5

E/(1 + ν) bulk (GPa) d

142.3 219.4 144.2

213 (Cu) a

147.0

200 420 (Cu) 002

Titanium (hcp)

E/(1 + ν) Hill (GPa) c

87.6

83.6

Anisotropy is calculated for cubic materials as 2c44 /(c11 − c12 ), where cij are the elastic constants from [28.9] and [28.10]. Suggested reflection/radiation combinations and corresponding Bragg angles for laboratory stress measurements are shown. Calculated using cij from [28.9] and [28.10] and using the Hill–Neerfeld average (Sect. 28.3.3). From [28.11]

28.2 Methodology Stress measurement with diffraction involves measuring lattice d-spacings in different directions in a specimen utilizing Bragg’s law and then using them to compute strains through ε=

sin θ0 d − d0 = −1 . d0 sin θ

(28.7)

Here d0 and θ0 are the unstressed d-spacing and the corresponding Bragg angle. We wish to obtain the stress and strain components in the specimen with respect to the specimen coordinate system shown in Fig. 28.1. By observing the diffraction with a diffraction vector oriented along the X 3 axis, we can determine the strain in this direction with (28.7). The X 3 axis is oriented with respect to the specimen coordinate system by the an-

gles φ and ψ. Applying tensor transformation rules to the specimen coordinate system we may write the measured strain, εφψ , in terms of the strains in the specimen coordinate system: dφψ − d0 d0 = ε11 cos2 φ sin2 ψ + ε22 sin2 φ sin2 ψ

εφψ =

+ ε33 cos2 ψ + ε12 sin 2φ sin2 ψ + ε13 cos φ sin 2ψ + ε23 sin φ sin 2ψ . (28.8) By measuring strains with diffraction in at least six independent directions, the strains in the specimen coordinate system can be determined by a least-squares procedure [28.12]. After the strains in the specimen co-

X-Ray Stress Analysis

ordinate system have been computed, the stresses in this coordinate system can be determined with Hooke’s law as   S1 1 εij − δij 1 εii , (28.9) σij = 1 2 S2 2 S2 + 3S1

28.2.1 Measurement Geometry Placing the diffraction vector along a particular X 3 axis can be accomplished in an infinite number of ways by rotating the diffraction plane (containing the incident and diffracted beams) around the diffraction vector. In practice this is accomplished in two ways, termed Ω goniometry and Ψ goniometry, which are illustrated in Fig. 28.1b,c. In Ω goniometry the specimen is rotated by an angle ψ about an axis perpendicular to the diffraction plane, while for Ψ goniometry this rotation is performed about an axis within the diffraction plane. These rotations orient the diffraction vector to the specimen coordinate system by the angle ψ in Fig. 28.1a. For both goniometry methods the orientation angle φ is obtained by rotating the specimen about the X 3 axis. The penetration depth into the sample, z, depends on the x-ray path length into and out of the sample, and thus on measurement geometry. This depth is given for both types of goniometry as [28.13]

σ23 , and σ33 , which have a component perpendicular to the surface. It is therefore common to assume a biaxial stress state and measure the stress in a particular in-plane direction (given by the angle φ in Fig. 28.1a). If we drop the σi3 components and define the stress in the φ direction as σφ = σ11 cos2 φ + σ12 sin 2φ + σ22 sin2 φ

(28.11)

and solve for dφψ we obtain dφψ =

1 S2 d0 σφ sin2 ψ + S1 d0 (σ11 + σ22 ) + d0 . 2 (28.12)

This equation gives the measured d-spacing as a function of the tilt angle. For a stressed material, at ψ = 0◦ the strain is given by the Poisson effect of the stress. As the specimen is tilted, the measured strain varies according to strain transformation rules, varying linearly with sin2 ψ. If the measured d-spacing is plotted versus the variable sin2 ψ, there should be a linear relationship. It is good to check experimentally that this relationship is indeed linear by measuring at multiple ψ tilts. Nonlinear relations can reveal a number of problems and can invalidate the analysis. Case studies illustrating both linear and nonlinear behavior are presented below. If a line is fitted to the d versus sin2 ψ data, the resulting slope will be proportional to the stress σφ . The stress can be obtained by multiplying the experimentally determined slope by 12 S2 and d0 . The stress-free lattice spacing, d0 , is generally unknown. Since it is simply a multiplier in (28.12), the intercept of the d versus sin2 ψ plot or another approximation can be used with only a small error.

28.2.3 Triaxial Analysis ln II0 sin2 θ − sin2 ψ + cos2 θ sin2 ψ sin2 η , z= 2μl sin θ cos ψ (28.10)

where η is the sample rotation around the diffraction vector and equal to 0◦ (90◦ ) for Ω (Ψ ) goniometry, respectively, and other quantities are as previously defined. The maximum penetration depth is thus at ψ = 0◦ and θ ⇒ 90◦ such that z max ⇒ ln(I0 /I )/2μl .

28.2.2 Biaxial Analysis Because conventional x-rays only penetrate a few tens of microns into most engineering materials, the stresses measured are usually biaxial, lacking components σ13 ,

805

When using high-energy x-rays or neutrons, the penetration depth is larger, so that all the components of the stress tensor may be present. In addition, as we will see in Sect. 28.3, triaxial analysis may be needed for conventional laboratory-based x-ray sources if multiphase materials are investigated. To determine the complete stress tensor the lattice spacing is measured in a number of directions/angles, and strains in these directions are calculated with the stress-free lattice spacing d0 . The triaxial strain components εij are computed using (28.8), which can be rewritten as εφψ =

6  k=1

ak f k (φ, ψ) ,

(28.13)

Part C 28.2

where δij is the Kronecker delta function and S1 = (ν/E)hkl and S2 /2 = ((1 + ν)/E)hkl are the diffraction elastic constants (DEC), which are discussed in Sect. 28.3.3. Equations (28.7)–(28.9) encapsulate the basic model of stress measurement by diffraction. Practical implementation of this model will be discussed in the following sections.

28.2 Methodology

806

Part C

Noncontact Methods

Part C 28.2

where ak are the six strain components determined by fitting the measured data, and the functions f k (φ, ψ) are functions of the two variables φ and ψ. At least six independent strain measurements are needed to solve (28.13) for the strain components. Greater accuracy is obtained by increasing the number of strain measurements and by using measurement directions that have a wide angular distribution. It is good practice to measure at a number of ψ values (both positive and negative) for several different constant φ values. Plots of the measured strain values for each of the φ values versus sin2 ψ help reveal the quality of the measured data. Examining (28.8), we see that if the shear strains ε13 and ε23 are zero, the ε versus sin2 ψ plots will be linear with the positive and negative ψ data overlapping on this line. If the shear strains ε13 and ε23 are present, the positive and negative branches of the plot will take different paths following two sides of an ellipse. If the data do not follow one of these shapes, it indicates a problem and fitting the data to (28.8) is unlikely to give accurate stress values. Plots of ε versus sin2 ψ which can be properly fitted are shown as a case study below. An important issue for accurate triaxial stress measurement is obtaining an accurate value of the stress-free lattice spacing. In contrast to the biaxial case where it is just a multiplier, the stress-free lattice spacing is subtracted from the measured d-spacings to obtain a strain. Consequently an accurate unstressed lattice spacing d0 is needed for triaxial analysis [28.14]. An error in the stress-free lattice spacing results in an error in all the resulting computed strain values, which yields an erroneous hydrostatic stress/strain component appearing in the final stress/strain tensor determination. If an accurate stress-free lattice spacing is unavailable, the deviatoric components of the stress and strain tensors will still be accurate even though the hydrostatic components are not [28.15].

28.2.4 Determination of Diffraction Peak Positions Lattice spacings are determined from diffraction by careful determination of diffraction peak positions. As a stressed specimen is tilted and diffraction peaks are recorded, the position of the peak will shift as a different component of strain is resolved. Strains in crystalline materials are small (usually 10−4 or less) and, thus, peak shifts will be small, often less than half the width of the peak. Therefore, the diffraction peak must be carefully recorded and the peak position precisely determined. The position of the peak can be defined as the

2θ value of the centroid of the diffraction intensity or the position of maximum diffracted intensity. For symmetric diffraction peaks these quantities will be the same, but in general, they will be different. The same definition should be used throughout a stress measurement. Theoretically, the peak centroid is more appropriate, but is not widely used in practice. In order to determine the peak position, the intensity versus 2θ data is usually fitted to a mathematical function representing the peak profile [28.3,16]. A Gaussian peak with a linear background is given by     2θ − 2θp 2 + 2θm + b , I = I0 exp −4 ln 2 W (28.14)

where I0 is the diffracted intensity over the background level, 2θp is the peak position, W is the peak full-width at half-maximum, m is the slope of the background, and b is the background intercept. A Gaussian often fails to fit the tails of the diffraction peak well. Adding a Lorentzian component to the peak function gives a pseudo-Voigt function, ⎧  ⎪    ⎨ 2θ − 2θp 2 I = I0 η exp −4 ln 2 ⎪ W ⎩ + (1 − η)

1+

1

⎫ ⎪ ⎬

 2θ−2θp 2 ⎪ ⎭ W

+ 2θm + b ,

which will usually better fit the tails. Here η is the fraction of the Gaussian component in the peak and (1 − η) is the fraction of the Lorentzian component. With conventional x-ray tubes the characteristic x-rays usually contain two closely spaced wavelengths, so the diffraction peaks are actually a doublet of two closely spaced peaks. For a Kα1 –Kα2 doublet, the higher-angle peak will have approximately half the height of the lower-angle peak due to the intensity ratio of the respective wavelengths in the incident beam. Constraining the peak height ratio to 0.5 can improve the fitting of the doublet. With sufficient broadening, the two peaks in the doublet can be treated as a single peak and fitted with one of the above equations. If the two peaks do not sufficiently overlap (i. e., they are distinct), they should be fitted with two peaks separated in 2θ by δ. Note that adding the second peak to the fit function does not require the addition of any new fit parameters. The separation between the two peaks δ is computed

X-Ray Stress Analysis

from the known x-ray wavelengths and Bragg’s law   λ2 sin(2θ1 /2) − 2θ1 . δ = 2 sin−1 (28.15) λ1

Intensity (Counts) 450 I0 = 203.44 ± 2.67 2θ = 144.071 ± 0.00275 W = 0.463429 ± 0.00546 m = – 8.7897 ± 2.03 b = 146.46 ± 293

400 350 300 250 200 150

143.5

144

144.5

145 2θ (Degrees)

Fig. 28.2 211 diffraction peak from the β phase of a 60–40 brass

along with fit to a Gaussian doublet. The best fit parameters from (28.14) are given in the inset box. The value of δ was calculated prior to fitting to have a value of 0.60409

advantage is that the degree of doublet overlap can vary with ψ, causing shifts in the location of maximum intensity due to varying doublet overlap, which will erroneously be recorded as due to residual stress. Fitting a doublet peak function to the data automatically accounts for this potential problem. With linear or area detectors, the whole peak is automatically recorded and there is little need to fit just the top portion of the peak.

28.3 Micromechanics of Multiphase Materials In Sect. 28.2, methods for measuring stress states with diffraction were outlined. In order to properly interpret these stresses, we now distinguish between types of stresses, outline key equilibrium conditions, and provide information on diffraction elastic constants needed to compute stresses from measured strains.

28.3.1 Macrostresses and Microstresses Macrostresses are the stresses that appear in a homogeneous material. They vary slowly on the scale of the microstructure and are the stresses that are revealed by dissection techniques. Residual stresses, as they have been traditionally treated, are macrostresses and arise due to nonuniform deformations on a macro-

807

scopic scale. Microstresses, in contrast, arise from the microstructural inhomogeneities in the material and nonuniform deformations on this scale. While microstresses can often be an impediment to determining the true residual macrostresses in a material, they are increasingly becoming of interest in their own right, with the recognition that they can also influence material behavior [28.18]. Residual stresses have also been distinguished in the literature as types I, II, and III, as illustrated in Fig. 28.3. Type I stresses are macrostresses that are constant over many grains in the material. Type II stresses are microstresses and can vary between phases, as well as between grains within a given phase. Typical diffraction data provide the grain-averaged type II stress value within a given

Part C 28.3

Figure 28.2 shows a 211 diffraction peak from the β phase of a 60–40 brass, measured with Cr Kα1 –Kα2 radiation and fitted with a Gaussian peak profile, including a second peak to account for the doublet. With an area or linear detector, the background away from the peak position is automatically recorded and should be included in the fitting. A simple line is usually sufficient, though a higher-order polynomial or other forms can also be used to represent the background accurately. If the diffraction peak is recorded by point counting, the background may not be sufficiently well recorded away from the peak to accurately fit. Recording the background can be time consuming and may not be worthwhile, particularly if the background is not changing as the specimen is tilted. In these instances the background should be assumed constant in 2θ and only this constant included in the fit function. If a background function with a slope is used and intensity data away from the peak that would allow an accurate determination of the slope is absent, erroneous peak positions can result from the fitting, due to correlations between the slope and peak position. Fitting the top 15% of the diffraction peak to a parabola has also been utilized for x-ray stress measurements [28.17]. For point counting this method has the advantage that time is not spent collecting the whole diffraction peak, speeding up the measurements. A dis-

28.3 Micromechanics of Multiphase Materials

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Noncontact Methods

phase. Type III stresses are also microstresses but represent stress variations within single grains. Residual stresses of types I and II will result in diffraction peak shifts, as they change the mean lattice spacing, while type III stresses result in diffraction peak broadening rather than peak shifts. Measurement of type III stresses is outside the current scope and will not be further discussed. For a two-phase material, the measured stress from peak shifts can be written as   t α  I σij = σij + II σijα , t β  I  β σij = σij + II σij , (28.16)

Part C 28.3

where the superscripts t, I, and II stand for the total stress in a phase (α or β), the type I macrostresses, and the type II microstresses, respectively. The total stresses and microstresses are represented as averages (denoted by carets) as they can vary significantly within the diffracting volume, whereas the macrostress does not. Note also that the macrostress component is common to both phases and thus does not have a phase designation. These equations hold for each component of the stress tensor. Using (28.16), the microstresses in the α and β phases must balance to zero when weighted by their volume fractions, thus   β  (28.17) (1 − f ) II σijα + f II σij = 0 . Here f is the volume fraction of the β phase. The total stress components in the α and β phases can be σ

α

σ β

σ II

I

σ II

β

σ II

measured with diffraction measurements as outlined in Sect. 28.2.3. If the volume fractions of the phases are known, the components of the macrostress tensor and microstress tensors can be then determined using (28.16) and (28.17).

28.3.2 Equilibrium Conditions As measured with conventional (low-energy) x-rays, residual macrostresses with a component perpendicular to the surface, σ13 , σ23 , and σ33 (with the x3 axis normal to the surface) are typically negligible. This can be seen via the following equilibrium relations for residual macrostresses: ∂σ11 ∂σ12 ∂σ13 + + =0, ∂x1 ∂x2 ∂x3 ∂σ21 ∂σ22 ∂σ23 + + =0, ∂x1 ∂x2 ∂x3 ∂σ31 ∂σ32 ∂σ33 + + =0. (28.18) ∂x1 ∂x2 ∂x3 These relationships show that the stress components σi3 can only change from their value of zero at the free surface if there are gradients in the other components of stress within the surface. These components of stress cannot normally have high enough gradients within the surface to cause the components σi3 to reach measurable levels within the penetration depth of conventional x-rays (the biaxial assumption, see Table 28.1). Another equilibrium relation that proves useful is that the stress integrated over a cross section of a solid body must be zero  σij dA = 0 . (28.19) A

This relation can be used as a check on experimental results and can also be used to determine the stress-free lattice spacing.

σ III

28.3.3 Diffraction Elastic Constants

Fig. 28.3 Definition of macrostress (σ I ) and microstresses in a two-phase material. The measured peak shift in a given phase is proportional to the total mean elastic stress in that   phase (σ I + σ II ). Variations in stress within a grain (σ III ) cause peak broadening rather than peak shifts

Elastic constants are needed to compute stresses from measured diffraction strains in polycrystalline materials. Due to the selective nature of the diffraction process and presence of crystalline elastic anisotropy, the so-called diffraction elastic constants (DEC) differ, in general, from the bulk elastic constants. Here, two methods for attaining DEC are described: 1. computing them using single-crystal elastic constants and a given micromechanical (grain-averaging) model, and

X-Ray Stress Analysis

2. measuring the constants in situ using known elastic loads.

•

•

The higher the crystalline anisotropy, the larger the deviation between DEC and bulk values for a given diffraction peak. Thus, use of bulk values will give a corresponding error in absolute stress value for a given diffraction measured strain (28.12). These deviations are especially important for lowmultiplicity peaks (e.g., 111/200). For highermultiplicity peaks (e.g., 732 in ferrite) the DEC approach the bulk values. Such peaks have the added benefit of being less sensitive to texture, as shown by Hauk in textured steel [28.25].

A second method is to measure the DEC in situ using known loads. This represents the inverse problem to stress analysis described throughout the rest of the chapter, where stresses are unknown and DEC are assumed known. Loads should be applied using either uniaxial

tension or four-point bending, since the applied stress distributions in such cases can be determined accurately from elasticity theory, and the diffraction strains are recorded for reflections of interest. The American Society for Testing and Materials has developed a standard test method for the measurement of the diffraction elastic constant S2 /2 for a biaxial stress measurement, which can give some practical guidance [28.26]. Such methods are particularly important when elastic constants of the material are unknown or in question. For example, alloying can modify DEC considerably from elemental values, as found by Dawson et al. [28.27] on aluminum alloys. Tables of measured DEC for various materials (including common alloys) can be found in [28.3, 7]. Taking this method one step further, Daymond [28.28] found that, by using strain from multiple diffraction peaks and a weighted Rietveld method, it was possible to calculate bulk elastic properties E and ν in polycrystals. He found good matches to bulk properties for both textured and untextured cubic (steel) and hexagonal (titanium) crystal symmetries. We note that the use of bulk elastic constants (E, v) in place of hkl-dependent DEC is common. While this can cause an error in the absolute stress values, relative changes in stress (e.g., between samples of the same composition, or positions in a given sample) may still be valid. Finally, we note that the grain-interaction models described above are appropriate for bulk samples, but van Leeuwen et al. [28.29] found that in the case of thin films (e.g., vapor-deposited nickel) the Vook– Witt [28.30] model is more appropriate to describe the measured strains. The latter model takes the special geometry of thin films into account for specifying the strain and stress state, and can be used to calculate nonlinear d–sin2 ψ behavior even in the absence of crystallographic texture. For additional information on strain measurements in thin films two reviews on the subject are noted [28.31, 32].

28.4 Instrumentation A variety of equipment can be used for diffraction stress measurements. Conventional x-ray diffractometers are often used, as well as special-purpose stress diffractometers which may be laboratory-based or portable. At synchrotron x-ray and neutron facilities the working area is typically larger than with laboratory-based diffraction systems, permitting larger samples and/or

ancillary equipment. Associated instrumentation is generally specialized and outside the scope of this chapter.

28.4.1 Conventional X-Ray Diffractometers Conventional x-ray diffractometers are widely available and used for a variety of applications. These can of-

809

Part C 28.4

The most widely used micromechanical models are those from Voigt [28.19], Reuss [28.20], Neerfeld– Hill [28.21,22], and Eshelby–Kroner [28.23]. The Voigt and Reuss models assume that all grains in a polycrystalline aggregate have, respectively, the same strain and stress, and represent, respectively, the upper and lower bounds of the elastic constants. It is generally found that the Neerfeld–Hill (which is the arithmetic average of the Voigt and Reuss values), and Eshelby–Kroner models best match experimentally determined DEC. Equations for determining DEC for various crystal symmetries can be found in [28.3, 4], and stress-analysis software packages may provide computational tools (e.g., [28.24]). In Table 28.1, Hill model values of the inverse DEC, 2/S2 = (E/(1 + ν))hkl , are compared with bulk values (i. e., those expected for an isotropic body) for various materials and hkl peaks. We note the following trends from the table:

28.4 Instrumentation

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ten be utilized for stress measurement, though software to control the data acquisition and analysis will often be deficient for this task, requiring a greater degree of operator expertise. Conventional x-ray diffractometers may have several limitations, making stress measurement difficult or impossible. They often have limited space for specimens, have limited or no ability to achieve ψ-tilts, have incompatible beam defining slits, and provide poor access to the raw data for analysis. If Ψ -goniometry is to be used with a diffractometer having a χ-circle, the x-ray tube needs to be oriented for point focus to limit the length of the beam along the diffractometer axis.

Part C 28.5

28.4.2 Special-Purpose Stress Diffractometers Various stress diffractometers have been specifically designed to perform these measurements. They have the ability to handle larger and heavier specimens and have software to automate the data collection and analysis. Most modern stress diffractometers use linear or area detectors to increase the measurement speed, and utilize the Ψ -goniometry because it does not defocus the x-ray optics. More information and a photograph of a stress diffractometer are given in a case study in Sect. 28.6.1. Laboratory-based units will be more flexible diffraction instruments, usually having texture measurement capabilities and the ability to do general powder diffraction. Portable stress diffractometers can be taken into the field to make stress measurements on very large objects without having to move them.

28.4.3 X-Ray Detectors X-ray detectors can be broadly classified into point, line, or area detectors. Point detectors include proportional, scintillation, and solid-state detectors. For each the x-ray energy is proportional to the measured pulse height, so that energy discrimination can be accomplished through electronic filtering. The energy resolution is highest for solid-state detectors (ΔE/E ∼ 3–5%) and lowest for scintillation

counters (ΔE/E ∼ 30–40%). Thus, solid-state detectors generally have lower background levels and are the standard choice for energy-dispersive measurements. If necessary, secondary monochromators and/or foils can be used in conjunction with scintillation or proportional detectors to reduce background levels. As their name implies, linear and area detectors measure, respectively, one- and two-dimensional angular sections of x-ray diffraction cones. As this information is collected simultaneously, these detectors allow for significantly faster strain measurements than point counters, which require scanning over the 2θ region of interest. Area detectors have the added benefit of providing simultaneous grain size and texture information through intensity variations around a diffraction cone. These detectors generally have limited energy resolution, and incident-beam filters can be employed to limit background levels.

28.4.4 Synchrotron and Neutron Facilities Both synchrotron and neutron facilities provide unique capabilities for strain and stress analysis. Synchrotrons offer x-ray fluxes several orders of magnitude higher than x-ray tubes, and as such can be used for timeresolved experiments. In addition, these sources have much lower divergence and source sizes, so focal spots from mm down to tens of nm can be used for spatially resolved measurements. Furthermore, thirdgeneration facilities [the Advanced Photon Source (APS), USA; the European Synchrotron Radiation Facility, France; and SPRing-8, Japan] are excellent sources of high-energy x-rays, with concomitant high penetration depths (over 1 mm in most materials). This, in turn, enables many in situ and bulk studies that are either not practical or possible with laboratory-based sources. Even higher penetration depths are provided by neutrons (up to the cm level), with similar in situ capabilities. Access to synchrotron and neutron facilities is typically granted (for free) based on peer-reviewed proposals, but proprietary (fee-based) work may be arranged at certain facilities.

28.5 Experimental Uncertainties With proper care, diffraction stress measurements can readily be made with sufficient accuracy and precision to solve many problems. There are, however, ran-

dom and systematic errors in the measurement process that should be carefully considered when evaluating diffraction stress measurement results. The best way

X-Ray Stress Analysis

28.5.1 Random Errors Random errors in x-ray measurements arise from the statistical nature of x-ray counting and the random component of proper alignment of the specimen and instrument. The statistical errors will typically be the dominant error in measurements with laboratory xrays or neutrons. These errors can be driven arbitrarily low by collecting data for longer periods of time, but the improvement varies as the square root of the time. When a diffraction peak is recorded and fitted to find its position, there will be an uncertainty associated with this position due to the statistical nature of recording x-ray intensities. Fitting programs usually give an estimate of the uncertainty in the fitted parameters, including the peak position. This uncertainty can be propagated through the analysis to estimate the uncertainty in the stresses. The uncertainty in a d-spacing is given by differentiating Bragg’s law [28.3]  STD(dψ ) = dψ cot θ

STD(2θp ) 2



π 180

 . (28.20)

Here, STD(dψ ) represents an estimated standard deviation in the value of dψ and STD(2θp ) is an estimated uncertainty in the peak position arising from the fitting of the diffraction peak. The π/180 factor is a conversion of degrees to radians, assuming that the uncertainties in peak position are in degrees. Similarly, the uncertainty

in a strain is given by STD(εφψ ) = cot θ



STD (2θ) 2



π 180

811

 .

(28.21)

These uncertainties can then be used in the fitting of the d-spacing or strains εφψ to the biaxial or triaxial models ((28.12) and (28.13)). A proper fitting package will propagate the uncertainties in the input data to the fitted parameters. A goodness-of-fit parameter can be computed to help determine if the measured data suitably fits the model used [28.36]. The goodness of fit is the probability that the observed deviations from the model fit arise from the estimated errors. A low goodness of fit can indicate that the errors are underestimated or that the data do not fit the model and the resulting stresses should not be considered valid. It has been recommended that goodness-of-fit parameters should be better than 10−3 –10−5 before considering the model fit to be valid [28.36]. Poor goodness-of-fit parameters may indicate the presence of systematic errors and/or samplerelated issues, both of which are described below. Measurements made under computer control can take some quick preliminary measurements of the x-ray intensities and control the measurement time to give a specified statistical error in the final measurements.

28.5.2 Systematic Errors Systematic errors in diffraction stress measurement consist primarily of imperfect optics and misalignments in the diffractometer, both of which cause shifts in the diffraction peak position from its true position. For conventional x-ray stress measurement using reflection geometries, a number of researchers have given equations to estimate the uncertainties in stress due to these sources, and these are summarized in Table 28.2. The size of these systematic peak shifts generally increases Table 28.2 References for systematic instrumental errors in diffraction stress measurements Source of error

Refs.

Specimen displacement (ω-goniometry) Specimen displacement (Ψ -goniometry) Horizontal beam divergence Vertical beam divergence Missetting of diffractometer angles ψ-axis displacement (ω-goniometry) ψ-axis displacement (Ψ -goniometry) Curvature of specimen surface

[28.3, 6, 34] [28.3, 6] [28.3] [28.3, 34] [28.35] [28.3, 6, 34] [28.3, 6] [28.3, 6]

Part C 28.5

to evaluate the errors present is to repeatedly measure a stress-free standard, such as an annealed powder from the material of interest. The average stress measured should give a measure of the systematic errors present, while the variation in results should give a measure of the random errors present. The American Society for Testing and Materials ASTM outlines a standard test method for verifying the alignment of an x-ray stress diffractometer [28.33] and specifies that an annealed powder measured according to accepted procedures should give a value of stress less than 14 MPa (2 ksi). Measuring an annealed powder is a time-consuming process and may not be practical. It is desirable to obtain good estimates of the random and systematic errors in a measurement based on just that single measurement. It is also desirable to be able to estimate the errors so that the measurements can be made to a specified accuracy. Finally, it can be a good practice to repeat measurements to check reproducibility.

28.5 Experimental Uncertainties

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Part C 28.5

with the tilt angle ψ and with decreasing diffraction peak position 2θ. Care should be taken if extreme values of ψ are to be used (greater than 60◦ ) or with diffraction peaks below 2θ of 130◦ . For transmission methods with high-energy x-rays or neutrons, successful measurements can be successfully made with low-2θ peaks, in part because they often have limited (or no) requisite sample movement. Generally the largest instrumental error is that due to specimen displacement, i. e., not having the sample surface at the center of the diffractometer axis. This results in a diffraction peak shift that depends on ψ and thus introduces an error in the measured stresses. This error can be minimized by using parallel beam geometry in which vertically aligned Soller slits are used to define the diffraction angles. This geometry only works for point detectors that are scanned across the diffraction peak, which is much slower than using linear or area detectors. This component of error can change as specimens are moved for measurements at different locations or as specimens are changed. This error will also change for a specimen from the value recorded on a stress-free powder. Placement of the specimen surface on the diffractometer axis is typically achieved with special alignment gages for dedicated instruments. Additionally, one can measure the specimen displacement for cubic materials with diffraction [28.3]. The lattice parameter a is determined from a series of diffraction peaks and plotted versus cos2 θ/ sin θ. The slope of this plot gives the specimen displacement Δx according to the relation Δx cos2 θ ahkl − a0 = , a0 RG sin θ

(28.22)

where RG is the goniometer radius and a0 is the true lattice parameter, which can be approximated with the intercept of the plot. For triaxial analysis an accurate value of the stressfree lattice spacing must be used. This should be measured on the same diffractometer used for the stress measurements. Inaccuracies in the stress-free lattice spacing lead to an error in the hydrostatic component of the stress tensor [28.15]. If a single stress-free lattice spacing is used for a series of measurements, this error will be systematic and changes in the hydrostatic component between measurements will be meaningful. In some situations, notably welds, the stress-free lattice spacing can vary from point to point in the specimen due to composition gradients. If the full tensor is then desired, stress-free reference values must be measured

for each point sampled, which can be accomplished by sample sectioning.

28.5.3 Sample-Related Issues The error analysis described above applies to ideally behaved specimens, and thus represents a lower limit on the error in stress values. Here we describe potential sample-related issues that can modify strain data relative to this ideal behavior. Texture Effects Crystallographic texture is common in materials and is due to directional processing methods, such as rolling in bulk materials or directional deposition in thin films and coatings. The presence of texture causes systematic variations in the peak intensity as a function of tilt. This can be distinguished from random (stochastic) variations in intensity, which indicate improper grain averaging (addressed below). If strong texture is known or suspected to be present, it can be beneficial to measure a pole figure on the peak intended for strain analysis, which both provides quantitative texture information and aids determination of the most appropriate orientations to measure. Pole figure measurements are outside the scope of this chapter; the reader is referred to [28.37] for details. The presence of texture, coupled with elastic anisotropy, can lead to oscillations in d versus sin2 ψ plots. In such cases the diffraction elastic constants vary on a macroscopic scale with orientation, so grains with different DEC are sampled at each (φ and ψ) tilt setting. Methods to account for texture have been developed by various researchers [28.4, 38, 39]. One particularly straightforward method is to use highmultiplicity planes, such as the (732) reflection in steel, which Hauk [28.25] showed provides more linear d versus sin2 ψ behavior than lower-multiplicity peaks (e.g., (211)). Drawbacks to the use of these peaks are that they are less intense, and thus subject to higher statistical error, than higher multiplicity peaks, and require higher-energy x-ray sources (e.g., Mo tubes) than those commonly used and available. Grain Averaging As illustrated in Fig. 28.3, the measured strain can vary substantially from grain to grain in multiphase materials, and even within a single phase due to elastic and/or plastic anisotropy. Thus, in (typical) cases where the macrostress is the desired quantity, it is desirable to sample many grains such that these intergrain variations

X-Ray Stress Analysis

Stress Gradients The fact that the penetration depth changes with tilting (28.10) can be important if strain/stress gradients exist within the penetration depth, as different (depthaveraged) strain values will be sampled at different tilts. This effect can cause curvature in d versus sin2 ψ plots, which can be analyzed by assuming a particu-

Stress error (MPa) 200 180 160

Error Envelope

140 120 100 80 60 40 20 0 400

900 1400 1900 Number of grains sampled in x-ray spot

Fig. 28.4 Difference between measured (using x-ray

diffraction) and known stresses as a function of the number of sampled grains (after [28.40]). Each point represents a single diffraction measurement, taken across a bent austenitic stainless-steel specimen containing a gradient both in residual stress and grain size (variation from 62–248 grains/mm2 ). Note the inverse relationship between sampled grains and stress error

lar depth dependence of the stress [28.3]. This issue can be of particular concern for coatings where microstructural gradients on the μm level are common, and measurement schemes have been developed for such cases [28.41]. If the stress variations over larger ranges than the penetration depth are of interest, these can be measured by destructive layer-removal methods (Sect. 28.6.2) or by nondestructive profiling with higher-energy x-ray or neutron sources (Sect. 28.6.4).

28.6 Case Studies In this section a series of case studies are presented. Laboratory x-ray diffraction is used for biaxial and triaxial stress analysis, for both well-behaved and oscillatory d versus sin2 ψ data. In addition, synchrotron high-energy x-ray diffraction is used for nondestructive stress profiling.

28.6.1 Biaxial Stress Biaxial residual stress measurements with the sin2 ψ method are the most commonly performed diffraction

813

stress measurements. In this example, biaxial x-ray stress measurements were used to help optimize the processing of coil springs for improved fatigue resistance. Large coil springs were heat treated to form tempered martensite, shot peened, and then preset. Figure 28.5 shows the stress diffractometer used to make biaxial residual stress measurements on the springs. Cr-Kα x-rays were used and the tempered martensite 211 diffraction peak was recorded with a linear position-sensitive detector. Specimen tilting was performed with the Ω-goniometry. The positions of the

Part C 28.6

are averaged out. The need for averaging is especially important in the case of large grain sizes and/or small x-ray beam sizes, and is exacerbated by the fact that only a small fraction of the irradiated grains diffract. Common methods to enhance averaging are to oscillate (in φ and/or ψ) and/or translate the sample during data collection. The importance of grain averaging was illustrated by Prime et al. [28.40], who examined a bent austenitic steel sample using both neutron and x-ray diffraction. The bending provided a gradient in both residual stress and grain size (determined from optical microscopy), and laboratory x-ray stress measurements were taken along the gradient direction. These x-ray stresses were compared with the calculated macrostresses (which agreed well with bulk neutron results), and the error in x-ray stresses (difference from calculations) is plotted versus number of sampled grains in Fig. 28.4. While there is considerable scatter in the data, there is clearly an inverse trend between stress error and the number of grains sampled. Finally it is noted that the other extreme to polycrystalline averaging is to measure strain in single grains. Such measurements provide fundamental insights into polycrystalline deformation behavior, as well as unique input to deformation models. New methods are being developed to perform such measurements in individual grains within polycrystals (Sect. 28.6.5).

28.6 Case Studies

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Part C 28.6

These measurements were used to study processing methods for spring manufacturing. Figure 28.7 shows the residual stresses with depth below the shot-peened surface for two different heat treatment methods. The stresses below the surface were measured after layer removal and then later corrected for the stress relaxation that occurs with layer removal. One heat treatment method leads to decarburization of the surface, which diminishes the compressive residual stresses the material will support. It should be noted that, with decarburization, the stress-free lattice parameter will vary with depth because of its dependence on the carbon content. Because the stresses are biaxial in nature, accurate stress-free lattice parameters are not needed, simplifying the analysis.

28.6.2 Triaxial Stress Fig. 28.5 Stress diffractometer making measurements on a segment of a coil spring (photo by Yunan Prowato of NHK International Corporation)

diffraction peaks were determined from the average of the two half-maximum points on the peak. The d versus sin2 ψ plot is shown in Fig. 28.6 and is seen to be linear. The goodness-of-fit is 0.57, indicating that the estimated errors from the diffraction peak position determinations account for the observed error to a high level of certainty. The slope and intercept from the d versus sin2 ψ plot were used, along with the DEC E/(1 + ν) = 176 GPa, to compute the residual stress as −329 MPa.

The data plotted in Fig. 28.8 were obtained from the surface of a ground steel bar [28.36]. The data were collected with a conventional x-ray diffractometer utilizing Cr-Kα radiation. The grinding direction is aligned along the φ = 0◦ direction. Along the grinding direction we see ψ-splitting, indicating the presence of shear stresses, σ13 . Perpendicular to the grinding direction, φ = 90◦ , ψ-splitting is absent, indicating that the shear stresses σ23 are negligible. Along φ = 45◦ , there is moderate ψ-splitting, indicating a component of shear stress perpendicular to the surface along this direction. From the discussion in Sect. 28.3, we know that these stresses arise as microstresses balanced by stresses of opposite sign in another phase. Here, the shear microstresses

d-spacing (Å) 1.1706 1.1704

sin2 ψ

2θ

d-spacing (Å)

0.041819 0.189426 0.325714 0.479062

156.296 156.439 156.598 156.742

1.170401 1.170096 1.169758 1.169455

Slope Intercept Stress

–0.0021916 1.1704953 –328.71 MPa

1.1702 1.17 1.1698 1.1696 1.1694 GOF = 0.573 1.1692

0

0.1

0.2

0.3

0.4

0.5

0.6 sin2 ψ

Fig. 28.6 Plot of d versus sin2 ψ for a shot-peened steel coil spring, with data and linear fit parameters included (data by

Yunan Prowoto of NHK International Corporation)

X-Ray Stress Analysis

are balanced by stresses in the carbide phases of the steel. They arise from the shear deformations imposed on the material from the grinding process. Low volume fraction, low crystal symmetry, and considerable peak broadening from the large deformations make measurement of the stresses in the carbide phase extremely difficult and they were not measured here. Fitting this data along with the known d0 and DEC to (28.8)–(28.9) gives the stress tensor [28.12]: ⎛

⎞ ⎛ ⎞ 527 −8 −40 7 5 1 ⎜ ⎟ ⎜ ⎟ σij = ⎝ −8 592 5 ⎠ ± ⎝5 7 1⎠ MPa . −40 5 102 1 1 3

28.6 Case Studies

815

Residual stress (MPa) 0 –100 –200 –300 – 400

Decarburized

–500 –600 –700

(28.23)

Here we confirm the presence of the shear stress σ13 and the negligible σ23 .

28.6.3 Oscillatory Data Not Applicable to the Classic Model Frequently, the strains measured in specimens will not behave with orientation as predicted by (28.8). This equation assumes that the strain measured in different directions transforms as a tensor quantity. Because the volume of material diffracting in each orientation is different, this does not have to hold true. If the stresses and strains partition themselves inhomogeneously on average, the measured strains will not behave according to (28.8) and numerically forcing the fit to compute stresses is questionable. Materials with texture due to deformation or growth, such as cold-rolled sheet or thin films [28.42], will frequently show this behavior. Figure 28.9 shows a set of biaxial data on a specimen of ground, cold-rolled steel [28.36]. The data clearly does not show the linear behavior as a function of sin2 ψ, which is partly attributed to the presence of sample texture (not shown). A goodness-of-fit value of 6.2 × 10−49 was computed, indicating an extremely low probability that the deviations from the model fit can be accounted for by the estimated errors on the data points [28.36]. It should be noted that, even if the data do not adequately fit the models, the computed stresses may prove useful as a quality control tool or nondestructive test, even though the interpretation of the results as stresses may not be warranted. Advanced analysis methods, described in Sect. 28.6.5, may in certain cases be used to more accurately determine mean stresses from such data.

–900

Not decarburized

0

0.05

0.1

0.15

0.2 Depth (mm)

Fig. 28.7 Residual stresses measured with x-ray diffraction on the outside of two coil springs as a function of depth. One spring has experienced decarburization during heat treatment while the other did not. Both raw data (solid lines) and layer-removal-corrected data (dashed lines) are shown

28.6.4 Synchrotron Example: Nondestructive, Depth-Resolved Stress Here we illustrate the use of high-energy x-rays from a synchrotron source to nondestructively measure strain and stress versus depth [28.43]. A schematic of the experiments is shown in Fig. 28.10a. Cylindrical steel specimens of 9 mm diameter were heat treated to form a tempered martensite matrix and nanosize M2 C strengthening precipitates. After heat treatment, specimens were laser peened and then subjected to rolling contact fatigue (RCF). Measurements were performed at the 1-ID beamline at the APS, Argonne National Laboratory. An x-ray energy of 76 keV and a conical slit [28.44] were used to create a diffraction volume of ≈ 20 × 20 × 150 μm3 . The high penetration power at this x-ray energy (τ1/e ≈ 3 mm) allowed for transmission measurements, wherein an area detector was placed after the conical slit to collect diffraction over a plane encompassing (nearly) the axial (ε11 ) and normal (ε33 ) strain directions in a single exposure. Thus, sample tilting was not required to evaluate two principal strain components, as opposed to the reflection-geometry methods cited above. Specimens were translated along the vertical (x3 ) direction, in 20 μm increments, relative to the fixed probe volume, to obtain strain and stress information versus depth, and translated in the

Part C 28.6

–800

816

Part C

Noncontact Methods

d-spacing (Å) 1.1705

1.17

1.1695

1.169

Part C 28.6

θ = 0°, θ = 0°, θ = 90°, θ = 90°, θ = 45°, θ = 45°,

1.1685

1.168

0

0.1

0.2

0.3

0.4

ψ (+) ψ (–) ψ (+) ψ (–) ψ (+) ψ (–) 0.5 sin2 ψ

Fig. 28.8 Plot of d versus sin2 ψ for a ground steel spec-

imen showing the presence of ψ-splitting, indicating the presence of shear stresses with a component normal to the surface

horizontal (x1 ) direction to evaluate different RCF wear tracks. The axial stress σ11 was determined from the strains from the martensite (211) reflection, using DEC for martensite and assuming an equibiaxial strain state (ε11 = ε22 ) (Fig. 28.10b). Significant compressive residd-spacing (Å) 1.1714 1.17135

ual stresses were observed near the surface after heat treatment, and these stresses were further increased after peening, with a maximum value near the surface. Furthermore, the RCF was found to change the residual stress profile, with a subsurface maximum (≈ 100 μm deep) observed under a wear track. It should be noted that both white-beam (energydispersive) x-ray [28.45] and neutron [28.46] techniques can also be used for such deeply penetrating, nondestructive measurements, with some limitations. As in the case study shown, three-dimensional volumes are defined by the incident- and diffracted-beam slits, with neutrons generally having larger probe volumes ( 1 mm3 ) due to flux limitations. White-beam measurements can suffer from high background levels, which can limit the measurable signal due to the limited dynamic range of typical solid-state detectors.

28.6.5 Emerging Techniques and Studies Both neutrons and synchrotron x-ray sources carry the obvious disadvantages of not being located on-site and having limited beamtime availability. However, the results of these measurements can be useful in a general sense, both to check against conventional x-ray stress measurements [28.40,47] and to provide critical tests of, and input to, deformation models [28.48]. Furthermore, they possess unique aspects such as being well suited for in situ measurements, especially under mechanical and/or thermal loading, and permitting nondestructive strain mapping such as in the case study described above. Here, we note emerging studies and techniques in the field of diffraction strain analysis, largely made possible by these sources.

1.1713 1.17125 1.1712 1.17115 1.1711 1.17105 1.171 GOF = 6.2 ×10– 49

1.17095 1.1709

0

0.1

0.2

0.3

0.4

0.5 sin2 ψ

Fig. 28.9 Plot of d versus sin2 ψ for a ground steel specimen, illustrating data that does not fit the biaxial stress model, with a correspondingly low goodness-of-fit (GOF) value

Single-Grain Studies In these studies individual grains within polycrystalline aggregates are evaluated (e.g., [28.49, 50]). The basic concept is to use beam sizes on the order of the grain size (typically 1–100 μm), spatially locate diffracting grains using single-crystal orientation methods, and determine strain using peak shifts. As these techniques rely on small beam sizes, they have so far been done at synchrotron sources. These techniques can also be used for structural characterization, including grain boundary mapping [28.51] and evaluating dislocations quantitatively [28.52] as well as dynamically [28.53]. Finally, these techniques can be combined with spectroscopy techniques such as extended x-ray absorption fine structure (EXAFS) and/or fluorescence for both materials and environmental science applications [28.54].

X-Ray Stress Analysis

Combined Strain and Imaging Studies For these studies, typically a relatively large (mm-sized) x-ray beam and an area detector are used to image heterogeneities/defects, and then a smaller beam is used for localized diffraction/strain measurements. For x-rays, the typical imaging contrast mechanism is absorption (radiography), but the coherence of synchrotron sources also allows for more sensitive phase-contrast imaging [28.55]. Examples of work in this area includes deformation studies on composite systems [28.56] and evaluation of creep damage [28.57].

Advanced Quantitative Analysis There have been several recent efforts to extend some fundamental concepts of quantitative texture analysis to describe orientation-dependent elastic strains/stresses in polycrystalline materials [28.39, 65, 66]. Strain measurements from multiple sample orientations are represented on so-called strain pole figures, which are analogous to pole figures used in texture analysis, and which may be inverted to obtain the underlying strain/stress distribution [28.39, 65, 66]. Using these

Top view, specimen x1

Peened surface

x2

9 mm

Conical slit

RCF wear tracks 20 µm APS x-rays E = 76 keV

x3 2θ~ 8° x2 Beam stop

100 µm

9 mm

Side view, experiment

Area detector

b) Axial residual stress σ11 (MPa) 0

–200 – 400 –600 –800 –1000 –1200

Unpeened, outside wear track Peened, outside wear track Peened, under wear track

–1400 –1600 –1800

0

0.5

1 1.5 Depth into sample x2 (mm)

Fig. 28.10 (a) Setup for three-dimensional spatially resolved strain

measurements using high-energy synchrotron x-rays, a conical slit, and area detector, with specimen photo shown in inset. (b) Measured residual stresses in the axial (x1 ) direction for three cases

techniques, it is possible to examine the micromechanical states of specific crystallite populations within the aggregate in greater detail than with standard methods. Such techniques also provide the modeling community with a powerful motivation and validation tool.

28.7 Summary •

X-rays are capable of measuring elastic strain in all crystalline phases present, using shifts in diffraction peak positions. These strains can be converted to elastic stresses using elastic constants. Diffraction also provides information on peak intensities and widths that can be used to analyze texture and plastic deformation, respectively.

817

• •

With laboratory x-rays the probed depth is typically microns. Deeper depths can be sampled either by (destructive) layer-removal methods or (nondestructively) using higher-energy x-rays or neutrons. Errors in strain values can arise from many sources, including counting statistics, instrument misalignment, and sample-related sources. Methods to assess experimental error include measuring an

Part C 28.7

Strain in Nontraditional Materials While diffraction has traditionally only been used to measure strain in crystalline materials, Windle et al. [28.58] showed that strain can be extracted from amorphous materials (polymers), by measuring changes in peak positions and/or radial distribution functions with applied load. This work has recently been extended with synchrotron studies [28.59, 60] on bulk metallic glasses. Another emerging area is biomaterials studies, including bone [28.61, 62], teeth [28.63], and synthetic coatings on implants [28.64]. These materials (e.g., bone) often contain additional ordering on longer length scales than typically evaluated by wide-angle scattering (up to the μm level [28.62]), and deformation on these levels can be evaluated by small-angle scattering [28.61] (28.1).

a)

28.7 Summary

818

Part C

Noncontact Methods

annealed sample and/or repeating a given measurement. In addition, the use of incorrect diffraction elastic constants to convert strain to stress will give a proportional error in absolute stress values.

•

With the maturation and anticipated growth of both synchrotron and neutron facilities, their use for advanced strain and stress analysis is expected to continue well into the future.

28.8 Further Reading General information on diffraction stress measurements can be found in [28.3–8].

Part C 28

References 28.1

28.2

28.3

28.4

28.5

28.6

28.7

28.8

28.9

P.J. Withers: Use of synchrotron x-ray radiation for stress measurement. In: Analysis of Residual Stress by Diffraction Using Neutron and Synchrotron Radiation, ed. by M.E. Fitzpatrick, A. Lodini (Taylor and Francis, London 2004) pp. 170–189 H.F. Poulsen, S. Garbe, T. Lorentzen, D. Juul Jensen, F.W. Poulsen, N.H. Andersen, T. Frello, R. Feidenhans’l, H. Graafsma: Applications of high-energy synchrotron radiation for structural studies of polycrystalline materials, J. Synchrotron Radiat. 4, 147–54 (1997) I.C. Noyan, J.B. Cohen: Residual Stress: Measurement by Diffraction and Interpretation (Springer, New York 1987) V. Hauk: Structural and Residual Stress Analysis by Nondestructive Methods (Elsevier Science B.V., Amsterdam 1997) M.E. Hilley, J.A. Larson, C.F. Jatczak, R.E. Ricklefs (Eds.): Residual Stress Measurement by X-ray Diffraction. In: SAE Information Report J784a, ed. by M.E. Hilley (Society of Automotive Engineers, Warrendale 1971) F. Lecroisey, B. Miege, A. Saint-Etienne: La Mesure de contraintes residuelles: methode de determination par rayons X, Memoires Techniques du CETIM, Vol. 33 (Centre Technique de Industries Mechaniques, 1978) P. Prevey: X-ray diffraction residual stress techniques. In: Metals Handbook 9th Edition, Vol. 10 (American Society for Metals: Metals Park, Ohio 1986) pp. 380–392 M.E. Fitzpatrick, A.T. Fry, P. Holdway, F.A. Kandil, J. Shackleton, L. Suominen: Good Practice Guide No. 52: Determination of Residual Stresses by X-ray Diffraction (National Physical Laboratory, Teddington, UK 2005) pp. 1–68 R.W. Hertzberg: Deformation and Fracture Mechanics of Engineering Materials (Wiley, New York 1996)

28.10

28.11 28.12

28.13

28.14

28.15

28.16

28.17

28.18

28.19 28.20

28.21

K. Inal, R. Pesci, J.L. Lebrun, O. Diard, R. Masson: Grain and phase stress criteria for behaviour and cleavage in duplex and bainitic steels, Fatigue Fract. Eng. Mater. Struct. 29(9,10), 685–696 (2006) G.W.C. Kaye, T.H. Laby: Tables of Physical and Chemical Constants (Longman, London 1995) R.A. Winholtz, J.B. Cohen: Generalized leastsquares determination of triaxial stress states by x-ray diffraction and associated errors, Aust. J. Phys. 41, 189–199 (1988) C. Genzel: X-ray stress gradient analysis in thin layers – problems and attempts at their solution, Phys. Status Solidi A 159, 283–296 (1997) I.C. Noyan: Determination of the unstressed lattice parameteraofor (triaxial) residual stress determination by x-rays, Adv. X-ray Anal. 28, 281–288 (1985) R.A. Winholtz, J.B. Cohen: Separation of macroand micro-stresses in plastically deformed 1080 steel, Adv. X-ray Anal. 32, 341–353 (1989) W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling: Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge 1986) M.R. James, J.B. Cohen: Study of the precision of x-ray stress analysis, Adv. X-ray Anal. 20, 291–307 (1977) R.A. Winholtz: Macro and micro-stresses. In: Encyclopedia of Materials, ed. by K.H.J. Buschow et al. (Pergamon, Oxford 2001) W. Voigt: Lehrbuch der Kristallphysik (Teubner, Leipzig, Berlin 1910) A. Reuss, Z. Agnew: Calculation of flow limits of mixed crystals on the basis of plasticity of single crystal, Math. Mech. 9, 49–52 (1929) H. Neerfeld: Zur Spannungsberechnung aus rontgenographische Dehnungsmessungen, In: Mitt. K.-Wilh.-Inst. Eisenforschg. 24, 61–70 (1942)

X-Ray Stress Analysis

28.22 28.23

28.24

28.25 28.26

28.28

28.29

28.30 28.31

28.32

28.33

28.34

28.35

28.36

28.37

28.38

28.39

28.40

28.41

28.42

28.43

28.44

28.45

28.46

28.47

28.48

28.49 28.50

28.51

J.V. Bernier, M.P. Miller: A direct method for the determination of the mean orientationdependent elastic strains and stresses in polycrystalline materials from strain pole figures, J. Appl. Cryst. 39, 358–368 (2006) M.B. Prime, P. Rangaswamy, M.R. Daymond, T.G. Abeln: Several methods applied to measuring residual stress in a known specimen. In: SEM Spring Conference on Experimental and Applied Mechanics (Society for Experimental Mechanics, Houston 1998) pp. 497–499 C. Genzel, W. Reimers: Depth-resolved X-ray residual stress analysis in PVD (Ti, Cr) N hard coatings, Z. Metallkunde 94(2), 655–661 (2003) J. Almer, U. Lienert, R.L. Peng, C. Schlauer, M. Oden: Strain and texture analysis of coatings using high-energy X-rays, J. Appl. Phys. 94(2), 697–702 (2003) Y. Qian, J. Almer, U. Lienert, G.B. Olson: Nondestructive Residual Stress Distribution Measurement in Nano-structured Ultra-high Strength Gear Steels, Fifth Int. Conf. Sychrotron Radiat. Mater. Sci. (Chicago 2006) pp. 27–28 U. Lienert, S. Grigull, Å. Kvick, R.V. Martins, H.W. Poulsen: Three Dimensional Strain Measurements in Bulk Materials with High Spatial Resolution, ICRS-6 (Oxford 2000) pp. 1050–1057 A. Steuwer, J.R. Santisteban, M. Turski, P.J. Withers, T. Buslaps: High-resolution strain mapping in bulk samples using full-profile analysis of energy-dispersive synchrotron X-ray diffraction data, J. Appl. Cryst. 37, 883–889 (2004) M.A.M. Bourke, D.C. Dunand, E. Ustundag: SMARTS – a spectrometer for strain measurement in engineering materials, Appl. Phys. A-Mater. Sci. Process. 74, S1707–S1709 (2002) D. Stefanescu, A. Bouzina, M. Dutta, D.Q. Wang, M.E. Fitzpatrick, L. Edwards: Comparison of residual stress measurements using neutron and X-ray diffraction around cold expanded holes, J. Neutron Res. 9(2), 399–404 (2001) P.R. Dawson, D.E. Boyce, R. Rogge: Issues in modeling heterogeneous deformations in polycrystalline metals using multiscale approaches, CMES-Comput. Model. Eng. Sci. 10(2), 123–141 (2005) G.E. Ice, B.C. Larson: 3D X-ray crystal microscope, Adv. Eng. Mater. 2(2), 643–646 (2000) H.F. Poulsen, S.F. Nielsen, E.M. Laurdisen, S. Schmidt, R.M. Suter, U. Lienert, L. Margulies, T. Lorentzen, D. Juul Jensen: Three-dimensional maps of grain boundaries and the stress state of individual grains in polycrystals and powders, J. Appl. Cryst. 34, 751–756 (2001) H.F. Poulsen, X. Fu, E. Knudsen, E.M. Lauridsen, L. Margulies, S. Schmidt: 3DXRD – Mapping grains and their dynamics in 3 dimensions, Mater. Sci. Forum 467, 1363–1372 (2004)

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28.27

R. Hill: The elastic behavior of polycrystalline aggregate, Proc. Phys. Soc. London 65, 349–354 (1952) F. Bollenrath, V. Hauk, E.H. Muller: On the calculation of polycrystalline elasticity constants from single crystal data, Z. Metallkunde 58, 76–82 (1967) A.C. Vermeulen: An elastic constants database and XEC calculator for use in XRD residual stress analysis, Adv. X-ray Anal. 44, 128–133 (2001) V. Hauk: In: Residual Stress and Stress Relaxation, ed. by E. Kula, V. Weiss (Plenum, New York 1982) ASTM: Standard test method for determining the effective elastic parameter for x-ray diffraction measurements of residual stress, E1426. In: Annual Book of ASTM Standards (ASTM, West Cohshohocken 2007) P. Dawson, D. Boyce, S. MacEwen, R. Rogge: On the influence of crystal elastic moduli on computed lattice strains in AA-5182 following plastic straining, Mater. Sci. Eng. A 313, 123–144 (2001) M.R. Daymond: The determination of a continuum mechanics equivalent elastic strain from the analysis of multiple diffraction peaks, J. Appl. Phys. 96(2), 4263–4272 (2004) M. v. Leeuwen, J.-D. Kamminga, E.J. Mittemeijer: Diffraction stress analysis of thin films: Modeling and experimental evaluation of elastic constants and grain interaction, J. Appl. Phys. 86, 1904–1914 (1999) F. Witt, R.W. Vook: Thermally induced strains in cubic metal films, J. Appl. Phys. 39, 2773–2776 (1968) A. Segmüller, M. Murakami: X-ray diffraction analysis of strains and stresses in thin films, Treatise Mater. Sci. Technol. 27, 143–197 (1988) I.C. Noyan, T.C. Huang, B.R. York: Residual stress/strain analysis in thin films by x-ray diffraction, Crit. Rev. Solid State Mater. Sci. 20(2), 125–177 (1995) ASTM: Standard test method for verifying the alignment of x-ray diffraction instrumentation for residual stress measurement, E915. In: Annual Book of ASTM Standards (ASTM, West Cohshohocken 2007) R.H. Marion: X-Ray Stress Analysis of Plastically Deformed Metals, PhD Thesis (Northwestern University, Evanston 1973) J. Jo, R.W. Hendricks: Diffractometer misalignment errors in x-ray residual stress measurements, J. Appl. Cryst. 24, 878–887 (1991) T.A. Lohkamp, R.A. Winholtz: Assessing the validity of diffraction stress data with the goodness-of-fit statistic, Adv. X-ray Anal. 39, 281–289 (1997) U.F. Kocks, C.N. Tome, H.-R. Wenk: Texture and Anisotropy (Cambridge Univ. Press, Cambridge 1998) S. Matthies, H.G. Priesmeyer, M.R. Daymond: On the diffractive determination of single-crystal elastic constants using polycrystalline samples, J. Appl. Cryst. 34, 585–601 (2001)

References

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Noncontact Methods

28.52

28.53

28.54

Part C 28

28.55

28.56

28.57

R. Barabash, G.E. Ice, B.C. Larson, G.M. Pharr, K.-S. Chung, W. Yang: White microbeam diffraction from distorted crystals, Appl. Phys. Lett. 79(2), 749–51 (2001) B. Jakobsen, H.F. Poulsen, U. Lienert, J. Almer, S.D. Shastri, H.O. Sorensen, C. Gundlach, W. Pantleon: Formation and subdivision of deformation structures during plastic deformation, Science 312(2), 889–892 (2006) N. Tamura, R.S. Celestre, A.A. MacDowell, H.A. Padmore, R. Spolenak, B.C. Valek, N.M. Chang, A. Manceau, J.R. Patel: Submicron x-ray diffraction and its applications to problems in materials and environmental science, Rev. Sci. Instrum. 73(2), 1369–1372 (2002) J. Baruchel, P. Cloetens, J. Hartwig, W. Ludwig, L. Mancini, P. Pernot, M. Schlenker: Phase imaging using highly coherent X-rays: radiography, tomography, diffraction topography, J. Synchrotron Radiat. 7, 196–201 (2000) R. Sinclair, M. Preuss, P.J. Withers: Imaging and strain mapping fibre by fibre in the vicinity of a fatigue crack in a Ti/SiC fibre composite, Mater. Sci. Technol. 21(2), 27–34 (2005) A. Pyzalla, B. Camin, T. Buslaps, M. Di Michiel, H. Kaminski, A. Kottar, A. Pernack, W. Reimers: Simultaneous tomography and diffraction analysis of creep damage, Science 308(2), 92–95 (2005)

28.58

28.59

28.60

28.61

28.62

28.63

28.64

28.65

28.66

M. Pick, R. Lovell, A.H. Windle: Detection of elastic strain in an amorphous polymer by X-ray scattering, Nature 281, 658–659 (1979) H.F. Poulsen: Measuring strain distributions in amorphous materials, Nat. Mater. 1, 33–36 (2005) T.C. Hufnagel, R.T. Ott, J. Almer: Structural aspects of elastic deformation of a metallic glass, Phys. Rev. B 73(6), 64204 (2006) J.D. Almer, S.R. Stock: Micromechanical response of mineral and collagen phases in bone, J. Struct. Biol. 157, 365–370 (2007) H.S. Gupta, P. Messmer, P. Roschger, S. Bernstorff, K. Klaushofer, P. Fratzl: Synchrotron diffraction study of deformation mechanisms in mineralized tendon, Phys. Rev. Lett. 93(15), 158101 (2004) T. Kallaste, J. Nemliher: Apatite varieties in extant and fossil vertebrate mineralized tissues, J. Appl. Cryst. 38, 587–594 (2005) T. Pirling: Stress determination with high lateral resolution using neutron diffraction, Thermec’2003, Vol. 1–5 (2003) pp. 3975–3980 Y.D. Wang, R.L. Peng, R.L. McGreevy: A novel method for constructing the mean field of grainorientation-dependent residual stress, Philos. Mag. Lett. 81(2), 153–163 (2001) H. Behnken: Strain-function method for the direct evaluation of intergranular strains and stresses, Phys. Stat. Sol. (a) 177(2), 401–418 (2000)

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Applicati Part D Applications

29 Optical Methods Archie A.T. Andonian, Akron, USA 30 Mechanical Testing at the Micro/Nanoscale M. Amanul Haque, University Park, USA Taher Saif, Urbana, USA 31 Experimental Methods in Biological Tissue Testing Stephen M. Belkoff, Baltimore, USA Roger C. Haut, East Lansing, USA 32 Implantable Biomedical Devices and Biologically Inspired Materials Hugh Bruck, College Park, USA 33 High Rates and Impact Experiments Kaliat T. Ramesh, Baltimore, USA 34 Delamination Mechanics Kenneth M. Liechti, Austin, USA 35 Structural Testing Applications Ashok Kumar Ghosh, Socorro, USA 36 Electronic Packaging Applications Jeffrey C. Suhling, Auburn, USA Pradeep Lall, Auburn, USA

823

Archie A.T. Andonian

Fundamental issues in engineering, manufacturing, medical, and other fields are now being addressed and solved by automated and userfriendly optical methods. Optical methods can provide full-field displacement and deformation fields of structures under static or dynamic conditions. Because of their noncontact nature, optical methods do not alter the response of the objects being studied. Thus, results obtained by optical methods represent reality and can be used to validate computational models. This chapter contains many examples drawn from research applications and has been prepared to give an overview of optical methods frequently used in industry and academia.

29.1 Photoelasticity ..................................... 824 29.1.1 Transmission Photoelasticity.......... 824 29.1.2 Reflection Photoelasticity .............. 827

29.2.2 Quantification of Dynamic 3-D Deformations and Brake Squeal ......................... 829 29.3 Shearography and Digital Shearography . 830 29.4 Point Laser Triangulation ...................... 831 29.5 Digital Image Correlation ...................... 29.5.1 Measurement of Strains at High Temperature ...... 29.5.2 High-Speed Spin Pit Testing .......... 29.5.3 Measurement of Deformations in Microelectronics Packaging ........

832 832 832

833

29.6 Laser Doppler Vibrometry ...................... 834 29.6.1 Laser Doppler Vibrometry in the Automotive Industry ........... 834 29.6.2 Vibration Analysis of Electron Projection Lithography Masks ....................... 834 29.7 Closing Remarks ................................... 835

29.2 Electronic Speckle Pattern Interferometry 828 29.2.1 Calculation of Hole-Drilling Residual Stresses ... 828

29.8 Further Reading ................................... 836

Sending and retrieving light signals to convey information probably dates back to prehistoric times. The use of light in optical metrology, however, can be traced back to Thomas Young’s experiments in the early 1800s in which he demonstrated that very small dimensions can be measured by the interference of two coherent rays of light [29.1]. Over a period of two centuries numerous optical methods were developed based on fundamental principles of interferometry. Not many appreciated the importance of interferometry as a tool of metrology until Theodore Maiman demonstrated the first laser in 1960 [29.2]. For several decades optical methods were developed and used by scientists, primarily in academia and industrial laboratories. Many advances in the opti-

cal methods have been stimulated by the requirements of engineering research and challenging problems in areas such as fracture mechanics, materials science, nondestructive inspection, manufacturing, biomechanics, and miniaturized structures and sensors. Although, much of the development of new optical methods have been carried out by scientists in well-equipped laboratories, most of the applications are currently performed by engineers who are not specialized in optics but own the problems to be solved in their fields. This chapter on optical methods contains many examples drawn from industrial applied research. It is not a comprehensive encyclopedia of techniques used in optical metrology, but a collection of frequently used

References .................................................. 836

Part D 29

Optical Metho 29. Optical Methods

824

Part D

Applications

Part D 29.1

methods in various fields of engineering. Instead of exploiting the underlying theory, real problems of engineering are introduced to illustrate and explain the application of various optical methods. Through these examples the full-field nature of the results, which is one of the advantages of the optical methods, is emphasized. Because of this attribute, there is no need to know the location of critical regions before studying the local response of the structure. The noncontact nature of the optical methods is another advantage in engineering applications since the concern of altering the local or global response of the test structure is eliminated. Optical methods require a meticulous approach in preparing test samples, conducting experiments, recording the events, and postprocessing the optical data for final interpretation. Recent advances in computing hardware and software, digital cameras, video recorders, and image processing systems have transformed optical methods into user-friendly tools in industrial environments. Specific details on theory and mathematical treatment of various methods have been described earlier and will not be repeated here. The chapter starts with photoelasticity, which is generally accepted as one of the earliest forms of fullfield stress analysis. In recent years, this topic has seen a rebirth of interest in transparent structures such as monitor screens, recordable discs, and windshields, providing staggering advancements in automated data

collection systems. Recent advances in digital recording and image processing have helped electronic speckle pattern interferometry (ESPI) emerge as a powerful tool for measuring structural deformations under controlled loading conditions. Some pulsed ESPI systems can produce three-dimensional (3-D) surface strains on structures during dynamic processes including transient events. The section on shearography provides applications of this method in screening structural components for surface irregularities and internal defects. Shearography is a practical measurement technique suitable for field and production applications since it does not require vibration isolation. A newer variation of this optical method, digital speckle pattern shearing interferometry, which is faster and more user friendly than conventional shearography, is also discussed. The section on laser point triangulation presents the applications of this optical method in industry for automating the reverse engineering of mechanical parts, component dimensioning, and machine vision. The section on digital image correlation emphasizes the capabilities of this relatively new method by giving multiple examples. Finally, laser Doppler vibrometry is presented as a high-precision measurement tool, and its use in vibration, noise, and modal analysis is discussed. Due to space limitations most of the basic papers and related literature could not be included in the list of references. Only recent applications and reference books are cited in the bibliography.

29.1 Photoelasticity Photoelasticity is one of the oldest optical metrology tools for engineering purposes. It makes use of stress-induced birefringence in transparent materials and provides a relationship between the change in state of stress or strain at a given point on the structure and the optical data. Photoelasticity can only provide the principle stress difference or maximum in-plane shear stress directly. Stress separation techniques need to be used to obtain the individual stress and strain components [29.3].

29.1.1 Transmission Photoelasticity

may not be valid if the structure being modeled exhibits very large deformations. Results may not be representative if the structure being modeled is made of a nonisotropic, nonhomogeneous or nonlinear material. In 3-D photoelasticity the mismatch of Poisson’s ratio between the photoelastic model and structure need to be compensated to eliminate the possibility of erroneous results. Engineering structures made of birefringent materials such as glass present a distinct advantage since the structure itself can be analyzed under polarized light, eliminating the need for photoelastic models.

In transmission photoelasticity if a birefringent plastic model is used to represent the structure care must be taken not to violate basic assumptions in the process. The results obtained from photoelastic analysis

Characterization of Buttress Threads Buttress threads [29.4] are commonly used to attach large cylinders together [29.5]. The loaded side of the buttress thread is effectively perpendicular to the load

Optical Methods

b)

Fig. 29.1 (a) Photoelastic test specimen in the test setup. (b) Image of the specimen under load

axis, minimizing the radial loading of the joint. An experimental procedure was used to validate the modeling efforts to capture the compliance and damping behavior of buttress threads under extreme axial loading conditions. A photoelastic study was conducted to visualize the stress field in a linear thread structure to compliment the mechanics study on the linear thread. Figure 29.1 shows the photoelastic test specimen in the test setup, as well as an image of the specimen under load. Figure 29.2a shows a magnified area corresponding to the center tooth and four points of interest. Figure 29.2b shows the stress levels at these target points as a function of tension acting on the buttress thread. a)

The photoelastic test results show stress fields in the form of contour lines plotted for significant stress levels, as well as the stress gradient that occurs during load application. The highest stresses are found near the edge of the tooth, and decay gradually toward the center of the specimen as expected. Evaluation of Dynamic Crack Propagation Natural-gas wells may require completion after drilling to increase the flow into the borehole. This may be accomplished by hydraulic fracturing, or by explosively loading the borehole. This is a critical procedure since the collapse of the well bore may make reentry

b) Stress (psi)

1000 900

Point 1 Point 2 Point 3 Point 4

800 Pt 4 Pt 3 Pt 2 Pt 1

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825

Part D 29.1

a)

29.1 Photoelasticity

0

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Load (lbs)

Fig. 29.2 (a) Magnified area corresponding to the center tooth and four points of interest. (b) Stress levels at the target points as a function of tension acting on the buttress tread

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for future stimulation or clean-out nearly impossible. Detailed experiments were conducted with twodimensional plate specimens fabricated from Homalite 100 to optimize the parameters of the explosive detonation. A borehole was machined into the specimen and a crack was then cut from the side of the borehole extending toward the center of the specimen. All machining was performed on a high-speed milling machine with sufficient cooling to minimize edge effects in the Homalite. The crack geometry was varied to investigate the effect of crack length (L) and crack face angle (α) on the driving energy and propagation [29.6]. Explosive charges were detonated at either the center of the borehole or at the midpoint of the crack length for each of the tests. At the moment of detonation, the firing pulse was sent to a time delay generator that in turn triggered a 24-frame Cranz–Schardin high-speed camera system. The system produced 24 snapshots of the specimen during the 400–500 μs time interval of the dynamic event. Polaroid sheets were placed on either side of the specimen along the optical path to convert the camera into a dynamic polariscope. Figure 29.3 shows a series of pictures from the camera for a typical test with the explosive in the borehole. In the first frame, at t = 50.5 μs after detonation, the compressive stress wave (P-wave) caused by detonation appears as a series of hemispherical fringes to the right of the crack tip. The complex fringe pattern to the left of the P-wave is due to a slower shear wave (S-wave) that is also generated. In the second frame, at t = 98 μs, both

t = 50.5 μs

t = 190.5 μs

t = 98 μs

t = 290 μs

of the previous stress waves have clearly propagated to the right, and the P-wave has even been reflected from the far boundary as indicated by the hemispherical fringes moving back towards the crack. In addition, a surface wave (Rayleigh wave) appears as an inclined, vertical fringe that propagates along the top and bottom edges of the specimen. Frame 3 shows the crack continuing to propagate and exhibits fringe loops on either side of the crack plane that indicate a classic tensile (mode I) loading. Frame 4 at t = 190.5 μs reveals the crack branching phenomenon. The fringe loops in frame 3 immediately before crack branching are quite large, confirming the presence of a large crack driving force. To dissipate this excess energy, the crack separates into two parts. Each of these new cracks continues to propagate to the right surrounded by independent sets of fringe loops, as seen in frames 5 and 6. Quantification of 3-D State of Strain by Stress Freezing Three-dimensional photoelasticity by the stress-freezing method is a competitive and powerful tool for determining the state of stress in complex structures and analyzing 3-D problems in fracture mechanics. Materials used for stress freezing are thermosetting polymers that experience a sudden change in properties above their glass-transition temperatures [29.7]. The example considered here focuses on the state of stress in a car engine bracket. The model, made from an optically sensitive epoxy resin (Fig. 29.4), was heated to a tem-

t = 181 μs

t = 375.5 μs

Fig. 29.3 Series of pictures from a camera for a typical test with the explosive in a borehole at different times t after detonation

Optical Methods

29.1 Photoelasticity

Peak tension Edge compression

Fig. 29.6 Compression and tension zones at the edge of a windshield Fig. 29.4 Model made from an optically sensitive epoxy

resin

perature above the stress-freezing temperature [29.8]. The model was thermally soaked for some time to ensure uniformity of temperature and then loaded. The model was slowly cooled to room temperature and the loads were removed. The optical anisotropy which can be related to the 3-D state of stress remains fixed in the model. Thin slices were removed from the stress frozen model in a predetermined orientation. The optical response of the material is not disturbed if cut with a diamond-tippted wheel rotating at very high speed. The wheel must be cooled to prevent overheating the model, which would alter the stress-frozen fringe pattern. Figure 29.5a shows a slice of interest and the frozen fringe pattern as viewed by a polariscope. Three stages of slicing and subslicing were performed and photoelastic data was recorded in each level to quantify 3-D state of stress. Figure 29.5b shows the geometric configurations for the subslices and sub-subslices. a)

The use of stress freezing in industry has declined in recent years because it is not cost effective when compared with numerical methods.

29.1.2 Reflection Photoelasticity Stress-sensitive birefringent coatings extend the classical procedures of conventional photoelasticity to the measurement of surface strains on any structure made of any engineering material [29.9]. The birefringent coating is bonded to the flat or curved surface of the structure being analyzed. When the structure is loaded, the surface deformations are transmitted to the birefringent coating. Coating stiffness is adjusted to reduce the stiffening effect of the coating layer on the structure being analyzed. The coating is usually bonded to the structure with reflective cement. Full-field isochromatic and isoclinic patterns can be analyzed quantitatively to produce the maximum shear strain contours and the direction of principle strains on the structure being evaluated.

b)

Fig. 29.5 (a) Slice of interest and the frozen fringe pattern as viewed by a polariscope. (b) Geometric configurations for sub-slices and sub-sub slices

Part D 29.1

Line profile

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Transmission and reflection photoelasticity have recently experienced a rebirth due to new advances in digital imaging and automated data acquisition procedures [29.10]. In particular, the gray-field polariscope has been applied to measure residual stress in glass in the automotive industry [29.11]. More recently, this methodology was developed into a realtime poleidoscope system for online glass inspection [29.12]. Residual Stresses in Windshields The band of edge compression around the perimeter of a windshield [29.11] is vital to the integrity of the unit. Insufficient edge compression can lead to cracking during installation or service. To satisfy equilibrium, a tension zone develops inside the compressive edge. The magnitude and location of this tension zone, however, need to be optimized to eliminate susceptibility to unstable crack growth from a stone chip. Figure 29.6

Fig. 29.7 Press form fold line in a windshield (af-

ter [29.12])

shows the compression and tension zones at the edge of a windshield. Undesirable residual stresses are frequently produced when large pieces of glass are shaped with a sag or press-form process. These unpredictable regions of stress in the windshield may make the window susceptible to stone impact or body twist loads. Figure 29.7 shows a press form fold line in a windshield [29.12].

29.2 Electronic Speckle Pattern Interferometry Electronic speckle pattern interferometry (ESPI) is a full-field optical method for measuring deformation components under static and dynamic conditions on complex surfaces. It offers advantages for cost-effective nondestructive evaluation (NDE) and measurement of surface strains and displacements [29.13]. There is no need for special specimen preparation, such as application of a grating or a coating. Different states of the structure being evaluated can be monitored and compared almost instantly [29.14]. The principle is quite simple: the surface to be measured is illuminated by laser light from different directions and the speckle images are recorded by a charge-coupled device (CCD) camera. Since the speckles change when the surface is deformed, comparisons of images under different loading conditions allow the quantification of the 3-D movement of points in the area of interest with high accuracy. From the 3-D displacement field, the complete strain field can be calculated. Examples in the literature clearly demonstrate that ESPI can be used in noisy industrial environments as a user-friendly tool by nonexperts. Applications of ESPI range from relatively simple surface inspections to 3-D strain measurements under dynamic conditions, some of which will be presented next.

29.2.1 Calculation of Hole-Drilling Residual Stresses The hole-drilling method is frequently used to quantify residual stresses in a structural element by drilling a hole [29.15] perpendicular to the surface (Fig. 29.8a) and relieving the residual stresses. Complex computational algorithms are developed to calculate residual stresses from ESPI data. Thousands of displacement data are postprocessed to produce surface deformations. Material constitutive relations and boundary conditions are then used to calculate the associated residual stresses. The light from a laser source is split using a half-silvered mirror. One part is directed through a piezoelectric actuator to provide a phase-stepped reference light to a CCD camera. The other part of the laser light is used to illuminate the specimen. When the two beams of laser light interfere they produce a speckle pattern on the image plane of the digital camera, the local phase of which varies with displacement of the specimen surface. By taking a series of phase-stepped images before and after surface deformation, it is possible to evaluate both the size and sign of the deformation at every pixel in the CCD image. Figure 29.8b shows the measured phase map which represents deformations

Optical Methods

σy θ

σx

a

τxy

a)

r σx

τxy σy

b)

b)

Fig. 29.8 (a) Hole drilling method used to quantify residual stresses in a structural element and relieving the residual stresses. (b) Measured phase map which represent deformations around the hole shown in (a) after residual strains were relieved by drilling the hole

around the hole shown in Fig. 29.8a after residual strains were relieved by drilling the hole. The light and dark fringes in Fig. 29.8b correspond to sequential half-wavelength displacements along the sensitivity direction. The central curve in Fig. 29.3b indicates the hole of radius a. The two dashed curves indicate the annular integration boundaries at two different radii. Although all these curves are concentric circles on the specimen surface, they appear as ellipses in the figure because the object beam was not normal to the specimen surface. This arrangement was chosen to provide optical clearance for the drill used to cut the hole. Perspective errors due to oblique illumination, however, can be completely eliminated by coordinate transformations during postprocessing.

Fig. 29.9 (a) Disc and caliper under acceptable operating conditions with no squeal. (b) Different disc with a caliper producing transient vibrations (break squeal)

29.2.2 Quantification of Dynamic 3-D Deformations and Brake Squeal For static or quasistatic conditions the standard ESPI system is sufficient. The investigation of vibrating structures, however, requires a pulsed ESPI system. The short illumination time of a pulsed laser freezes the object movement and can resolve deformations during steady-state dynamic or transient processes [29.16]. Pulsed ESPI techniques have been improved during the past few years [29.17, 18] and led to the development of user-friendly systems. Recent developments have extended the capabilities of pulsed ESPI techniques to 3-D measurement of dynamic behavior [29.19], providing the components of displacements and strains. To produce the 3-D deformations, the object is illuminated with short light pulses and recorded by three digital

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a)

29.2 Electronic Speckle Pattern Interferometry

830

Part D

Applications

Part D 29.3

cameras from different directions. The use of a doublepulse laser allows the acquisition of two data sets within a very short duration (a few nanoseconds). The two sets of optical information captured by three cameras are postprocessed and the complete 3-D-deformation vectors are calculated at every point on the surface being investigated. The 3-D pulsed ESPI technique allows the inspection of large areas up to several square meters. Recent industrial applications have been carried out in the fields of automotive and aerospace engineering. In [29.16], electronic speckle pattern interferometry (ESPI) was used with a pulsed laser to provide complete maps of the vibration components of the brake and

quantify brake squeal characteristics. A rotating brake disc with an engaged caliper was illuminated by two laser pulses. The images at these two laser illuminations were recorded with special high-speed cameras. The deformation maps were generated for the brake disc and caliper, describing the movement of the individual points between the two laser pulses. Special algorithms were used to remove the rotation component of the brake disc and extract the in-plane and out-ofplane components of vibrations. Figure 29.9a shows the disc and the caliper under acceptable operating conditions with no squeal. Figure 29.9b shows a different disc with a caliper producing transient vibrations (brake squeal).

29.3 Shearography and Digital Shearography Shearography or speckle pattern shearing interferometry (SPSI) is primarily used in industry to screen the structural components for surface irregularities and internal defects. It provides a full-field map of the first derivative of deformation [29.20]. Flaws usually induce strain concentrations around them, and shearography can be employed to detect those flaws. Conventional shearography uses a speckle-shearing camera, in which a thin glass wedge covers one half of the lens [29.21]. The structure being studied is illuminated by laser light, and the camera produces two sheared images that interfere with each other, producing a speckle pattern. The speckle pattern changes when the object is deformed. High-resolution film is exposed before and after the object is deformed. The two speckle patterns interfere to produce a fringe pattern that provides the gradient of surface deformations. Shearography is a practical measurement technique with a simple optical setup. It is suitable for field and production applications since it does not require vibration isolation. In the real-time mode, a live fringe pattern representing the derivatives of the surface displacements can be observed and recorded. This allows the deformations in the object to be continuously monitored under dynamic loading [29.22]. A newer variation of this optical method, digital speckle pattern shearing interferometry (DSPSI), uses a digital camera and computer image processing to capture and process the interferometric fringe patterns. The digital version is faster and more user friendly than conventional shearography. The optical arrangement for DSPSI makes use of the Michelson interferometer as

the shearing device. Laterally sheared images of the test object are produced directly on the image plane of the camera by moving one of the mirrors of the Michelson interferometer by a small amount using a piezoelectric translator. Various image processing routines are often used to improve the quality of the fringe pattern. Phase shifting is used to quantify displacement derivatives with high accuracy. To illustrate the use of shearography in the detection of internal defects, a crack was machined inside a steel pipe [29.23]. The pipe was pressurized by a hydraulic pump to unveil the effect of internal discontinuity. Figure 29.10 shows the fringe pattern due to the internal flaw as imaged by conventional shearography. The outline shows the flaw’s shape clearly, but due to the shearing effect, it appears elongated.

Fig. 29.10 Fringe pattern due to the internal flaw as imaged by conventional shearography

Optical Methods

29.4 Point Laser Triangulation

the phase map with DSPSI (AUTHOR Text to distinguish a and b, left/right)

With DSPSI (Fig. 29.11), the defect is clearly revealed in the phase map. The fringe patterns in Fig. 29.11 correspond to the out-of-plane displacement derivatives of the deformed pipe with shearing along the pipe’s longitudinal axis. In DSPSI, a 10 mW helium–neon laser was used as the light source. The resolution of the image processing system was 512 × 512 × 8 bits. Before pressurizing the pipe, three images of the speckle patterns were obtained with different phase shifts. The fourth image was captured after the pressure was applied. Real-time sub-

traction was performed by a pipeline image processor to obtain the fringe patterns of the test specimen when loaded. The images were displayed at 25 frames per second. As shown in this example, a simple yet effective optical setup with an image processing system forms a useful on-line nondestructive testing apparatus for industrial applications. The phase-shift technique provides numerical values of the displacement derivative at all points on the surface of the specimen, allowing rapid determination of the location, size, and shape of defects.

29.4 Point Laser Triangulation Laser triangulation sensors are frequently used in various engineering fields for accurately measuring distances [29.24]. The laser beam is projected from the sensor and is reflected from a target surface to a recording medium through a collection lens. This lens is typically located adjacent to the laser emitter such that the sensor enclosure, the emitted laser, and the reflected laser light form a triangle. The lens focuses an image of the spot on a linear array camera (CCD array). Depending on the design of the collection lens system, the line array views the measurement range from an angle that varies from 45◦ to 65◦ degrees at the center of the measurement range. Figure 29.12 illustrates the principle of laser scanning by triangulation. The included angle between the axes of illumination and observation, the distance between the CCD array and the light source, and the design of the collecting lens system are all critical parameters for determining the position of the target. The position of a target spot on the CCD array is processed to determine the distance to the target. The beam is viewed from one side so that the apparent location of the spot changes with the distance to the target. Triangulation devices are ideal for measuring distances of a few inches with high accuracy. Triangulation de-

CCD array Lens system Target

d α

Laser emitter

Axis of observation

Fig. 29.12 Principle of laser scanning by triangulation

vices may be built on any scale, but their accuracy falls off rapidly with increasing range [29.25]. Laser triangulation is frequently used for automating the reverse engineering of mechanical parts, component dimensioning, and machine vision. Manufacturing professionals use laser triangulation systems to create new patterns for prototype tooling and computer numerical control (CNC) machining [29.26]. Medical professionals use laser triangulation to prepare 3-D profiles of bones and other body parts for the development of implants. Laser triangulation devices are relatively inexpensive, off-the-shelf items and can be easily calibrated and set up for various applications.

Part D 29.4

Fig. 29.11 Defect clearly revealed in

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29.5 Digital Image Correlation Two- and three-dimensional digital image correlation methods are full-field displacement and strain measurement tools increasingly used for static and dynamic testing [29.27]. Sample preparation consists of applying a random high-contrast dot pattern to the surface. This step can be omitted if the sample surface has a highcontrast granular morphology. Two images obtained by cameras from different directions are processed by special image processing algorithms to identify the location of each dot on the surface and calculate the displacement of each target point after loading the object of

interest. The displacement field is finally postprocessed to produce the full-field state of strain on the surface. Thousands of overlapping unique correlation areas are defined in each digital image. The centers of correlation areas are tracked in each successive pair of images. Thus a 3-D image correlation system can provide three layers of information: the topography of the component, the displacement fields in three directions, and the in-plane strains. The method is extremely robust since the surface topography and strain measurements are not affected by rigid-body motions. In high-speed applications, as long as pairs of nonblurred pictures can be captured, 3-D coordinates, 3-D displacements, and the in-plane strains can be measured [29.28]. Applications of digital image correlation range from relatively simple surface strain measurements in material test laboratories to 3-D shape and strain measurements under dynamic conditions, as illustrated by the following examples.

29.5.1 Measurement of Strains at High Temperature

05096

Three-dimensional image correlation can be used in hazardous environments as long as the components of the system are properly protected from thermal or chemical hazards. If the component to be studied needs to be in an environmental chamber, the window becomes part of the optical system and needs to be in the optical path during calibration. Surface preparation is particularly important in such applications since the modification or deterioration of the surface pattern can lead to significant errors. A bimetallic component to be tested under tension was heated to elevated temperatures [29.28]. Figure 29.13 shows the shape of the part and the strains from the tensile load at high temperature.

0.4395

29.5.2 High-Speed Spin Pit Testing

Fig. 29.13 Shape of the part and the strains from the tensile load at high temperature

a)

b)

Strain X (Technical %) 0.5797

0.3694 0.2993 0.2292 0.1591 0.0890 0.0189

Fig. 29.14 (a) Image correlation camera mounted in a massive steel cylinder before insertion into the containment vessel. (b) Radial

strain result at 35 000 rpm showing significant banding effects and a maximum radial strain of 5797 microstrain

Spin testing is an essential step in the prevention of centrifugal burst disasters [29.29]. Developers and manufacturers of turbo machinery components need to test for centrifugal strength, to verify stress analysis and establish fatigue lifetimes. Pulsed two-dimensional (2-D) image correlation was used to study a composite flywheel with a flat surface at high speed and measure the in-plane strains. Since no out-of-plane displacements were expected, the test setup was greatly simplified by mounting a single camera in the spin pit. Figure 29.14a shows the image correlation camera mounted in a massive steel cylinder, before insertion

Optical Methods

29.5 Digital Image Correlation

Part D 29.5

EMC

Gold wire Photosensitive resist (PSR)

Die (face-up) Die adhesive

0.027

8 mm

0.025

0.033

BT core (BT resin + 0.033 fabric glass fibers)

0.25

0.027

Copper pattern 9.02 mm

Solder ball

Plated through hole (PTH)

Fig. 29.15a,b Cross section (a) and top view (b) of a FBGA package with 2 layer PCB (Courtesy of Samsung); epoxy molding compound (EMC); bismaleimide triazine (BT); (all units are in mm)

a)

Disp. z (mm)

b)

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c)

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d)

Disp. z (mm)

0.0020

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Fig. 29.16 Series of images of the same FBGA package (Fig. 29.15) showing the progression of warping during cool-

down

into the containment vessel. To protect the camera in the event of a flywheel burst, it was placed behind a thick polycarbonate viewing window. A pulsed yttrium aluminum garnet (YAG) laser was used to reduce the exposure time for each image acquisition to 6 ns, thus eliminating any smearing of the pattern at high speeds. This allowed the use of a standard 20 frame per second camera. A once-perrevolution signal from a tachometer provided the input to a trigger module, which opened the camera shutter within 5 μs. A delayed sync output signal from the camera shutter was used to trigger the pulsed laser. Figure 29.14b provides radial strain result at 35 000 rpm, showing significant banding effects and a maximum radial strain of 5797 microstrain.

29.5.3 Measurement of Deformations in Microelectronics Packaging

tions and straining during thermal cycling. Resultant fatigue cracking or bond failure is often the limiting Displacement Z (mm) 0.004 0.002 0 –0.002 –0.004 –0.006 –0.008 –0.01 –0.012 –0.014 –0.016

Thermally induced warpage is a significant cause of failure in electronics packages [29.30]. Intrinsic coefficient of thermal expansion mismatches between chips, substrates, and solder bumps lead to deforma-

0

1.18

2.36

3.53

4.71

833

5.89

7.07

8.24

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10.6

Surface distance (mm)

Fig. 29.17 Plot of out-of-plane displacements along the diagonal

section line shown in Fig. 29.16 indicating warpage

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Part D 29.6

factor in device lifetime. A 3-D image correlation system was used to quantify thermally induced strains on an fine-pitch ball grid array (FBGA) package during cool-down. Figure 29.15 shows the crosssection and top view of a FBGA package with two-layer printed circuit board (PCB) (courtesy of Samsung). Figure 29.16 provides a series of images of the same FBGA package showing the progression of warp-

ing during cool-down. The spreading of the maximum displacement to the top edge on the right side was consistent during the cool-down process. Nonuniform out-of-plane displacements may indicate possible defects such as internal debonding. Figure 29.17 plots out-of-plane displacements along the diagonal section line shown in Fig. 29.16, indicating warpage. Each line corresponds to a different temperature during cool down.

29.6 Laser Doppler Vibrometry Laser Doppler vibrometry is a noncontact measurement technique for measuring velocity and vibrations on moving surfaces. Laser Doppler vibrometers (LDVs) are well suited for measuring vibrations where alternate methods simply cannot be applied. Contacting transducers may fail when attempting to measure high amplitudes. LDVs can measure vibrations up to the 30 MHz range with very linear phase response and high accuracy. Laser Doppler vibrometry can be applied in complex applications, where contact sensors cannot make a measurement, such as making measurements on very hot objects. The laser Doppler vibrometer uses the principle of optical interference, requiring two coherent light beams to overlap [29.31]. The resulting interference relates to the path length difference between both beams.

to predict and optimize the vibration characteristics of automotive components and systems. Laser Doppler vibrometers are used to acquire vibration data and validate computational models. Scanning vibrometers can characterize the dynamic response of an entire surface driven by a broadband mechanical excitation such as an electrodynamic shaker and map the out-of-plane vibration. By combining a single-point laser vibrometer with 2-D fast-scan mirrors, vibrations can be characterized over an entire surface. The deflection shapes in the frequency or time domain can be calculated and superimposed on the structure being analyzed. Figure 29.18 illustrates the dynamic response of various components on a car.

29.6.1 Laser Doppler Vibrometry in the Automotive Industry

Electron-beam projection lithography (EPL) masks [29.33] are fabricated to carry the patterns for massproducing integrated circuit chips. Dynamic analysis of the mask is important when the mask is mounted in a cleaning device. To characterize the dynamic response of the mask, the effect of an annular mount and a fourpad mounting scheme have been analyzed. The mask was mounted on an electromagnetic shaker as shown in Fig. 29.19.

Optimization of vibration and acoustic characteristics has become a high priority in the automotive industry [29.32]. Laser vibrometry is frequently used in design development and experimental modal analysis of automotive components. Computational tools are used

29.6.2 Vibration Analysis of Electron Projection Lithography Masks

Fig. 29.19 Mask Fig. 29.18 Dynamic response of various components on

a car

mounted on an electromagnetic shaker (see text)

Optical Methods

Experiment (in air)

Simulation (vacuum)

1

2

3

4

5

6

The root-mean-square (RMS) velocity at the surface of an 8-inch mask was recorded by measuring the Doppler shift in the laser point source. The response of the mask was observed as the frequency was swept from 100 to 800 Hz. At resonant frequen-

Fig. 29.20 The first six out-of-plane natural frequencies and corresponding mode shapes determined experimentally to validate the FE models

cies, a large increase in the amplitude of the resulting oscillations indicated a natural mode of vibration. At these points, the frequency was recorded and fixed, and the entire surface of the mask was mapped using a scanning LDV. The first six out-of-plane natural frequencies and corresponding mode shapes (Fig. 29.20) were determined experimentally to validate the finite element (FE) models. The intersections of the grid lines on the vibrometer plots from the experimental investigations indicate the points of observation. The lines on the plots from the numerical simulations identify the orientation and location of the grillage structure. The FE plots illustrate the mode shapes with the maximum positive and negative values. Since the vibrometer determines the RMS surface velocity, the vibrometer maps are the absolute values of the displacements. The effects due to the elasticity of the aluminum fixture, friction between the gasket and the mask, structural damping of the mask, and the mass of the air enveloping the mask were ignored in the FE simulations. For mask cleaning purposes, it is essential to produce an acceleration field that provides a uniform and substantial level of excitation over the pattern region to ensure that there are no dead spots. By numerically superimposing modes, a synthesized acceleration field that is larger and more uniform within the pattern area can be produced. In this manner, larger force transmission to the mask surface for debris removal can be achieved with smaller excitation energy levels than would be needed for individual mode excitation.

29.7 Closing Remarks The role of optical methods in mature and emerging fields of technology is growing and producing new demands on automation, resolution enhancements, miniaturization, and speed. Sophisticated computational capabilities, new image processing methods, advanced sensors of recording media, and better illumination sources have contributed considerably to the evolution of optical methods. Driven by the continued miniaturization in nanotechnologies, the demand for resolution enhancement techniques will occupy an important place in optical methods. Because of their

noncontact nature, optical methods are expected to play an important role in inspection routines for nanotechnology, studying live subjects in medical fields, and characterizing the response of structures in extreme environments in addition to mature technical fields. The author’s wish is that this chapter will be useful to the reader in choosing an optical method appropriate for the application at hand. This introductory chapter does not provide a comprehensive list of optical methods for obvious reasons. Some effort has been expended in choosing methods which are reasonably automated

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and easy to use by application engineers in an industrial environment. Given the breadth of coverage and limited space, however, it is inevitable that some statements may be misunderstood or misinterpreted. The reader

who intends to use this chapter as a starting point to adopt an optical method in an application is strongly advised to become familiarized with the technical details provided in the related references.

29.8 Further Reading •

G. Cloud: Optical Methods of Engineering Analysis (Cambridge Univ. Press, Cambridge 1998)

•

N. A. Halliwell: Optical Methods in Engineering Metrology. In: Laser Vibrometry, ed. by D.C. Williams (Chapman Hall, London 1993)

References 29.1 29.2 29.3 29.4

29.5

29.6

29.7 29.8

29.9

29.10

29.11

F. Träger (Ed.): Springer Handbook of Lasers and Optics (Springer, Berlin, Heidelberg 2006) J. Hecht: Understanding Fiber Optics, 4th edn. (Prentice Hall, Upper Saddle River 2002) J.F. Dally, W.F. Riley: Experimental Stress Analysis (McGraw-Hill, New York 1991) B.R. Rogillio, J.D. Sanders: An experimental procedure to validate a 2D plane stress buttress thread loaded in tension, Proceedings of the 2005 SEM Annual Conference on Experimental and Applied Mechanics (Portland 2005), paper 199 R.E. Green, C.J. McCauley, E. Oberg, F.D. Jones, H.L. Horton, H.H. Ryffel: Machinery’s Handbook, 25th edn. (Industrial, New York 1996) R.J. Bonenberger, H.U. Leiste, S.J. Spencer, W.L. Fourney: An inverse problem approach to model studies of well stimulation, Proceedings of the 2004 SEM International Congress on Experimental Mechanics (2004), paper 352 J. Cernosek: Three dimensional photoelasticity by stress freezing, Exp. Mech. 20, 417–426 (1980) U. Galietti, C. Pappalettere, V. Quarta: Photoelastic characterization of stereolithographic epoxy resins and study of a car engine bracket 3-D model, Proceedings of the 2002 SEM Annual Conference on Experimental and Applied Mechanics (Milwaukee 2002), paper 183 F. Zandman, S. Redner, J.W. Dally: Photoelastic coatings, Soc. Exp. Stress Anal. Monogr. 1, 111–156 (1977) E.A. Patterson, Z.F. Wang: Simultaneous observation of phase-stepped images for automated photoelasticity, J. Strain Anal. Eng. Des. 33(1), 1–15 (1998) J. Lesniak, M.J. Zickel, D.J. Trate, R. Lebrecque, K. Harkins: Residual stress measurement of automobile windshields using gray-field photoelasticity, Proceedings of the 1999 SEM Annual Conference on Experimental and Applied Mechanics (Cincinnati 1998) pp. 860–862

29.12

29.13

29.14

29.15

29.16

29.17

29.18

29.19

29.20

29.21

29.22

29.23

J. Lesniak, S.J. Zhang, E.A. Patterson: Design and evaluation of the poleidoscope: A novel digital polariscope, Exp. Mech. 44(2), 128–135 (2004) R. Jones, C. Wykes: Holographic and Speckle Interferometry, 2nd edn. (Cambridge Univ. Press, Cambridge 1989) A. Ettemeyer: Non contact and whole field strain analysis with a laser-optic strain sensor, VIII International Congress on Experimental Mechanics (Nashville 1996) G.S. Schajer, M. Steinzig: Full-field calculation of hole-drilling residual stresses from ESPI data, Exp. Mech. 45(6), 526–532 (2005) R. Krupka, T. Waltz, A. Ettemeyer: New techniques and applications for 3-D brake analysis, 2000 Society of Automotive Engineers Brake Colloquium (San Diego 2000), paper 00BRAKE-37 G. Pedrini, B. Pfister, H.J. Tiziani: Double pulse electronic speckle interferometry, J. Mod. Opt. 40, 89–96 (1993) G. Pedrini, H.J. Tiziani: Double pulse electronic speckle interferometry for vibration analysis, Appl. Opt. 33, 7857–7863 (1994) Z. Wang, T. Walz, A. Ettemeyer: 3D-PulsESPI technique for measurement of dynamic structure response, XVIII Conference & Exhibition on Structural Dynamics (San Antonio 2000) J.A. Leendertz, J.N. Butters: An image-shearing speckle-pattern interferometer for measuring bending moments, J. Phys. 6, 1107–1110 (1973) Y.Y. Hung, R.E. Rowlands, I.M. Daniel: Speckleshearing interferometric technique: a full-field strain gauge, Appl. Opt. 14(3), 618–622 (1975) Y.Y. Hung, S. Tang, J.D. Hovanesian: Real time shearography for measuring time-dependent displacement derivatives, Exp. Mech. 34(1), 89–92 (1994) S.L. Toh, F.S. Chau: Using shearography to find the flaws, 1997 ASME Asia Congress & Exhibition in Singapore (1997), paper 97-AA-63

Optical Methods

29.25

29.26 29.27

29.28

29.29

D. Ravaiv, Y.H. Pao, K.A. Loparo: Reconstruction of 3-D surfaces from 2-D binary images, IEEE Trans. Robot. Autom. 5(5), 701–710 (1989) J. Clark, A.M. Wallace, G.L. Pronzato: Measuring range using a triangulation sensor with variable geometry, IEEE Trans. Robot. Autom. 14(1), 60–68 (1998) M.P. Groover: Fundamentals of Modern Manufacturing, 2nd edn. (Wiley, New York 2002) M.A. Sutton, S.R. McNeill, J.Y. Helm: Threedimensional Image Correlation for Surface Displacement Measurement, Proceedings of SEM Spring Conference (1995) pp. 438–446 T. Schmidt, J. Tyson, K. Galanulis: Full-field dynamic displacement and strain measurement using advanced 3-D image correlation photogrammetry – part I, Exp. Techn. 27(3), 47–50 (2003) T. Schmidt, J. Tyson, K. Galanulis: Full-field dynamic displacement and strain measurement

29.30

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29.33

using advanced 3-D image correlation photogrammetry – part II, Exp. Techn. 27(4), 44–47 (2003) J. Tyson, T. Schmidt: 3-D image correlation for micron-scale measurement in microelectronics packaging, Proceedings of the 2004 SEM International Congress on Experimental Mechanics (Costa Mesa 2004), paper 447 E.M. Lawrence, C. Rembe: Micro motion analysis system for MEMS characterization, Proceedings of the 2004 SEM International Congress on Experimental Mechanics (Costa Mesa 2004), paper 244 Polytec GmbH: Laser Measurement Systems Application Note VIB-C-02 (Polytec GmbH, Waldbronn 2005) C. Chen, R. Mikkelson, R.L. Engelstad, E.G. Lovell: Vibrational analysis of 8-in. electron projection lithography, Proceedings of the 2002 SEM Annual Conference on Experimental & Applied Mechanics (Milwaukee 2002), paper 139

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30. Mechanical Testing at the Micro/Nanoscale

Mechanical testing of micro- and nanoscale materials such as thin films, nanotubes and nanowires, and cellular and subcellular biomaterials is a significant step towards the realization of nanoscale devices and is essential for the commercialization of microscale integrated systems. The challenges in mechanical testing at these smaller length scales emanate from the very basic (specimen preparation and manipulation, high-resolution force and displacement sensing) to complex (enhanced multiphysics coupling, specimen–environment interaction) experimental issues. In this chapter, we attempt to focus on these issues in light of the existing and potential solutions as we review the state of the art in micro- and nanoscale mechanical testing as well as our understanding of materials behavior at these length scales. This chapter is divided into six sections. The first section introduces the research problem in micro/nanomechanical testing and describes the evolution for the need of this research area with examples in micro/nanoscale materials phenomena. The second section describes the challenges in nanomechanical testing and commonly studied materials. The third section provides a broad overview of the available tools and techniques. The vastness of the relevant literature means that this review essentially is incomplete and instead, the section focuses on the techniques that are most commonly used or have very high potential.

This chapter is an account of the recent progresses in the development of characterization tools for materials with at least one dimension at the micro/nanoscale. These are bulk three-dimensional materials with ultrafine grains, two-dimensional materials such as thin films with micronand sub-micron-scale thickness, one-dimensional nanotubes and nanowires with diameters on the order of a few to few tens of microns, and biological materials that

30.1 Evolution of Micro/Nanomechanical Testing........... 840 30.2 Novel Materials and Challenges ............. 841 30.3 Micro/Nanomechanical Testing Techniques ............................... 30.3.1 Classification ............................. 30.3.2 Challenges in Micro/Nanomechanical Testing . 30.3.3 Micro/Nanomechanical Testing Tools .............................. 30.3.4 Nontraditional (MEMS) Tools ........ 30.3.5 AFM-Based Tools ........................ 30.3.6 Nanoindentation ....................... 30.3.7 Other Tools ................................

842 842 843 844 847 851 853 856

30.4 Biomaterial Testing Techniques ............. 856 30.5 Discussions and Future Directions .......... 859 30.6 Further Reading ................................... 862 References .................................................. 862

The fourth section is devoted to bio-materials, which are ultra compliant and sensitive to the environment and thus present the ultimate challenge in mechanical testing. Section five provides a critical analysis of the available techniques and the emerging problems in mechanical testing and opportunities and challenges therein. Finally, section six suggests further readings.

range from single cells (diameter on the order of tens of microns) to nanoscopic protein filaments. Mechanical testing is a vigorously active research area, yet only a few comprehensive reviews [30.1–6] exist in the literature. Even though it is difficult to dissect mechanics of materials from the micro/nanocharacterization research, this chapter is intended to focus primarily on characterization techniques.

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30.1 Evolution of Micro/Nanomechanical Testing

Part D 30.1

Mechanical testing of materials has been a critical research area since the era of industrial revolution. At the macroscale, many standardized techniques have been developed to measure elastic modulus, yield strength, fatigue strength, fracture toughness, and other properties such as adhesion and residual stress which are essential for designing reliable machines and structures. The preoccupation of materials properties with the microstructural dimensions and features (grain size, dislocation density) is well established in the literature [30.9], without any consideration of the specimen size itself. This is because, for all practical purposes, macroscopic specimens are orders of magnitude larger than the microstructural features. The steady growth of the microelectronic industry since the 1960s led to the development and application of microscale materials (mainly thin films) in which the microstructural feature sizes are on the same order as the material’s structural dimensions. However, there was limited interest in exploring the mechanical properties of thin films or microscale materials. This is probably not because mechanical properties of materials were deemed to be irrelevant to microelectronic processing, but rather because of the focus on improving performance and reliability of the purely electrical aspects of the semiconductor devices. The advent of micro-electromechanical systems (MEMS) in the 1980s resulted in microscale sensors and actuators that included mechanical structures as moving parts, producing or experiencing mechanical force and displacement [30.10]. As MEMS continually

grew to be a truly multidisciplinary research area, the mechanical properties of materials at the microscale appeared to be more relevant since the performance a) E/E0 and G/G0 1.1

1

0.9

0.8

0.7

0

20

40

60

80 100 Grain size (nm)

b) τ (MPa) 1000 100 nm

25 nm

10 nm

800

600

400

200

0

Fig. 30.1 Warping due to intrinsic stresses in thin-film materials renders fabrication of even the simplest mechanical structures (a cantilever beam) challenging

0

0.1

0.2

0.3

0.4 d 1/2 (nm1/2)

Fig. 30.2a,b Breakdown of scaling laws at the nanoscale c MRS) for (a) elastic (after [30.7], with permission  c Elseand (b) plastic (after [30.8], with permission  vier) behaviors in iron and copper. In (a), E 0 and G 0 are Young’s and the shear moduli, respectively, for largegrained polycrystalline iron. The dashed and the solid lines are predicted Young’s modulus with grain boundary thicknesses of 0.5 and 1 nm, respectively. Open circles are experimental values of E/E 0 . In (b), different symbols represent different sources of experimental data. These results highlight the significance of mechanical testing at micro/nanoscale

Mechanical Testing at the Micro/Nanoscale

1. Elastic properties of materials: The Young’s modulus E of a material is a function of fundamental properties such as the atomic weight, atomic size, and packing factor and is generally regarded to be independent of the specimen size at the bulk scale. Contrary to this concept, the Young’s modulus has been observed to be lower than the bulk values for nanocrystalline materials [30.15–19] with grain size (or specimen dimension) less than about 100 nm. This is shown in Fig. 30.2a. This anomaly has been attributed to synthesis defects [30.20, 21], grain boundary compliance [30.17, 22], and anelasticity [30.23–25], which are characteristics of polycrystalline materials. Interestingly, the same effect has also been observed in nanoscale singlecrystal [30.26] and amorphous [30.27] materials. 2. Plastic properties of materials: Yield strength (σ y ) is related to grain size (d) as σ y = σ0 + kd −1/2 ,

(30.1)

where σ0 and k are material-dependent constants. This relationship, known as the Hall–Petch equation, breaks down for grain sizes below 1 μm [30.28, 29]. This is shown in Fig. 30.2b, where the peak in the strength–size curve indicates that materials behave softer than predicted by the Hall–Petch equation at the nanoscale [30.30]. This anomalous behavior has been attributed to specimen preparation defects [30.20], suppression of dislocation pileups [30.31, 32], enhanced Coble creep [30.8, 33, 34], grain boundary sliding [30.19, 35, 36], dislocation-transparent grain boundary [30.37–39], and high volume fraction of disordered grain boundaries [30.40].

30.2 Novel Materials and Challenges In the last decade, synthesis and application of novel nanostructured materials such as nanotubes and nanowires have received remarkable attention. Unlike bulk three-dimensional (3-D) nanocrystalline materials or two-dimensional (2-D) thin films, nanotubes and nanowires are truly one-dimensional (1-D) nanoscale materials with both dimension and microstructural length scales at the nanoscale. For example, carbon nanotubes are single- (SWNT) or multiwalled (MWNT) are seamless cylinders of carbon atoms, a few nanometers in diameter. Nanowires are solid materials, tens of nanometers in diameter and a few micrometer long.

Extreme dimensional confinements, predominance of interfacial phenomena, unique atomic configuration, and novel structural characteristics lead to unusual materials properties. For example, carbon nanotubes are deemed to have the best performance of any known material: electrically (1 × 109 A/cm2 currentcarrying capacity, quantum conductance, field emission at 3 V/μm), mechanically (20 times stronger and 6 times lighter than steel), and thermally (6000 W/mK thermal conductivity and stable up to 2800 ◦ C in vacuum) [30.41]. Nanowires show remarkable electrical, mechanical, and optical properties [30.42,43]. Although

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and reliability of MEMS devices now encompass the mechanical as well as the electrical aspects of the devices. Many applications evolved, where device performance depends on the mechanical properties of the materials used; for example, residual stresses and interfacial adhesion may affect the reliability of the multilayer electronic structures, while mechanical components such as cantilevers, gears, and micromirror arrays may fail due to fatigue, fracture or other phenomena, none of which are well understood at the microscale [30.11,12]. An example is given in Fig. 30.1, where intrinsic stresses developed during the deposition of micromechanical structures make even manufacturing of the simplest structure (a cantilever beam) difficult. The rapid development of MEMS and the emergence of the new class of bulk nanocrystalline (NC) materials led to the exploration and exploitation of size effects (mainly scaling of physical laws) in electrical, mechanical, and thermal domains [30.13] and very soon it was realized that, as specimen dimensions approach their microstructural length scale (grain size, dislocation spacing), materials exhibit very different properties that may not be extrapolated from that observed at the macroscale [30.1]. The problem is further complicated by the fact that mechanical properties of microscale (and nanoscale) materials are significantly affected by the fabrication processes, and are very sensitive to the influences of interfaces and adjoining materials [30.14]. The indispensability of mechanical testing at these length scales is obvious from the following two classic examples of the breakdown of the simple scaling approach to the mechanical behavior of materials.

30.2 Novel Materials and Challenges

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much of the focus in nanotechnology is on the electronic and magnetic properties of nanoscale devices, device performance in these applications depends on the mechanical properties of the materials used. Therefore, the ability to understand and ultimately manipulate the mechanical behavior of a wide range of materials and devices at this scale has become a major research theme. However, reproducible, accurate, and reliable mechanical testing techniques in this realm are still rare and a tremendous opportunity exists for nanomechanical characterization tools. Yet another opportunity is on the horizon for the nanomechanical characterization community. Catalyzed by the multidisciplinary aspects of nanotechnology, the bridging between the physical and biological sciences has received attention as a high-priority research area. The relevance of mechanical testing of micro- and nanoscale biomaterials is evident from our own body components; our bone, heart, muscle, lung, blood, and kidney cells experience forces that are transduced to biochemical responses (mechanotransduction). It is now accepted that mechanical

properties of cells (deformability, adhesion, mobility) critically influences the cell’s normal health and functioning [30.44–47]. Testing of biomaterials, ranging from biomolecules to single cells, pose the ultimate challenge in mechanical testing because they 1. are at least six orders of magnitude softer than engineering materials, requiring the tools to have piconewton force and nanometer displacement resolutions [30.48, 49], 2. make discrete contacts with the substrate (called focal adhesion complex, FAC) that depend on chemical cues and external mechanical load [30.50], and 3. reorganize their structure and counteract external forces as a living being [30.51], the magnitude of which depends on the substrate and the nature (shear, normal or both) of loading [30.52]. Also, mechanical properties of biomaterials in this realm are governed by multifield (electrical, mechanical, thermal, chemical) physics [30.53–55] and characterization in the individual domains provides only one facet of the entire problem.

30.3 Micro/Nanomechanical Testing Techniques The field of mechanical testing has been vigorously active in adaptation of existing techniques and development of new ones to meet the challenges mentioned above. The most commonly measured properties are Young’s modulus, yield strength, ultimate strength and strain, hardness, fracture toughness, and fatigue strength. All of these properties can be measured for microscale specimens. However, measurement of even the Young’s modulus remains to be challenging for one-dimensional (nanotubes and nanowires) and soft biological specimens.

30.3.1 Classification Both conventional and modern techniques can be classified into two broad categories, as follows. Uniform Deformation In this category, the stress (and strain) field is uniform throughout the specimen cross-section and gage length. In a tensile test, precautionary measures such as constant specimen cross-section, longer gage length with respect to diameter, and dog-bone-shaped ends help obtaining truly uniform stress field in the gage length

portion. This allows direct and accurate measurement of stress and strain. Uniform deformation field can also be obtained by axisymmetric (three-point) bending if the flexural rigidity of the specimen is negligible and the specimen is long enough to confine bending moments to the specimen ends only [30.56]. The advantage is that data interpretation is relatively straightforward and both elastic and plastic properties can be measured without any modeling efforts. Gradient-Dominated Deformation In this category, the stress (and strain) field is not uniform throughout the specimen cross-section and length. Rather, a gradient (zero stress and strain in the neutral axis of the specimen and maximum at the outer surfaces) exists. The type of stress depends on the type of loading: pure shear and flexural stress in cases of torsional and bending loading, respectively. Gradientdominant deformation is also observed in fracture testing (stress field tapering down from the highest value at the crack tip to the far-field value) and indentation tests, where a small volume of material just below the indenter tip experiences the highest stress. Model development (often working satisfactorily in the

Mechanical Testing at the Micro/Nanoscale

30.3.2 Challenges in Micro/Nanomechanical Testing The basic concepts of tests such as tensile, bending, torsion, and indentation, well standardized [30.59] and applied at the macroscale, do not change at these smaller length scales. Instead, the way specimens are prepared and gripped and the way forces or displacements are applied and measured change. The major reason for this is the invisibility of the specimens to the naked eye and the resulting inability to manipulate objects at the smaller length scales. Whether it is adaptation of macroscale tools or the development of a new tool, the following challenges are often encountered. Specimen Preparation of a micro/nanoscale specimen requires considerable effort and caution because specimens must be similar in size to structural components, and produced by the same manufacturing processes as the components [30.57]. Residual stress in specimens may appear to be a menace, as seen in Fig. 30.1. Positioning the specimen on to the test bed (with perfect alignment) and gripping the specimen with the desired boundary conditions is difficult. For example, how does one grip a carbon nanotube that is about 10 nm in diameter and 2 μm long?

Visualization Although not a requirement for macroscale experiments, visualization may be critical to the successful characterization and interpretation at smaller scales. This is because of the numerous possible artifacts in specimen and experimental setup and unexpected events or modes of deformation arising from the difference in the length scales. For example, Fig. 30.3 shows optical micrographs of axisymmetric and cantilever bending testing where the load is being applied with a MEMS probe. Etching of the substrate material during specimen preparation alters the position of the edges, which cannot be readily accounted for unless the experimental setup is tilted by some angle. The unaccounted for change in specimen length or support compliance introduces errors in experimental results and analysis. As a result, microscopy will play an indispensable role in the future of micro- and nanoscale mechanical testing. a)

MEMS probe

Bending specimen

Apparent length True length

b) MEMS probe

Bending specimen Apparent edge

Force and Displacement Measurement Smaller specimens produce even smaller deformation under smaller applied forces. This necessitates new force and displacement sensors that are preferably smaller in size and of very high resolution. This is particularly challenging because the best resolution offered by modern off-the-shelf load cells is about 1 mN, whereas resolutions on the orders of micro-, nanoand piconewtons are required for thin films/nanowires, nanotubes and single cells, and single biomolecules (DNA or other protein filaments), respectively.

True edge

Fig. 30.3a,b Optical micrographs of an experimental

setup (tilted view) that bending experiments to be carried out on a 100 nm-thick aluminum specimen ((a) three-point and (b) cantilever bending). Visualization can be a critical requirement in micro/nanomechanical testing to ascertain correct experimental conditions

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elastic regime only) is a required step to couple the stress field to strain field and thus measure the elastic properties. For this reason, these techniques are also known as inverse techniques [30.57]. Advantages of these techniques, on the other hand, are the ability to locally interrogate heterogeneous material, smaller force requirement, and larger values of displacement in the transverse direction compared to the larger force and smaller displacement in the axial direction in tensile tests [30.58].

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As mentioned earlier in this section, the basic concepts or techniques for mechanical testing change very little with the specimen length scale. Most of the literature reviews [30.1–6] classify and describe these techniques (tensile, bending, torsion, and indentation) and the reader is encouraged to go through these references or handbooks for mechanical testing [30.59]. Since major changes become rather necessary in specimen preparation techniques and force and displacement measurement tools, this chapter takes an alternate approach by highlighting the tools for micro/nanomechanical testing. It is expected that this approach will be more beneficial for a researcher to determine the most effective tool(s) to carry out the testing. Tools for micro/nanoscale biomaterials testing is presented in a separate subsection because of the inherent differences in the experimental environment. Adapted Classical Tools In conventional tensile testing, the specimen is gripped by mechanical jaws that are given relative motion through cross-heads [30.61]. A load cell is used to measure force in the specimen and foil gage extensometers are attached directly to the specimen to measure displacement. One of the earliest adaptations of this technique (probably coining the term nanotensilometer) is due to Andeen et al. [30.60], first published in 1977. Figure 30.4 shows a schematic design. Two separate platforms (used as mechanical jaws for the specimen) are supported by thin shims on a base. This allows the platforms to displace as guided beams whose spring constant is given by the geometry and elas-

Microscope objective Specimen Motion

Platform

Base Shims

Fig. 30.4 Schematic design of one of the earliest nanotensilome-

ters [30.60]

tic properties of the shims. The specimen (separately fabricated on a glass cover slip) is mounted on these platforms. One of the platforms is given an external displacement, which applies the load to the specimen and induces displacement in the other platform. Specimen elongation is measured directly with an optical microscope. The force on the specimen is obtained by measuring the induced displacement in the platforms with a capacitive sensor (0.1 nm resolution) and multiplying by the effective spring constants of the shims. The nanotensilometer could measure forces in the range 5 × 10−1 –10−8 N. This tool has been used to measure Young’s modulus and ultimate tensile strength of freestanding aluminum and aluminum/alumina multilayers as thin as 50 nm [30.62]. Specimen preparation involved coating of water-soluble film on a cover slip and subsequent evaporation of aluminum. Putting the cover slip in water would result in floating off of the aluminum films, which were collected on a specimen holder. The major drawbacks of the technique are the difficulties in preparing a virgin flat specimen (due to fluidic loading) and measuring the elongation in the specimen gauge section. A modern, representative tool for tensile testing of microscale specimens is due to Sharpe and coworkers [30.63,64]. The setup (shown in Fig. 30.5a) employs a load cell with 5 μN resolution to measure force in the specimen. Strain is directly measured using an interferometric strain/displacement gage (ISDG). Here, gold pads or lines are patterned on the specimen as markers. When the two markers are illuminated with a laser, the diffracted reflections from each one overlap and interfere to produce fringes. As the specimen is strained, the two markers move relative to each other, as do the fringe patterns. The fringes are converted to electrical signals using linear diode arrays. The 0.33 με resolution of the diode array is limited by noise and vibration, and effectively 5 με resolution can be obtained. The setup has been used to measure tensile, fatigue, fracture, and high-temperature properties of polysilicon, nitride, carbide, and metallic thin films. For example, phosphorus-doped polysilicon samples (3.5 μm × 600 μm in cross section) showed brittle behavior with Young’s modulus of 170 GPa and fracture strength of 1.2 GPa. Several other variants of the aforementioned experimental setup are available in the literature. For example, the Spaepen group used a photoresist grid micropatterned on the specimens to diffract a laser focused onto the specimens [30.22]. A twodimensional position sensor array is used to pick the diffracted signal to measure both axial and transverse

Mechanical Testing at the Micro/Nanoscale

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30.3 Micro/Nanomechanical Testing Techniques

Fringe detector

Piezoelectric translator

Incident laser beam

Load cell Fringe pattern

Part D 30.3

Air bearing

Fringe pattern

Grips

Specimen

Base

Specimen

Gold lines

Fig. 30.5 (a) Schematic design of a microtensile testing equipment (b) Direct strain measurement using an interferometc ric technique (after [30.63] 1997 IEEE)

strains in the specimen, giving the tool unique capability for Poisson’s ratio measurement. The Sharpe group also developed a process for specimen microfabrication. This is shown in Fig. 30.6. Here a silicon wafer is coated with layers of silicon nitride, oxide, and the specimen material polysilicon and gold. The gold film is patterned and subsequently etched to realize the fringe-producing markers. The top polysilicon layer is then patterned and etched to create the specimen (step a). In the next step, the top side is protected by a ceramic carrier attached to the wafer by adhesive wax. Polysilicon and oxide layers are then removed from the bottom side of the wafer. Also an etch hole is patterned on the nitride layer (step b). The exposed silicon is anisotropically etched using NaOH solution (step c). The polysilicon specimen is then made freestanding by reactive ion etching of the nitride and hydrofluoric-acid-based wet etching of the oxide layer (step d). These steps exemplify typical specimen microfabrication process as almost all modern micro/nanospecimen fabrication methods employ thin-film deposition to lay down the material layer of interest, photolithography to pattern the specimen, and either gas- or liquid-based etching of the substrate material to release the specimen from the substrate. Another distinct feature of this experimental setup is that it uses electrostatic- and/or ultraviolet-curable glues to grip the specimens. Electrostatic gripping is also used by another experimental setup [30.65], and the general recommendation is that the method works very well for lower load levels (0.1 N or lower).

Two other techniques that could replace the ISDG strain measurement are electron microscopy and the digital image correlation technique (DIC). Electron microscopes can be used to visualize the specimen continuously and measure the specimen displacement at up to 100 000× magnification for scanning electron microa)

Gold lines

Patterned polysilicon

Oxide

Silicon die

Nitride Uniform polysilicon

b)

Ceramic carrier

Adhesive

Nitride etchmask

c)

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Grip end Tensile specimen

Fig. 30.6 Microtensile specimen fabrication process developed by c the Sharpe group (after [30.63] 1997 IEEE)

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Part D 30.3

scopes (SEM) and 1 000 000× for transmission electron microscopes (TEM). They allow in situ observation of the deformation but at the cost of miniaturization, because electron microscope chambers are very small compared to conventional mechanical testing equipment. Also, experiments are carried out in a vacuum environment for the electron beam to be sustained. An example of using electron microscopy to measure strain in tensile testing is given in [30.66]. The DIC technique, on the other hand, is a versatile displacement measurement technique that only requires time-series snapshots of the specimen surface, which can be applied in conjunction with a number of imaging techniques. It tracks the gray value patterns (speckles) in small neighborhoods called subsets, which have changed their positions during deformation. The image correlation processing searches the displaced points in a pair of these pictures by matching spots between them [30.67]. Figure 30.7a,b shows atomic force microscopy (AFM) scans of a MEMS actuator segment for two different actuation voltages. To obtain the in-plane displacement a)

y

components u and v (in the x- and y-directions) for such a pair of images, a subset of m × n pixels is used as a searching spot with a wider variation of grey levels than any individual pixel for identification. Image background influence is minimized with a normalized crosscorrelation coefficient C(u, v) of the grey intensity: n  m  

C(u, v) = 

i=1 j=1

  f (xi ,y j ) − f¯ g(xi ,yj ) − g¯

m  n  

,  m  n  2  2   f (xi ,y j )− f¯ g(x ,y )−¯g

i=1 j=1

i=1 j=1

i

j

(30.2)

where f (xi , yi ) is the grey level at point F(xi , yi ) in one of the images serving as the reference frame, g(xi , yi ) is that at point G(xi , yi ) in another image as the deformed frame, and f¯ and g¯ are the mean grey scale values of the pixels involved in the subsets, respectively. The correlation coefficient C(u, v) is computed against the parameters u, v and the matching spots are obtained when the correlation coefficient C reaches the max-

b)

4000 nm

C

4000 nm

A

B x

c) Correlation coefficient 1 0.9 0.8 0.7 0.6 0.5 260 250 y (pixel) 240 230 220 210 200 280

290

300

310

320

330 x (pixel)

Fig. 30.7 (a),(b) AFM scans of a MEMS actuator for two different applied voltages (after [30.68], with c IOP). (c) A typical corpermission  c 2004 relation peak (after [30.69]  IEEE)

Mechanical Testing at the Micro/Nanoscale

30.3.4 Nontraditional (MEMS) Tools Microfabricated testing tools can be classified in two broad categories. In the first one, passive devices are a)

Anchor

built with a cofabricated specimen portion. The basic concept is to suspend the freestanding specimens with micromechanical springs so that, when the specimen is released from the substrate, the springs are deflected from their original positions. The nature of the internal stress and the magnitude of gradient can be ascertained by modeling the specimen–spring system. Such devices are particularly useful in measuring residual stresses and stress gradients [30.71, 75–79]. Other than their simplicity in design and accuracy, these devices allow on-chip characterization for a wide array of specimen geometries. Figure 30.8a shows a representative device, where the entire device itself is a specimen and can be represented by a spring assembly (Fig. 30.8b). Figure 30.8c shows another variant of the concept that employs a mechanical amplifier to improve the device resolution [30.75]. It is important to note that residual stresses in thin films can also be very efficiently measured with the wafer curvature technique [30.80–82], which is a rather older tool not described here in detail. In the second category, active devices apply a known load to the specimens and measure the corresponding displacement. The research paradigm of testing MEMS materials with active MEMS devices was introduced

Anchor

c) Vernier gauge

Tension

Compression

Indicator beam

b)

k2

Unreleased

k1 Released

x1

k3

k4

x2

Slope beam

k5 Test beam Anchors

Fig. 30.8 (a) Schematic design of a passive MEMS device to measure intrinsic stress and (b) the equivalent spring c 1996 IEEE). (c) A passive MEMS device with built-in mechanical displacement amplifier network (after [30.71]  c Sage). Here, all the components are free-standing except the anchors. A tensile residual (after [30.4], with permission  stress causes the free-standing test beam to shrink giving a reading at the vernier gauge (in (c), shrinkage of the test beam causes rotation of the indicator beam counterclockwise)

847

Part D 30.3

imum, then u = xi − xi , and v = yi − yi , respectively. The displacement resolution can be enhanced by using nonintegral pixels. A detailed description of the various interpolation techniques and associated error estimates is given in [30.70]. Figure 30.7c shows a typical correlation peak. The location of the peak indicates how far the speckle has moved between the two images. The DIC technique has been applied to microtensile testing with AFM [30.72, 73] and optical interface [30.74] where strains were measured with 40 and 56 με resolution, respectively. The AFM interface is particularly interesting because the specimen surface can be scanned with nanoscopic resolution as the experiment progresses, thus providing in situ deformation visualization, which is extremely important in micro/nanomechanical testing. Noncontact-mode AFM operation can prevent external disturbances during experiments. A potential drawback is the low speed of AFM scans, about 10 min for a 512 × 512 pixel scan [30.72].

30.3 Micro/Nanomechanical Testing Techniques

848

Part D

Applications

Part D 30.3

in [30.83]. A plethora of MEMS sensors and actuators have now been described in the literature. A comprehensive and comparative study and selection criteria for a wide variety of applications (including mechanical testing) is given in [30.84]. One of the most commonly used technique is electrostatic actuation, which has been used to study tensile [30.85], bending [30.85, 86], torsion [30.87], fatigue [30.88–90], and fracture [30.83,91] properties of micro/nanoscale materials. The importance of in situ visualization (direct observation of deformation and relevant events) has applied a drive for highly miniaturized experimental tools that are compatible with electron microscopes. This becomes a daunting task for the TEM, where only 10 mm-diameter and 0.5 mm-thick grids are allowed for specimens. Figure 30.9 [30.17] shows a MEMS tensile test stage capable of performing in situ experiments in both SEM and TEM. The specimen is microfabricated with force and displacement sensors, which automatically resolves the issues of specimen gripping and alignment. Here, the specimen material is deposited on a silicon wafer, on which the device and the specimen

are subsequently patterned and etched. The specimen is made freestanding by backside etching of the silicon using deep reactive ion etching (DRIE). Details of the device design and fabrication are given in [30.92, 93]. The freestanding specimen is suspended with a force sensor beam at one end and a set of supporting beams at the other end. Quasistatic displacements are applied at one end of the chip, which are then transmitted to the force sensor beam through the specimen. The displacement sensors (in the form of two marker gaps F and D) read the specimen elongation. The force on the specimen is given by the spring constant and the displacement (marker F) of the force sensor beam. The supporting flexural beam assembly prevents misalignment of the specimen, and a finite element analysis of the stage shows that it suppresses any misalignment error in loading by six orders of magnitude. The spring constant of the force sensor beam is calibrated with a nanoindenter after the experiment. The clearance (X–X  ) between the fixed and moving parts allows spring-loading of the stage on a TEM straining stage and prevents any prestress on the specimen.

a) Displacement sensors D and F

Moving end

Specimen

XX '

Fixed end

TEM straining stage

500 μm

b)

Force sensing beam

Supporting beams

c) TEM stage motion Moving end

D Fixed end

200 μm F

Hole in the micro stage for TEM post

Fig. 30.9 (a) SEM image of an in situ TEM micro/nanotensile testing device. (b) Compatibility of the MEMS device with a TEM straining stage. (c) Zoomed view of the specimen and the two displacement sensing markers D and F (after [30.17] c 2004 National Acad. Sci. USA) 

Mechanical Testing at the Micro/Nanoscale

This experimental setup has a few unique features:

30.3 Micro/Nanomechanical Testing Techniques

1. Since the specimen is spring loaded, any residual stress due to deposition and/or fabrication processing (compressive or tensile) can be accounted for simply by recording the positions of the markers D and F immediately before the experiment and after the specimen fractures. Such residual stresses can be as high as megapascals (tensile) to gigapascals (compressive) for thin films [30.94, 95], and failure to take account of this in experimental results makes scatter in the data inevitable. 2. Any material that can be deposited on silicon and etched can be tested. 3. Force resolution is governed by the spring constant of the force sensor beam, and e-beam lithography can lead to nanonewtons resolution. Specimens as thin as 30 nm have been tested with the device [30.96].

The setup described does not include an automated data collection technique and one has to interrupt the

Stress (MPa) 800 Expt. 1 (TEM) Expt. 2 (TEM) Expt. 3 (TEM) Expt. 4 (SEM)

600

1 μm

400

200

0

G

Intergranular crack 0

0.25

0.5

H G

0.75

1

1.25

G

H

I

1.5 Strain (%)

x

y z

50 nm

H I

I

Crack J

A

A E

B D

E

50 nm

y E

B

C D

x

J

A

J

C 50 nm

M

D

C 50 nm

z

B 100 nm

Fig. 30.10 Quantitative measurement of the stress–strain response of 100 nm-thick freestanding aluminum film with

c 2004 National Acad. Sci. simultaneous in situ qualitative observation of the microstructure in TEM (after [30.17]  USA)

Part D 30.3

4. Displacement resolution is governed by the electron microscope resolution. 5. The entire setup is 3 mm × 10 mm in size, which allowed, for the first time, simultaneous qualitative and quantitative tensile experiments inside a TEM [30.17, 97]. Figure 30.10 shows representative experimental results on 100 nm-thick freestanding aluminum films. In situ visualization of microstructure shows little dislocation activity in grains less than 100 nm, even ahead of an advancing crack, and points to a grain-boundary-based mechanism to account for deformation. The MEMS-based tensile testing revealed unusual mechanical behavior in nanograined aluminum, such as nonlinear elasticity and a slight decrease in elastic modulus when the grain size is 20 nm and lower [30.17].

849

850

Part D

Applications

a)

Load sensor

Part D 30.3

200 μm

Specimen

Fig. 30.11a–c Integrated micro/nano

Thermal actuator

tensile testing device including actuator, load sensor, and specimen. (a) Testing chip for in situ TEM. (b) Experimental setup with actuation and data acquisition provisions (c) In situ TEM holder (after [30.98] c 2005 National Acad. Sci. USA) 

Folded beams Backside window

b)

Datachable socket

c) Clamps

Sensing IC chip

5 mm

Device chip Electric contacts In-situ TEM holder

PC board

MEMS chip

1 cm

experiment in order to record the data. This can be resolved by integrating actuators and sensors with dataacquisition system. Such an enhancement in force data collection is presented in [30.98]. Figure 30.11a shows a version of the setup, where thermal actuators (electrostatic actuators can be used if the load requirements are in the micronewton range) are used to apply tens of millinewtons of force on the specimen. Experiments have been performed on multiwalled nanotubes and nanograined polycrystalline nanowires using the setup. Another experimental setup with a similar (thermal) loading principle is described in [30.99]. Another experimental setup for nanoscale specimens that is not MEMS-based, yet which bears tremendous significance in this research area, employs cantilever beam(s) to load the specimen [30.100, 101]. Here a nanomanipulator is used to attach individual nanotubes to two cantilevers beams or to a fixed substrate and a moving cantilever beam. An applied displacement to the cantilever can be converted to force if the spring constant of the cantilever beam is known. Figure 30.12 shows SEM images and a schematic of the experimental setup. The setup enjoys the advantages of in situ experimentation, but specimen attachment is an extremely crucial step that requires considerable skill. This setup was used to obtain, for the first time, tensile properties of individual multiwalled carbon nanotubes (MW(C)NT) and ropes of single-walled carbon nano-

tubes (SW(C)NT). It was found that the outermost layer has an elastic modulus of 270–950 GPa and breaks first under tensile stress with a tensile strength of 11–63 GPa. All of the previous MEMS-based experimental tools employ micromechanical beams as force sensors. The force resolution of these devices depends on the spring constant and the displacement measurement resolution. These and other commercial probe tips can have a spring constant as low as 0.01 N/m and the springs are essentially linear. The spring constants cannot be arbitrarily reduced (to obtain better force resolution) because of factors such as precision of fabrication, stiffness calibration, thermomechanical noise, and structural instability (sample–tip interaction, snapin). Also, the choice of electrostatic, thermal and other actuation schemes practically dictate the experimental environment. Figure 30.13 shows an experimental setup that addresses these issues by employing nonlinear springs with ultra low stiffness. The device can achieve piconewton force resolution, even under an optical microscope (1 μm displacement resolution), which compares favorably with probe microscopes where angstrom-level displacement sensors are required to achieve the same force resolution. The device operation is purely mechanical in nature and is unaffected by the type of environment (air, vacuum, aqueous, electromagnetic, electrolytic). The extremely small size of

Mechanical Testing at the Micro/Nanoscale

Pspecimen = Pcd + Pc d − Pab − Pa b ,

(30.3)

where the load on any column (say ab) is given by Pab = ω2ab [1 − (ξab /Dab )]κab   2 1 − (ξab /Dab )3 κab /32 + ω4ab Dab

(30.4)

and the axial displacement of the specimen is given by 2 δ = π 2 Dab /(4L ab ) ,

(30.5)

which is also the axial displacement of the central platform, or the upper set of columns. Here, D is the lateral displacement, δ is the axial displacement, κ is the lowest flexural rigidity, ξ is the initial imperfection, and ω = 2π/L. Figure 30.13d shows an individual freestanding multiwalled carbon nanotube sandwiched between polymer layers and Fig. 30.13e shows a single mouse myeloma cell compressively loaded by the device. One critical concern in application of the aforementioned picotensilometer is that positioning a separately fabricated specimen on the device is a daunting task. This challenge can be overcome using commercially available nanomanipulators, such as the Omniprobe that comes with the focused ion beam (FIB). Figure 30.14

a)

851

b)

2 μm

Part D 30.3

200 nm

c)

d)

2 μm

100 nm

f)

Up XYZ stage Biomorph AFM probe Sample

S L+δL

L

nm

e)

1.7

the device makes it compatible with almost all the visualization tools in nanotechnology (AFM, SEM, TEM, STM) as well as fluorescence-based biotechnology. The aforementioned experimental setup exploits post-buckling structural stiffness attenuation along with displacement amplification. An SEM image of the setup is shown in Fig. 30.13a. Here, two sets of slender silicon columns (ab/a b and cd/c d ) with fixed–fixed support are connected to a center platform, all of which are freestanding. The columns ab and a b are slightly shorter than the cd and c d columns. A piezo-motor is used to apply displacement at the dd end, while keeping the aa end fixed. The cd and c d columns buckle first and, upon continued displacement, they apply load on ab and a b to buckle them. The center platform can then be assumed as a rigid body connected with two springs with ultralow stiffness values. Figure 30.13b shows an optical micrograph of the cascaded buckled system. Figure 30.13c shows a schematic view of the nanomechanical testing specimen. One end of this specimen is kept fixed to the substrate while the other end is attached to the movable center platform. The position of the fixed specimen end, with respect to the center platform motion, therefore determines the nature of the applied load (tensile or compressive). At any point in time, the force on the specimen is given by

30.3 Micro/Nanomechanical Testing Techniques

0.68 nm

Sample

Fig. 30.12 (a) A tensile-loaded SW(C)NT rope between an AFM tip and a SW(C)NT paper sample. (b) Close-up view showing the SW(C)NT rope on the AFM tip. (c) The same SW(C)NT rope after being loaded to the point where it broke. (d) Another close-up view of the attachment area after the rope was broken. (e) A schematic showing the tensile-loading experiment. (f) Schematic of the SW(C)NT rope with hexagonal cross section c 2000 APS) (after [30.101] 

shows a specimen (a ZnO nanowire) preparation and transfer technique that can be performed inside the FIB chamber.

30.3.5 AFM-Based Tools The superior displacement resolution of the AFM has resulted in its application in nanoscale imaging of virtually all types of materials specimens in ambient conditions. In one mode of operation, a sharp tip (on a compliant cantilever base) is scanned over the specimen, while the deflection of the cantilever base is

Part D

Applications

a)

d)

Fixed end

b)

Part D 30.3

a

a'

b

b'

c

c'

Verniers

c) 500 μm

852

Specimen

e) Autoalignment beams

d

d'

Displacement from piezo

Fig. 30.13 (a) SEM image of a tensilometer with piconewton force and nanometer displacement resolution. (b) Optical micrograph of device in operation. (c) Schematic diagram showing specimen fixture. (d) Free-standing carbon nanotube c 2005 Am. Inst. Phys.) specimen and (e) biological cell specimen (after [30.104] 

continuously monitored by shining a laser beam on the tip and collecting the signals with photodiodes. Figure 30.15a shows only a subset of the AFM’s capability. It also shows how reflected laser is collected in diode quadrant A and B to measure surface topography and/or force, and C and D to measure lateral b)

a)

Oxide grid

Nanowire

c) Grid aligned with posts

Tungsten probe

d)

Nanowire

FIB machined posts on jaws

Free-standing nanowire

Fig. 30.14 Nanomanipulation of separately fabricated specimens on the picotensilometer illustrated in Fig. 30.13 (after [30.105], with c Council Inst. Mech. Engineers) permission 

forces (friction or torsion). For surface scanning, one only needs to measure the cantilever tip deflections. However, for force measurement, one needs to find the relative displacement of the cantilever tip with respect to the specimen. Hence, the displacement of the cantilever base (mounted on a pizeo scanner) or alternatively the specimen stage must be measured. Detailed description of force measurement using AFM is given in [30.102, 103]. See Chap. 17 for further discussion of other aspects of this method. Explosive developments in the modeling and hardware of the AFM has transformed it into a standard yet versatile nanocharacterization tool. Since force and displacement are two generic quantities that need to be measured for any type of mechanical testing, the potential applications of AFM are endless. For example, bending of gold nanowires is shown in Fig. 30.15b. Here, focused ion beam (FIB)-assisted specimen deposition is used to grip a freestanding nanowire on a trench. The AFM tip is then used to push the nanowire laterally and measure the force-displacement response. Figure 30.15c,d shows the nanowire before and after deformation [30.106]. For details on bending tests (mostly on nanotubes and nanowires) by AFM, the reader is en-

Mechanical Testing at the Micro/Nanoscale

a)

D

A

Laser

Part D 30.3

C

Ma gne t mic force ic rosc opy

D y m fo nam icr rc ic os e co py

Lateral force signal

Scanning thermal microscopy Scanning force microscopy

ic son tra e Ul forc copy s cro mi

Photodiode

Laser source

B Ela m stic mic oduluity ros s cop y

A–B topography Friction force microscopy

z

Ele

Ch sen emi so cal rs

C –D torsion

y

in Kelvbe pro copy os micr

c)

d)

x

ctro s mic force tatic ros cop y

c eti gn ce Ma onane res forc copy s cro mi

Scanning capacitanc microscop e y

Force distance spectroscopy

c IOP). Fig. 30.15a–d The AFM has developed into a versatile nanoscale characterization tool (after [30.113], with permission  (b) AFM application on three-point beam bending, (c) gold nanowire before and (d) after bending. Scale bars are 500 nm c (after [30.106] 2005 Nature Mater.)

couraged to go through the references [30.107–111]. Extensions toward fatigue and fracture testing are discussed in [30.112].

30.3.6 Nanoindentation In a nanoindentation test, a hard tip, typically made of diamond, is pressed into the sample with a known a)

load. The deformation field is highly nonuniform, with the small volume of material immediately beneath the tip experiencing the highest load. Typical application is to measure materials hardness, which is the ratio of the applied load and the residual indentation area. Figure 30.16 shows a typical penetration and load– displacement curve obtained from nanoindentation. The loading curve is a combination of elastic and plas-

b) Load (P) Pmax

25 0 –25 0.1

0.5

0.2

S = dP/dh

0.4 0.3

0.3 0.2

0.4 0.5

0.1 μm

853

b)

Information storage Chemical force microscopy

30.3 Micro/Nanomechanical Testing Techniques

hmax ho Displacement (h)

Fig. 30.16 (a) Typical load– indentation profiles on a ZnS nanowire and (b) a typical load–penetration curve due to nanoinc dentation (after [30.114] 2005 AIP)

854

Part D

Applications

Part D 30.3

tic deformation and the unloading curve is related to the elastic properties of the specimen material. Since the indenter is not perfectly rigid, at first the effective modulus (E r ) is obtained. Using     2 2 1 − νspecimen 1 − νindenter 1 = + , E effective E specimen E indenter (30.6)

where ν is the Poisson’s ratio, the effective modulus is obtained from the slope of the unloading curve (Fig. 30.16a) using 2 √ dP (30.7) = √ β AE r , S= dh π where P is the load, h is the indenter penetration, A is the indenter–specimen contact area, and β is a dimensionless parameter that depends on the indenter profile. The critical part is the determination of the contact area.

It is now established that the contact area is a function of a critical contact depth h c , A = f (h c )

and h c is obtained from the following expression [30.118]: Pmax , dP/ dh

h c = h max − ε

(30.9)

where ε is a parameter dependent on the indenter geometry, and Pmax and h max are obtained from the load–displacement curve. Once these values are obtained, solid mechanics can be used to obtain other materials properties such as the hardness (H), the ultimate tensile strength (σUTS ), and the fracture toughness

b)

a)

(30.8)

LM PV θ Wafer

PM

Δ

PM

Step 1. Tip moves down in the middle of the ring Diamond tip Fixed end Step 2. Tip pulls the ring of the specimen until the specimen fractures

Optics

Mirau microscope objective

Free end

c)

d)

F Transducer (T) Δ

ET

Force applied to fiber = F = FT = Ff

Fiber (f) Ef

Elongation of fiber = Δf = Δ – ΔT 10 μm

5 μm

Fig. 30.17a–d Nontraditional applications of a nanoindenter in micron and submicron materials testing, (a) tensile c (b) membrane bending extesting of surface micromachined specimens (after [30.115], with permission Springer) c (c) tensile testing of nanofibers periments that produce tensile testing results (after [30.56], with permission Elsevier) c AIP 2004) and (d) compression testing of short single-crystal Ni superalloy columns (after [30.116], with permission  c Elsevier) (after [30.117], with permission 

Mechanical Testing at the Micro/Nanoscale

(K cr ): (30.10) (30.11)

(30.12)

where n is the work hardening exponent and α is an indenter-geometry-dependent constant. Modern nanoindenters have depth and load sensing features and calibrated tip shape function, which facilitates the difficult task of residual indentation area measurement. Significant progress has been made in the modeling and analysis of nanoindentation deformation and hardware, and the tool is recognized to be one of the most popular in hardness and Young’s modulus determination [30.121–136], even though it is suitable for residual stress [30.137–140], dynamic properties [30.141–143], fracture toughness [30.130, 144, 145], and wear resistance tests as well. Perhaps the greatest advantage of nanoindentation is the ease of specimen fabrication, where the only requirement is a hard substrate. However, data interpretation is not straightforward [30.146–148], and the technique is very sensitive to the hardness/softness of materials tested [30.147, 149–152], depth of indentation, type of indenter, surface preparation, presence of oxides, and other heterogeneous artifacts. As a result, sophisticated modeling and simulation is necessary [30.153–157]. Also, only a very small volume of material is tested, so significant data scatter is expected due to the existence of inhomogeneous elements (grain boundary, inclusions, precipitates, void, surface roughness) in specimens that may appear homogeneous in a bulk sense. The reader is encouraged to go through Chap. 16 of this Handbook and the following references for a better working idea on nanoindentation [30.132, 158, 159]. A focus issue on the latest developments in nanoindentation is also available [J. Mater. Res. 19(1) (2004)]. The most challenging applications in nanoindentation are found in ultrathin films [30.160] and nanowires or nanotubes [30.127, 161], since the rule of thumb is that the indentation depth should not be more than 10% of the film thickness, otherwise the substrate effect will contaminate the results. Figure 30.16b shows a nanoindentation impression on a ZnO nanowire. A new model has been proposed in the literature to

account for the deformation of such one-dimensional specimens [30.127]. The emergence of AFM and nanoindenter as sophisticated and well-accepted tools for force and/or displacement application has also resulted in nontraditional applications, examples of which are shown in Fig. 30.17. In Fig. 30.17a, the indenter is used to apply tensile loads to microfabricated specimens [30.115]. The nanoindenter can also be used to apply a transverse load to a beam to measure its tensile properties, provided that the specimen is long and flexible enough to have only tensile loads in the gage section; this is shown in Fig. 30.17b. Here displacements in the gage section are measured using the optics placed beneath the specimen, which uses interferometry. Figure 30.17c shows another tensile testing application of a nanoindenter [30.116]; the same technique has also been used to perform compression testing on submicron short columns. Figure 30.17d shows the a) Pressure p (MPa) 0.008

z0

0.006

p a

0.004 Experiment 'p-theory'

0.002

E = 3.0 GPa residual stress = 5.8 MPa 0 0

10

20

30

40

Mid-point deflection z0 (μm)

b)

Fig. 30.18a,b Nanoscale thin film and wire testing techniques. (a) Bulge testing of freestanding membranes c AIP 2002) and (b) Resonant vibrating (after [30.119] 

beam testing of nanowires and nanotubes (after [30.120], c Springer) with permission 

855

Part D 30.3

Pmax , A  σUTS 1 − n 12.5n n , = H 2.9 1−n  1/2  E P K cr = α , H c3/2 H=

30.3 Micro/Nanomechanical Testing Techniques

856

Part D

Applications

Part D 30.4

shear bands developed during such a compression test on single-crystal Ni superalloy where size effect on plastic behavior was observed for pillars with diameters below 5 μm [30.117]. A similar nanoindentation technique was applied to compress single-crystal gold pillars as small as 300 nm in diameter where the flow stress reached 4.5 GPa, a significant fraction of the theoretical strength. The high strength was explained by a dislocation starvation model [30.162]. A groundbreaking application on thin-film bending is described in [30.163].

30.3.7 Other Tools The challenges in specimen fabrication, gripping, and load application intensify as the specimen size decreases to the nanoscale. To cope with this, the biaxial bulge testing (mainly for freestanding thin membranes) [30.164–170] and resonant vibrating beam [30.120, 171–175] techniques have been applied very effectively in the literature. In bulge testing, a freestanding membrane is pressurized to deform it in the form of a bulge or a blister. The corresponding deformation is recorded with an interferometric or laser-based displacement measurement system. For example, the Young’s modulus (E) of a circular membrane can be modeled to as a function of Poisson’s ratio (ν), thickness (h), and radius (a) of the membrane, applied pressure

( p), and residual stress (σ0 ) [30.1],    σ0 h E h 3 8 z +4 z0 , p = (1 − 0.24ν) 3 1 − ν a4 0 a2 (30.13)

where z 0 is the midpoint deflection due to the applied pressure. Figure 30.18a shows a schematic of the experiment and parameter fitting with experimental data. The resonant vibrating beam technique is simpler to apply since no force application/measurement is necessary, and only a freestanding beam specimen is needed. For example, a cantilever beam specimen can be given alternating displacement at its base (if nonconductive) or subjected to an alternating electrical field, so that the beam resonates. The task of finding the resonant frequency of the specimen is usually performed in an SEM or TEM. This resonant frequency is then equated to the stiffness and mass of the material as

λ2 EI , (30.14) ω0 = 2π m L 4 where E, I , m, and L are the Young’s modulus, area moment of inertia, mass per unit length, and length of the cantilever, respectively, and λ is the eigenvalue corresponding to the resonance mode and boundary condition. The experiment is relatively straightforward to perform but requires caution in accounting for damping and the estimation of mass.

30.4 Biomaterial Testing Techniques We define biomaterials as single cells and subcellular components. These materials can be up to ten orders of magnitude softer than the engineering materials as described above. As a result, they can absorb very small forces (pico- to nanonewtons) and produce very large (on the order of microns) displacements. Mechanical testing of biomaterials is a relatively unexplored territory since the importance of mechanical properties at the cellular and subcellular level has only recently received considerable attention. Figure 30.19 shows the most commonly used tools for micro/nano-biomaterials testing. AFM and magnetic bead techniques are applicable for cellular and subcellular components. For example, AFM has been used to study unfolding of deoxyribonucleic acid (DNA) [30.176, 177] and other proteins [30.178–183] that govern the micromechanics of cell structural components. This is shown in Fig. 30.20. Another local

probe is the magnetic bead technique, where controlled mechanical stresses can be applied to specific cell surface receptors using ligand-coated ferromagnetic beads up to 5 μm in diameter. An advantage over the AFM is that the technique is massively parallel; 20 000–40 000 cells can be tested for different receptor site simultaneously [30.184]. It can also be used as an assay for ligand-receptor adhesion strength, mechanical linkages between a cell surface receptor and the cytoskeleton, and the rheological and remodeling properties of the cytoskeleton. Micropipette aspiration technique employs in situ microscopy to track the shape of a single cell as it is aspirated into a micropipette. Because cell deformation depends on the applied suction pressure, it is possible to evaluate the bulk viscosity of the cell by measuring the cell elongation into the pipette as a function of the suction pressure. This technique has

Mechanical Testing at the Micro/Nanoscale

a)

Magnetic bead

d)

Optical trap

aspiration Red blood cell

e)

f) Shear flow

Part D 30.4

c) Micropipette

Stretching

been used to measure cell adhesion, stiffness, and other rheological properties [30.185–191]. It is a useful approach for cell types that undergo large shear deformation. The technique is appealing because of its ease of operation and the available concepts of continuum mechanics for force calibration. However, the a)

Silica bead

Focal adhesion complex

applied stress state is relatively complex because of the coupling of solid and fluid mechanics. Figure 30.21a shows an example of the micropipette aspiration experiment. Like micropipette aspiration, optical trap/tweezer technique can be used to interrogate single cells mec) Typical traces with “fly fishing” feedback

Force Polymers with binding partners

b)

Displacement

Force

Repulsive contact

200 pN

200 pN 50 nm

857

Fig. 30.19 Various tools for single cell, subcellular components and cell population testing (after [30.52], c 2003 with permission Macmillan  Nature Mater.)

b)

AFM

30.4 Biomaterial Testing Techniques

50 nm

Elongation

Fig. 30.20 (a) Schematics of single-molecule force response measurements, (b) AFM tip approach and retract curves showing discrete molecular deformation and unbinding events, (c) AFM tip retract curves under conditions where only c Springer) single molecular interaction occurs (after [30.181], with permission 

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a)

b) Radiation

c)

Optical axis

pressure dominates Wave front λ

Part D 30.4

Colloidal particle Gradient force dominates

Laser beam

c Fig. 30.21 (a) Micropipette aspiration of a single chondrocyte (after [30.49], with permission Elsevier). (b) Tweezer action arising from the balance of radiation pressure and gradient forces in optical tweezers, and (c) manipulation of microbeads with c 2003 Nature) optical tweezers (after [30.195], with permission Macmillan 

a)

c)

δ

e)

y

F d

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x L

F=⎛

⎝

3EI ⎛ L3

⎝

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δ

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b)

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Piezo actuator

δ

F = kδ Backbone

Expose UV Photoresist

Sensor base

Si substrate

Silicon Develop

d)

Cast PDMS

f) Cure PDMS

g)

Cell Probe

Cast PDMS

Cure PDMS 36 μm

Fig. 30.22a–g Single-cell mechanical testing techniques, (a) Concept of the microneedle test. (b) Microfabrication sequence for c 2003 National the microneedles. (c) and (d) Muscle cells exerting forces on the needles, thereby deflecting them (after [30.202]  Acad. Sci. USA). (e) Schematic design of MEMS-based cell mechanics study. (f) and (g) MEMS probe loading a fibroblast cell c AIP 2005) in tension (after [30.203] 

Mechanical Testing at the Micro/Nanoscale

MEMS for Mechanical Testing of Cells One of the greatest and potentially long-lasting impacts of MEMS and nanotechnology is on the biomedical area [30.47]. However, there are only a few instances of MEMS applications in biomaterials testing [30.205, 206]; the reason could be the specialized environmental considerations required for cell and other biomaterial testing. One of the notable studies used passive microfabricated device to determine mechanical properties of cells. This is shown in Fig. 30.22a [30.202], where cells are allowed to attach to microfabricated poly-

mer microneedles. The cells attach to the tips and then exert forces on the highly compliant needles, which act as cantilever beams. The magnitude of the forces exerted by the cells is given by the product of the needle tip deflection and the needle spring constant. Optical microscopes are used to find the direction of the forces. MEMS force sensors, essentially micromachined cantilevers, with functionalized probes have been used to contact cells on substrates and form focal adhesion contacts. Hence the probes have a local handle on cell cytoskeletal structure. Cells are then stretched by a magnitude that is similar to the initial size of the cell [30.207] and their force response is measured. A more recent approach uses a micromachined spring, which is more amenable to the cell culture environment [30.203], and can perform multiaxis force measurement with nanonewton resolution. Figure 30.22f,g shows the MEMS probe attached to a cell. Experiments using these MEMS force sensors revealed that the force response of living cells is linear and reversible even under large stretch, contrary to the expected stiffening response due to alignment of the cytoskeletal actin network along the direction of stretch [30.208]. These results shed new light on cell mechanobiology, and highlight the potential role of micro- and nanomechanical sensors in probing biological samples at a cellular and subcellular scale. An important difference between testing a living and a nonliving sample is that, for the latter, the response is in general mechanical, whereas for the former the response is both biological and mechanical. For example, a cell when subjected to mechanical stimuli (e.g., stretch) may initiate a cascade of signals and corresponding structural changes (e.g., cell reorientation when the substrate is stretched). The measured timedependent response of the living sample, for example, cell force due to applied deformation, contains both the biological and the mechanical contributions. As yet, there is very little understanding of the biological processes induced by mechanical stimuli, and hence it is difficult to decouple the two contributions, an issue that needs to be carefully considered while modeling cell mechanical behavior.

30.5 Discussions and Future Directions Irrespective of the type and size of the specimen material, the following issues need to be considered by the researcher.

Deformation Type and Desired Properties First, one should determine what are the properties needed for the specific micro/nanomaterial. This de-

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chanically. In the most basic optical trap, a laser beam is focused by a microscope objective to a spot in the specimen plane. As shown in Fig. 30.21b, intensity gradients in the laser beam draw small objects, such as a colloidal particle, toward the focus, whereas the radiation pressure of the beam tends to blow them down the optical axis. Under conditions where the force gradient dominates, a particle can be trapped, in three dimensions, near the focal point [30.192–195]. The technique can apply forces in the tens of piconewton range on objects ranging in size from 10 nm to several microns. Typical applications are manipulation and mechanical testing of biological (viruses, bacteria, living cells, organelles) and nonbiological (nanotubes, small metal and dielectric particles) materials [30.196–199]. However, quantitative modeling and calibration of the process is very complex [30.200, 201]. Figure 30.21c shows manipulation of microbeads using optical tweezers. The remaining two techniques are applicable for both individual and cell populations. The shear flow technique is particularly appealing because of the compatibility of the test environment with the cell’s natural habitat. Shear flow methods (Fig. 30.19e) and stretching devices (Fig. 30.19f) are typically used to study the mechanical response of an entire culture of cells. Here, the cells are subjected to fluid flow that effectively applies shear stress on the cells. Substrate deformation is another classical tool for cell mechanobiological testing. The cells are attached to a flexible membrane, which is then given uniaxial, biaxial or pressure-controlled stretching [30.204].

30.5 Discussions and Future Directions

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pends on the type (brittle, ductile, bio) of material and the type of application. For example, static tests are rarely useful in dynamic environments, as in radiofrequency (RF) MEMS devices. Similarly, fracture strength is probably the most important property for brittle materials. The choice of the characterization tool becomes obvious when the materials type and specific application are pin-pointed. Interestingly, the techniques described in literature show a balance between operating ease and meaningfulness of the data. For example, tensile tests are highly desired for their straightforward interpretations yet very hard to perform at the micro/nanoscale. The opposite is true for now standardized tools such as the nanoindenter. As a result more and more nontraditional applications of the nanoindenter are appearing in the literature. Despite these challenges, tensile tests should be performed whenever possible and techniques requiring significant modeling of materials behavior should be avoided as much as possible in the case of exploratory studies. Specimen Preparation Specimen preparation makes the major difference in micro/nanomechanical testing. The first reason is the residual stress. Seemingly invisible at the macroscale, residual stress effects are very strong at these smaller length scales (see, e.g., Fig. 30.1). Macroscale techa)

niques for residual stress mitigation is not highly effective for nanograined films. Such stress strongly depends on the synthesis/deposition environment and can be on the order of a few hundred megapascals for tensile and a few gigapascals for compressive type and cannot be ignored. Therefore, the burden is on the researcher to account for the stress during experimentation, otherwise large scatter will pervade the data. The least effort required is to run some preliminary experiments with the wafer curvature technique [30.80–82]. Wrinkling and other ways of relieving the residual stress also change the specimen geometry, as shown in Fig. 30.23a. Here, buckling of a square membrane is shown, which could compromise the cleanliness of bulge testing data. Specimen fabrication technique is more standardized for MEMS and microelectronic materials for which deposition and etching technology are available in the literature. Here the specimen material is first deposited (evaporated, sputtered, chemical vapor deposited) onto the wafer. Lithographic techniques are used to pattern the specimen, after which the substrate is etched away to make the specimen freestanding. The researcher should make a careful examination of the etched profile of the substrate, especially for gradient-dominated tests, since uncontrollable etching features may change the boundary conditions and, in extreme cases, the specimen geometry as well.

2nd sandwich layer 1st sandwich layer Silicon substrate Embedded nanotube

b)

Exposure

Mask Slot to create freestanding specimen Device patterns Freestanding nanotube c)

Gripped end of the nanotube Device patterns on photoresist

200 nm

Fig. 30.23 (a) Wrinkling of a specimen due to residual stress c IEEE 1999). (after [30.77]  (b) Nanofabrication of freestanding one-dimensional specimens, (c) a free-standing multiwalled carbon nanotube tensile specimen c (after [30.209] IEEE 2005)

Mechanical Testing at the Micro/Nanoscale

1

Pt line

30.5 Discussions and Future Directions

Lift out probe

Pt line

861

Pt deposits

5a Cuts

Part D 30.5

2

4 Ion milled trench Ion milled trench

Lift out probe

3 Ion milled trench Tilt specimen

Pt deposits

5b Cuts

Top

Side

Front

Fig. 30.24a,b Schematic diagram showing the major steps in the lift-out method for producing micro/nanoscale specimens. In step 5, the specimen may be cut either (a) normal or (b) parallel to the surface of the specimen (after [30.213], c Elsevier) with permission 

One-dimensional solids, on the other hand, are fabricated in small quantities in entangled configuration, and are first sonicated in a liquid bath. Droplets of nanotube- or nanowire-containing solutions are then poured onto wafers, where the solvent is allowed to dry. This technique is used in almost all nanotubebased devices described in the literature. However, microfabrication techniques for thin-film specimens may be adapted to place and align nanotubes and nanowires at and along preferred orientations, as shown in Fig. 30.23b. It is important to note that the solvent drying process leaves the nanotubes and nanowires stressed, and often entangled or wavy after fabrication. Another specimen preparation technique that the reader should be aware of involves focused ion beam (FIB). A focused ion beam can be used to deposit and remove materials with up to 30 nm precision. This technique is gaining popularity, especially for TEM specimen fabrication, which has been a matter of art [30.210–213]. Special care must be given to contamination of the specimen from the ion beam, which worsens if the ion beam is used to deposit materials. Figure 30.24 shows some examples of FIB as a specimen fabrication tool.

Force and Displacement Measurement Even though there is a growing need for sensors and actuators at the same length scale as that of the specimen, the opposite trend is also observed, with AFM and nanoindenters being used due to the acceptance of these tools by the scientific community for their precision and resolution. These tools can measure forces and displacement above nanonewtons and nanometers, which is insufficient for testing nanoscale biomaterials. However, AFM and similar tools make the researcher design the experiment along the lines of these tools and leave little room for customization. MEMS tools, on the other hand, offer more customizability than resolution. A small MEMS testing chip (for example, the devices shown in Figs. 30.9, 11, 13) is compatible with any form of microscopy, which is not possible with off-the-shelf alternatives. This is extremely important for nanoscale materials testing, where high-magnification visualization is important not only to observe in situ events in exploratory experiments, but also to identify and eliminate artifacts and their effects. The future direction therefore heavily involves microscopy as an integral part of the experimental setup, and in situ experimen-

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tation will enjoy increasing popularity. The literature already shows signs of this, as scientific and commercial endeavors are targeted towards in situ TEM experimental tools. The burning issue for the present is testing techniques standardization. While this is well organized throughout the world for macroscale testing, only a few round-robin test schemes are available for microtesting of polysilicon, and almost none for other materials. This is critical for the research area, where residual stress, specimen fabrication, force and displacement measurement can drastically affect the experimental results. The evidence is in the results available for nanotubes, where the Young’s modulus value varies from 0.2 TPa to 5 TPa [30.214]. A governing body nucleated from researchers worldwide is needed to determine the acceptable codes for nanomaterials testing. The future of micro/nanomechanical testing will not be the tyranny of the residual stress, or the requirement for ultrahigh force resolution. Rather, it will be the need for multiphysics and multidomain characterization. Although yet to draw the entire community’s attention, nanoscale materials exhibit strong multifield coupling. This is predicted by both theory [30.215–

223] and preliminary experiments [30.109, 224]. Therefore, the future will see nanomaterials characterization with an emphasis on state variables such as temperature, strain, and electrical and magnetic fields. Characterization in individual domains will therefore mean little. The same is true for biomaterials, where the multifield coupling in biological cells [30.44, 45, 48, 49, 225] is already accepted and recent studies suggest that a better understanding of the cell structure, functioning, pathology, and drug discovery is possible if we study them simultaneously. However, the extreme size, sensitivity, and multifunctionality requirements of biosensors have hitherto rendered such unified experimental studies very difficult. It is imperative that we understand the fundamentals of these coupled-field sensitivities to exploit their good side (in ultrasensitive physical and biosensors) and take caution of the bad sides (unwanted vulnerability to device fabrication and operating environment) while designing nanoscale devices. The future of micro/nanomechanical testing will therefore require more severe degrees of miniaturization to accommodate highresolution multifield characterization, all at the same time.

30.6 Further Reading Further reading can be found in the following references:

•

mechanical properties of thin films [30.1–6, 10, 14, 115]

• • • •

mechanical properties of nanowires and nanotubes [30.43, 214] mechanical properties of biomaterials [30.44,48,49, 52, 53] atomic force microscopy [30.102, 113] nanoindentation [30.129, 143, 146, 158, 159]

References 30.1 30.2

30.3

30.4

R.P. Vinci, J.J. Vlassak: Mechanical behavior of thin films, Ann. Rev. Mater. Sci. 26, 431–462 (1996) O. Kraft, C.A. Volkert: Mechanical testing of thin films and small structures, Adv. Eng. Mater. 3, 99– 110 (2001) W.N. Sharpe Jr.: Mechanical properties of MEMS materials. In: The MEMS Handbook, ed. by M. Gadel-Hak (CRC, Boca Raton 2002) pp. 3–33, Sect. 1, Chapt. 3 V.T. Srikar, S.M. Spearing: A critical review of microscale mechanical testing methods used in the design of microelectromechanical systems, Exp. Mech. 43, 238–247 (2003)

30.5

30.6 30.7

30.8

M.A. Haque, M.T.A. Saif: A review on micro and nano-mechanical testing with MEMS, Exp. Mech. 43, 1–8 (2003) F.R. Brotzen: Mechanical testing of thin films, Int. Mater. Rev. 39, 24–45 (1994) T.D. Shen, C.C. Koch, T.Y. Tsui, G.M. Pharr: On the elastic moduli of nano-crystalline Fe, Cu, Ni, and Cu Ni alloys prepared by mechanical milling/alloying, J. Mater. Res. 10, 2892–2896 (1995) R.A. Masumura, P.M. Hazzledine, C.S. Pande: Yield stress of fine grained materials, Acta Mater. 46, 4527–4534 (1998)

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30.14 30.15

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30.21

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30.41 30.42

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Conf. Micro Electro Mechan. Syst. (2002) pp. 427– 430 N.G. Chechenin, J. Bøtigger, J.P. Krog: Nanoindentation of amorphous aluminum oxide films III. The influence of the substrate on the elastic properties, Thin Solid Films 304, 70–77 (1997) M.Y. Gutkin, I.A. Ovidko, C.S. Pande: Theoretical models of plastic deformation processes in nanocrystalline materials, Rev. Adv. Mater. Sci. 2, 80– 102 (2001) C.C. Koch, J. Naryan: The inverse Hall-Petch effect – Fact or artifact?, Proc. Mater. Res. Soc. Symp., Vol. 634 (2001) pp. B5.1.1–B51.11 S. Yip: Nanocrystals – the strongest size, Nature 391, 532–533 (1998) T.G. Nieh, J. Wadsworth: Hall-Petch relation in nano-crystalline solids, Scr. Mater. 25, 955–958 (1991) C.S. Pande, R.A. Masumura, R.W. Armstrong: Pile-up based Hall-Petch relation for nano-scale materials, Nanostruct. Mater. 2, 323–331 (1993) H. Chokshi, A. Rosen, J. Karch, H. Gleiter: On the validity of the Hall-Petch relationship in nanocrystalline materials, Scr. Mater. 23, 1679–1684 (1989) B. Cai, Q. Kong, L. Liu, K. Lu: Interface controlled diffusional creep of nano-crystalline pure copper, Scr. Mater. 41, 755–759 (1999) H.V. Swygenhoven, M. Spaczar, A. Caro: Microscopic description of plasticity in computer generated metallic nanophase samples: A comparison between Cu and Ni, Acta Mater. 47, 3117–3126 (1999) P. Heino, E. Ristolainen: Strength of nano-scale polycrystalline copper under shear, Philos. Mag. A 81, 957–970 (2001) R.Z. Valiev, F. Chmelik, F. Bordeaux, G. Kapelski, B. Baudelet: The Hall-Petch relation in submicrongrained Al-1.5%Mg alloy, Scr. Mater. 27, 855–860 (1992) J. Lian, B. Baudelet, A.A. Nazarov: Model for the prediction of the mechanical behavior of nanocrystalline materials, Mater. Sci. Eng. A 172, 23–29 (1993) K. Lu, M.L. Sui: An explanation to the abnormal Hall-Petch relation in nano-crystalline materials, Scr. Mater. 28, 1465–1470 (1993) H.S. Kim, Y. Esterin, B.M. Bush: Plastic deformation of fine-grained materials, Acta Mater. 48, 493–504 (2000) P.G. Collins, P. Avouris: Nanotubes for electronics, Sci. Am. 12, 62–69 (2000) C.M. Lieber: Nanoscale science and technology: Building a big future from small things, MRS Bull. 28, 486–491 (2003) M. Law, J. Goldberger, P. Yang: Semiconductor nanowires and nanotubes, Ann. Rev. Mater. Res. 34, 83–122 (2004)

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E. Arzt: Size effects in materials due to microstructural and dimensional constraints: A comparative review, Acta Mater. 46, 5611–5626 (1998) W.S. Trimmer: Micromechanics and MEMS: Classic and Seminal Papers to 1990 (Wiley-IEEE, New York 1997) R.O. Ritchie, C.L. Muhlstein, R.K. Nalla: Failure by fracture and fatigue in "nano" and "bio" materials, JSME Int. Ser. A: Solid Mech. Mater. Eng. 47, 238–251 (2004) S.M. Spearing: Materials issues in microelectromechanical systems (MEMS), Acta Mater. 48, 179–196 (2000) J. Judy: Microelectromechanical systems (MEMS): fabrication, design and applications, Smart Mater. Struct. 10, 1115–1134 (2001) W.D. Nix: Mechanical properties of thin films, Metall. Trans. 20A, 2217–2245 (1989) G.W. Nieman, J.R. Weertman, R.W. Siegel: Mechanical behavior of nano-crystalline Cu and Pd, J. Mater. Res. 6, 1012–1027 (1991) H. Mizubayashi, J. Matsuno, H. Tanimoto: Young’s modulus of silver films, Scr. Mater. 41, 443–448 (1999) M.A. Haque, M.T.A. Saif: Deformation mechanisms in free-standing nano-scale thin films: A quantitative in-situ TEM study, Proc. Natl. Acad. Sci., Vol. 101 (2004) pp. 6335–6340 D. Chen: Computer model simulation study of nano-crystalline iron, Mater. Sci. Eng. A 190, 193– 198 (1995) J. Schiotz, F.D. Di-Tolla, K.W. Jacobsen: Softening of nano-crystalline metals at very small grain sizes, Nature 391, 561–563 (1998) P.G. Sanders, C.J. Youngdahl, J.R. Weertman: The strength of nano-crystalline metals with and without flaws, Mater. Sci. Eng. A 234–236, 77–82 (1997) H.S. Kim, M.B. Bush: The effects of grain size and porosity on the elastic modulus of nano-crystalline materials, Nanostruct. Mater. 11, 361–367 (1999) H. Huang, F. Spaepen: Tensile testing of freestanding Cu, Ag, and Al thin films and Ag/Cu multilayers, Acta Mater. 48, 3261–3269 (2000) M.A. Haque, M.T.A. Saif: Thermo-mechanical properties of nanoscale freestanding aluminum films, Thin Solid Films 484, 364–368 (2005) A.J. Kalkman, A.H. Verbruggen, G.C. Janssen: Young’s modulus measurements and grain boundary sliding in freestanding thin metal films, Appl. Phys. Lett. 78, 2673–2675 (2001) S. Sakai, H. Tanimoto, H. Mizubayashi: Mechanical behavior of high-density nano-crystalline gold prepared by gas deposition Method, Acta Mater. 47, 211–217 (1999) X. Lin, O. Takahito, Y. Wang, M. Esashi: Study on ultra-thin NEMS cantilevers: High yield fabrication and size-effect on Young’s modulus, Proc. IEEE Int.

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30.47 30.48

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D.E. Ingber: Mechanobiology and diseases of mechanotransduction, Ann. Med. 35, 564–577 (2003) J.D. Humphrey: Stress, strain, and mechanotransduction in Cells, J. Biomech. Eng. 123, 638–641 (2001) H. Huang, R.D. Kamm, R.T. Lee: Cell mechanics and mechanotransduction: pathways, probes, and physiology, Am. J. Physiol. – Cell Physiol. 287, 1–11 (2004) R. Ritchie: Whither ’nano’ or ’bio’?, Mater. Today 8, 72 (2005) C. Zhu, G. Bao, N. Wang: Cell mechanics: mechanical response, cell adhesion, and molecular deformation, Ann. Rev. Biomed. Eng. 2, 189–226 (2000) K.J. Van Vliet, G. Bao, S. Suresh: The biomechanics toolbox: experimental approaches for living cells and biomolecules, Acta Mater. 51, 5881–5905 (2003) J.V. Small, B. Geiger, I. Kaverina, A. Bershadsky: How do microtubules guide migrating cells?, Nature Rev.; Mol. Cell Biol. 3, 957–964 (2002) P. Bursac, G. Lenormand, B. Fabry, M. Oliver, D.A. Weitz, V. Viasnoff, J.P. Butler, J.P. Fredberg: Cytoskeletal remodelling and slow dynamics in the living cell, Nat. Mater. 4, 557–561 (2005) G. Bao, S. Suresh: Cell and molecular mechanics of biological materials, Nat. Mater. 2, 715–725 (2003) D. Boal: Mechanics of the Cell (Cambridge Univ. Press., Cambridge 2002) U.S. Bhalla: Models of cell signaling pathways, Curr. Opin. Genet. Dev. 14, 375–381 (2004) B. Martinac: Mechanosensitive ion channels: molecules of mechanotransduction, J. Cell Sci. 117, 2449–2460 (2004) H.D. Espinosa, B.C. Prorok, M. Fischer: A methodology for determining mechanical properties of freestanding thin films and MEMS materials, J. Mech. Phys. Solids 51, 47–67 (2003) W.N. Sharpe Jr.: Tensile testing at the micrometer scale (opportunities in experimental mechanics), Exp. Mech. 43, 228–237 (2003) G.C. Johnson, P.T. Jones, R.T. Howe: Materials characterization for MEMS: A comparison of uniaxial and bending tests, Proc. SPIE, Vol. 3874 (2000) pp. 94–101 ASTM: Metals – Mechanical Testing; Elevated and Low-Temperature Tests; Metallography, Vol. 03.01 (ASTM, West Conshohocken 2005) C.G. Andeen, C.W. Hagerling, R.W. Hoffman: The nanotensilometer – An accurate, sensitive tensile test instrument, Proc. 7th Int. Vacuum Congr. and 3rd Int. Conf. Solid Surf. (Vienna 1977) J.R. Davis: Tensile Testing (ASM International, Materials Park 2004)

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30.194 Y. Liu, G.J. Sonek, M.W. Berns: Two-beam scanning optical tweezers for cell manipulation and force transduction, Proc. 15th Ann. Int. Conf. IEEE Eng. Med. Biol. Soc. (San Diego 1993) 30.195 D.G. Grier: A revolution in optical manipulation, Nature 424, 810–816 (2003) 30.196 S. Kulin, R. Kishore, K. Helmerson, W.D. Phillips: Studying biological adhesion using optical tweezers, Pacific Rim Conference on Lasers and Electro-Optics, CLEO - Technical Digest, Conference on Lasers and Electro-Optics (CLEO 2000) (2000) p. 590 30.197 J.E. Reiner, R. Kishore, C. Pfefferkorn, J. Wells, K. Helmerson, P. Howell, W. Vreeland, S. Forry, L. Locasio, D. Reyes-Hernandez, M. Gaitan: Optical manipulation of lipid and polymer nanotubes with optical tweezers, Optical Trapping and Optical Micromanipulation (Denver 2004) 30.198 P. Li, K. Shi, Z. Liu: Manipulation and spectroscopy of a single particle by use of white-light optical tweezers, Opt. Lett. 30, 156–158 (2005) 30.199 G.V. Shivashankar, A. Libchaber: Single DNA molecule grafting and manipulation using a combined atomic force microscope and an optical tweezer, Appl. Phys. Lett. 71, 3727 (1997) 30.200 W. Singer, S. Bernet, N. Hecker, M. Ritsch-Marte: Three-dimensional force calibration of optical tweezers, J. Mod. Opt. 47, 2921–2931 (2000) 30.201 A. Buosciolo, G. Pesce, A. Sasso: New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers, Opt. Commun. 230, 357–368 (2004) 30.202 J.L. Tan, J. Tien, D.M. Pirone, D.S. Gray, K. Bhadriraju, C.S. Chen: Cells lying on a bed of microneedles: An approach to isolate mechanical force, Proc. Natl. Acad. Sci., Vol. 100 (2003) pp. 1484–1489 30.203 S. Yang, M.T.A. Saif: Micromachined force sensors for the study of cell mechanics, Rev. Sci. Instrum. 76, 044301–8 (2005) 30.204 C. Neidlinger-Wilke, E.S. Grood, J.H.-C. Wang, R.A. Brand, L. Claes: Cell alignment is induced by cyclic changes in cell length: Studies of cells grown in cyclically stretched substrates, J. Orthop. Res. 19, 286–293 (2001) 30.205 Y.U. Sun, B.J. Nelson: MEMS for cellular force measurements and molecular detection, Int. J. Inf. Acquis. 1, 23–32 (2004) 30.206 N. Li, A. Tourovskaia, A. Folch: Biology on a Chip: Microfabrication for studying the behavior of cultured cells, Crit. Rev. Biomed. Eng. 31, 423–488 (2003) 30.207 M.T.A. Saif, C.R. Sager, S. Coyer: Functionalized biomicroelectromechanical systems sensors for force response study at local adhesion sites of single living cells on substrates, Ann. Biomed. Eng. 31, 950–961 (2003)

References

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30.224 J. Cao, Q. Wang, H. Dai: Electromechanical properties of metallic, quasimetallic, and semiconducting carbon nanotubes under stretching, Phys. Rev. Lett. 90, 157601/1–157601/4 (2003)

30.225 C. Oddou, S. Wendling, H. Petite, A. Meunier: Cell mechanotransduction and interactions with biological tissues, Biorheology 37, 17–25 (2000)

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31. Experimental Methods in Biological Tissue Testing

Stephen M. Belkoff, Roger C. Haut

31.1 General Precautions.............................. 871 31.2 Connective Tissue Overview ................... 872 31.3 Experimental Methods on Ligaments and Tendons ................... 873 31.3.1 Measurement of Cross-Sectional Area ................. 873 31.3.2 Determination of Initial Lengths and Strain Measurement Techniques .................................. 873

31.3.3 Gripping Issues in the Mechanical Testing of Ligaments and Tendons.. 31.3.4 Preconditioning of Ligaments and Tendons ............ 31.3.5 Temperature and Hydration Effects on the Mechanical Properties of Ligaments and Tendons ............ 31.3.6 Rate of Loading and Viscoelastic Considerations...... 31.4 Experimental Methods in the Mechanical Testing of Articular Cartilage ................. 31.4.1 Articular Cartilage......................... 31.4.2 Tensile Testing of Articular Cartilage 31.4.3 Confined Compression Tests ........... 31.4.4 Unconfined Compression Tests ....... 31.4.5 Indentation Tests of Articular Cartilage .....................

874 874

875 875 876 876 876 876 877 877

31.5 Bone ................................................... 31.5.1 Bone Specimen Preparation and Testing Considerations............ 31.5.2 Whole Bone................................. 31.5.3 Constructs ................................... 31.5.4 Testing Surrogates for Bone ........... 31.5.5 Outcome Measures .......................

878 878 879 880 880 880

31.6 Skin Testing ......................................... 31.6.1 Background ................................. 31.6.2 In Vivo Testing ............................. 31.6.3 In Vitro .......................................

883 883 883 883

References .................................................. 884

31.1 General Precautions Unlike most engineering materials, many biological tissues are considered biohazards and appropriate precautions must be taken in their handling. Human cadaveric tissue is considered potentially infectious, as is nonhuman primate tissue and occasionally other animal tissue. In the USA, anyone who may be exposed to potentially infectious tissue is required by

Occupational Safety and Health Administration regulation 29 CFR 1910.1030 to receive annual training in bloodborne pathogens exposure prevention. The training is provided free of charge by employers. Also, the employer must provide proper personal protective equipment, such as gowns, gloves, facemasks, etc. Laboratories in which human tissue is used are

Part D 31

The current chapter on testing biological tissue is intended to serve as an introduction to the field of experimental biomechanics. The field is broad, encompassing the investigation of the material behavior of plant and animal tissue. We have chosen to focus on experimental methods used to test human tissue, primarily the connective tissues ligament, tendon, articular cartilage, bone, and skin. For each of these tissues, the chapter presents a brief overview of the structure–function relationship of the tissue and then discusses some of the common tests conducted on the tissue to obtain various material and mechanical properties of interest. The chapter also highlights some of the stark differences in testing biological tissues compared with engineering materials with which the reader may be more familiar.

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registered as biosafety laboratory level 2 facilities Centers for Disease Control (CDC). Researchers are encouraged to check with their local authorities to be in compliance with laws and regulations governing the use and transportation of cadaveric tissue.

Employers in the USA are also required to offer Hepatitis B vaccinations to those at risk of exposure to human tissue. Most institutions also require investigators to complete courses on animal handling and care.

31.2 Connective Tissue Overview The human body is composed of four primary groups of tissues:

Part D 31.2

1. Epithelial tissues, characterized by having cells closely joined one to another and found on free surfaces of the body. 2. Muscle tissues, characterized by a high degree of contractility of their cells or fibers. Their primary function is to move the skeleton. 3. Nervous tissues, composed of cells specialized in the properties of irritability and conductivity. 4. Connective tissues, in which the cells are separated by large amounts of extracellular materials. Tissues are combined in the body to form organs. Organs are defined as structures composed of two or more tissues integrated in such a manner as to perform certain functions. The heart, for example, has its chambers lined with a special epithelium, its walls are primarily muscle, and there are connective tissues present in numerous forms throughout this organ. Connective tissues represent wide-ranging types, both in their variety and distribution. They are all characterized, however, by large amounts of extracellular material. Their functions are as varied as the tissues themselves. These tissues bind, support, and protect the human body and its vital organs. Structural integrity and function of vital organs are adversely affected when connective tissue surrounding them is damaged. Connective tissues give us the strength to resist mechanical forces, and provide a recognizable shape that persists in the face of these forces. Connective tissues can be classified by their extracellular constituents. Connective tissue proper is subdivided into loose, dense, or regular. Loose connective tissue is widely distributed. For example, it is found in the walls of blood vessels, surrounding muscles as fascia, and in the lung parenchyma. Dense connective tissues contain essentially the same elements as loose connective tissue, but there are fewer cells. This type of tissue is found in skin, many parts of the urinary ducts, digestive organs, and blood vessels. The stromas and capsules of the internal organs,

e.g., kidney and liver, are dense connective tissues that are responsible for maintaining the structural integrity of organs against mechanical forces. Other body components have primarily mechanical functions and are composed almost exclusively of connective tissue with few cells called regular connective tissue. Bone tissue forms the greater part of the skeleton and it primarily resists the forces of compression resulting from muscle contraction and gravity. Tendons and ligaments have a parallel arrangement of the extracellular component that allow these tissues to transmit tensile loads. All connective tissues are, in fact, complex fiberreinforced composite materials. The mechanical properties of soft connective tissue depend on the properties and organization of collagen fibers in association with elastin fibers, which are embedded in a hydrated matrix of proteoglycans. The constitution of each connective tissue is tuned to perform a specific function. Ligaments provide mechanical stability to joints. The function of tendons is to transmit high tensile forces to bone via muscles and to allow the muscles to function at their optimal length. Skin is a two-dimensional soft connective tissue that supports internal organs and protects the body from abrasions, blunt impact, cutting, and penetration, while at the same time allowing considerable mobility. Articular cartilage is a 1–5 mm layer of connective tissue that covers the articular surfaces of diarthrodial joints. The primary functions of the cartilage are 1. to spread loads across joints and in so doing to minimize contact stresses, and 2. to allow relative movement of the opposing surfaces with minimum friction and wear. Bone is the connective tissue that makes up the skeleton of the body. Like other connective tissues, bone has a significant amount of extracellular material, but in the case of bone there is a substantial mineral phase that imparts its characteristic strong and stiff structural properties. In biological tissue there is an intimate relationship between structure and function. Some consider

Experimental Methods in Biological Tissue Testing

the structure of a given tissue to have been optimized through evolution to provide a given function or set of functions. When subjected to trauma, i. e., impact or overuse types of injuries, or disease the structure of the connective tissue is altered or distorted. The altered structure expresses itself as tissue dysfunction. Therefore, tissue function/dysfunction is often used clinically as a diagnostic tool for tissue damage. Therefore it is

31.3 Experimental Methods on Ligaments and Tendons

873

important to characterize the mechanical behavior of all types of connective tissues. The current chapter introduces various experimental methods used to evaluate the behavior of native and damaged tissue. We will also introduce experimental methods important in the documentation of tissue subjected to repair processes, either by natural healing or clinical interventions such as bone plating, skin suturing, etc.

31.3 Experimental Methods on Ligaments and Tendons

31.3.1 Measurement of Cross-Sectional Area Accurate measurement of the geometry of ligaments and tendons is essential in order to determine the material properties of these tissues. In the body, these structures typically have irregular, complex geometries that make cross-sectional area measurements difficult. Generally, the techniques that have been documented in the literature for determination of cross-sectional areas of ligament and tendons involve either contact or noncontact methods. Contact methods include molding techniques, digital vernier calipers, and area micrometers [31.1, 2]. These methods often rely on the investigators ability to gently touch the specimen

without causing significant deformations that may expel water and alter the dimension of interest. And yet, in numerous studies, investigators have also developed contact methods of measurement that rely on compression of the specimen into a defined shape by the application of a known, standardized load or stress [31.3, 4]. One study has developed an area micrometer technique in which a compressive, external pressure of 0.12 MPa is applied to ligaments and tendons in the determination of their cross-sectional area [31.5]. Due to the viscoelastic nature of these tissues, in part resulting from fluid flow within the tissue, the use of such devices and techniques must incorporate time as a variable in the measurement of cross-sectional area. To minimize the distortion of tissue shape, other investigators advocate the use of noncontact methods. These techniques include the shadow technique [31.4], the profile method [31.6], and the use of a light source [31.7]. A more recent and often used measurement tool is laser microscopy for the measurement of cross-sectional area [31.8, 9]. This technique has been shown to be highly accurate and reproducible. In addition, to correct for errors inherent in the system due to specimen concavities, a low-cost laser reflectance system has been described [31.10].

31.3.2 Determination of Initial Lengths and Strain Measurement Techniques Another difficulty that must be dealt with in the determination of the mechanical properties of ligaments and tendons is the measurement of specimen length. Ligaments are attachments between bones, but the insertion points vary over an area. There are direct insertions through Sharpey’s fibers and there are indirect insertions, in which the ligament fibers merge with the collagenous tissue of the periosteum. Because

Part D 31.3

Ligaments and tendons are parallel-fibered, dense connective tissues. These complex, fiber-reinforced composite materials provide stability to joints and aid in the control of joint motion. The fibers, collagen and elastin, are embedded in various proportions depending on tissue function, in a gelatin-like matrix of macromolecules (proteoglycans) and water. The role of ligaments, which connect bone to bone, is to augment the mechanical stability of joints and help control joint function. Tendons, on the other hand, attach muscle to bone and typically transmit large tensile loads across joints to control motions of the body. Tendons also allow muscles to function at their optimal length. In general, the tensile mechanical response of ligaments and tendons is highly nonlinear and dependent on the rate of loading or stretch. Such complex mechanical behavior presents a number of challenges when conducting tissue tests. The following will attempt to discuss some of the basic concerns and methods needed in the evaluation of the mechanical (material) and structural properties of ligaments and tendons, based on previous studies performed with animal and human tissues.

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Video cassette recorder Video dimensional analyzer Load cell

Femur ACL Video camera Tibia

Part D 31.3

Computer ACL

Tensile load TV monitor

VDA system

Fig. 31.1 Typical experimental setup of displacement measurement using video dimension analysis (VDA)

insertions are not discrete, there can be substantial variance in the initial length of a specimen [31.8, 11]. Various researchers have used pins or wires to mark the insertions of ligaments into bones. The distances between the markers have been determined from roentgenographs [31.11–13] or directly with a ruler [31.14]. Computation of strain can be based on the deformation of the ligament over these lengths or grip to grip, however, one should be mindful that surface strain varies along the length of ligaments and tendons [31.15]. Local strains can be measured by segmenting the specimen by means of drawing or fixing reference markers on the surface of the tendon or ligament. The markers can be drawn or fixed on the surface of a tendon or ligament by means of Verhoeff’s stain [31.16], elastin stain [31.10, 17], or reflective tape [31.18]. A charge-coupled device (CCD) camera, which is part of the video dimension analysis (VDA) equipment [31.19–21], is then used to record the motion of these markers during tensile stretch and the data are converted to surface strains Fig. 31.1.

31.3.3 Gripping Issues in the Mechanical Testing of Ligaments and Tendons Another reason to use noncontact video systems in the measurement of ligament or tendon tissue strain is the potential for specimen slippage within the tissue grips. Actual grip-to-grip strain or surface strain

can be measured more accurately when one of these systems is employed during testing. When it is possible or appropriate to apply clamps to ligaments or tendons directly, a number of specially designed freezing, hydraulic, and pneumatic clamps with roughened gripping surfaces have been utilized. Freeze clamps have been successfully used in the mechanical testing of musculo-tendonous junctions [31.22] as well as bovine and human tendons [31.23]. These types of clamps maintain a constant pressure against the soft, deforming tissue during axial tensile stretch. In many cases, however, the substance of a ligament may be too short. In this case, the entire bone–ligament–bone preparation is utilized for testing. Typically, a normal vice-grip type of clamp may be sufficient, especially when the bones can be shaped adequately to fit snugly into standard clamps. In other cases, for example, when testing a patella–patellar tendon–tibia preparation in which the bones have a nonuniform shape, the bone end can be embedded in room-temperature-curing epoxy or bone cement polymethal methacrylate (PMMA) and inserted into a holder grip [31.2, 24].

31.3.4 Preconditioning of Ligaments and Tendons Biological tissues are viscoelastic and exhibit natural states in response to repeated application of load or stretch. Such a state in vivo is called a homeostatic state, whereas in vitro it is called a preconditioned state. Some consider preconditioning tissue a necessary step in rheological testing of biological tissues [31.25]. Because biological tissues are viscoelastic and have memory, preconditioning specimens means that the specimens all have the same recent history. If a certain procedure for testing (stressing or straining) is decided upon, that procedure should be followed a number of times until the response becomes steady before the mechanical response of the tissue is documented. If the protocol changes, for example the amplitude changes, then the specimen should be preconditioned at that new level. In most tendons and ligaments, preconditioning effects on the stress decay during consecutive cycles are assumed to reach a steady response after approximately 10–20 cycles of loading [31.26]. However, a recent more detailed study of the preconditioning phenomenon suggests that the effect can persist in some tissues for many more cycles. It has therefore been suggested that preconditioning has to be integrated into constitutive formulations of biological tissues [31.27]. However, while the nonlinear and viscoelastic aspects of many

Experimental Methods in Biological Tissue Testing

tissues such as ligaments and tendons are well documented in the literature, the effects of preconditioning or its mechanisms are not well understood [31.28].

31.3.5 Temperature and Hydration Effects on the Mechanical Properties of Ligaments and Tendons

31.3.6 Rate of Loading and Viscoelastic Considerations Ligaments and tendons are known to be exposed to varied loads of deformation (loading) during normal physiological activities and extremely high rates during traumatic injury [31.32]. Currently, most of the literature has assumed a constant strain rate of 100%/s for physiological studies and approximately 1000%/s or above for traumatic injury studies [31.32]. Thus, it is important to consider strain rate in the experimental

methods used for the study of ligaments and tendons. The time dependence that does exist in ligaments and tendons is largely due to the viscoelastic [31.33] or biphasic [31.34] nature of these tissues. For the description of viscoelastic properties, there are two relevant quantities of interest: creep and relaxation. Relaxation relates to the decrease in load in a tissue under repeated or constant elongation, while creep relates to the increase in elongation under repeated or constant load. The quasilinear viscoelastic theory (QLV) is the most widely accepted model of viscoelasticity for ligaments and tendons [31.21]. In experimental studies relaxation is the more commonly measured property. Studies on bone–ligament–bone preparations have shown that the failure characteristics of these structures are highly strain rate dependent. These experiments have indicated that generally high-strain-rate experiments will produce failure of the ligament substance, while low strain rates more typically produce failure of bone near the sites of insertion [31.35, 36]. However, more recent studies with animal models suggest that the rate sensitivity of the ligament substance itself may have been overstated in the early experiments [31.32,37] and the QLV theory takes the form t G(t − τ)

σ(t) =

dσ e (ε) dε dτ , dε dτ

(31.1)

0

where G(t) is the reduced relaxation function and ε(t) is the strain history parameter [31.38]. While G(t) theoretically must be determined under step changes in strain, an improved method following a finite ramp time has recently been documented [31.39]. The inherent elastic function σ e can vary slightly between tissues, but for tendon [31.40] and ligament [31.38] it takes the form σ e = A( e Bε − 1) ,

(31.2)

where A and B are constants that are typically determined during a fast constant-strain-rate test using a variety of least-squares-based fitting routines. Ligaments, however, probably function in normal daily activity under repeated low loads, thus they function through creep rather than relaxation. The QLV theory has also been formulated in creep [31.28]. However, experimental studies have shown that the stress relaxation response can only be predicted from creep if collagen fiber recruitment is also accounted for in this model [31.41]. Finally, recent studies suggest that these tissues are, indeed, nonlinear so the currently accepted theory needs modification to include nonlinear viscoelastic characteristics [31.42].

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Environmental conditions, including temperature and hydration, are important considerations when testing ligaments and tendons. Testing specimens in air at room temperature will yield different results than when immersing them in an aqueous bath of an isotonic solution where the pH and temperature are closely controlled. Generally, the stiffness of ligaments and tendons will increase slightly when the bath temperature is decreased [31.28, 29], and there will be a reduction in the amount of cyclic stress relaxation when these tissues are immersed in baths at reduced temperatures. Similarly the rate of cyclic stress relaxation will be significantly reduced as the concentration of water in the tissue is reduced [31.30]. A significant increase is noted in the modulus and strength of human patellar tendons when tested in a phosphate-buffered saline (PBS) bath versus when tested using a PBS drip onto its surface [31.2]. The notion that the extent of tissue hydration plays a significant role in the mechanical properties of ligaments and tendons has also been confirmed in experiments in which human patellar tendon is stretched at a high (50%/s) or low (0.5%/s) rate. At high rates of strain the structural stiffness of these human tendons is significantly higher when immersed in a hypotonic (high water content in the tendon) solution versus a hypertonic (low water content) solution [31.31]. In contrast, for a low rate of strain, the structural stiffness is not dependent on the tonicity of the bath solution. This suggests that the viscous response is related to the water content of the specimen and not some inherent viscoelastic property of the collagen fibers.

31.3 Experimental Methods on Ligaments and Tendons

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31.4 Experimental Methods in the Mechanical Testing of Articular Cartilage 31.4.1 Articular Cartilage

Part D 31.4

There are three broad classes of cartilaginous tissues in the body: hyaline cartilage, elastic cartilage, and fibrocartilage. These tissues are distinguished by their biochemical composition, their molecular microstructure, and their biomechanical properties and functions. Hyaline cartilage is normally glossy, smooth, glistening, and bluish-white in appearance. This tissue covers the articulating surfaces of long bones and sesamoid bones within synovial joints, e.g., the surfaces of the tibia, the femur, and the patella of the knee joint. Articular cartilage is vital to maintaining normal joint motion, and its degradation is key to degenerative diseases such as osteoarthritis. Articular cartilage in freely movable joints, such as the hip and knee, can withstand very large compressive loads while providing a smooth, lubricated, load-bearing surface. In order to understand the mechanical properties of normal articular cartilage and those of degenerated tissue, a number of methods have been documented in the literature. The following will attempt to describe some of these experimental methods. The specific choice of test depends however on the size, shape, and amount of tissue available for study and the objectives of each study.

From these sheets of tissue, dumbbell-shaped or rectangular specimens are cut with a stamping device. The tissue slices are then placed in grips which have the faces lined with fine sandpaper (approximately 1500 grit) [31.47]. The tensile response of specimens oriented parallel and perpendicular to split lines is exponentially stiffening, similar to most other soft biological tissues. These data can be fit with an equation of the form: σ = A( e Bε − 1) ,

(31.3)

where A and B are found to be significantly higher near the surface of the tissue and for those test specimens oriented parallel to the surface split lines [31.48]. As in most soft connective tissue testing, preconditioning is often performed before these experiments. In early studies, constant-strain-rate tests were performed [31.49]. More recently, however, ramp-step relaxation testing is the method of choice. Typically, the tissue slices are stretched using a moderately rapid ramp in 2% strain increments to approximately 10–15% strain. Following each ramp, stress relaxation is invoked. While earlier studies (i. e. Woo et al. [31.50]) have equilibrated the tissue for 10–30 min after each step, a recent study suggests that equilibrium requires several hours [31.47]. Equilibrium values of stress and strain are then documented for various layers of the tissue [31.51, 52].

31.4.2 Tensile Testing of Articular Cartilage 31.4.3 Confined Compression Tests The tensile properties of articular cartilage are extremely relevant to the compressive stiffness of the tissue [31.43]. When a strip of cartilage is stretched under a constant rate, the tensile stress–strain curve behavior is nonlinear. Specimen orientation is important, because articular cartilage is not isotropic. The primary strength and stiffness directions follow the so-called split or cleavage lines according to Hultkrantz, which can be observed by penetrating an India-inked needle into the surface layer of the tissue [31.44]. These and other studies have verified that the mapped lines follow the primary orientations of the strength-bearing collagen fibrils in the tissue. Since the concentration, content, and organization of collagen fibrils in articular cartilage varies significantly with depth into the tissue [31.45], tensile tests are most often conducted on thin layers of tissue cut parallel to the surface with a sledge microtome [31.46] or a Vibratome (Scott Scientific, Montreal).

The intrinsic compressive properties of cartilage are usually obtained by the confined compression creep test. Typically, a cylindrical plug of cartilage and underlying subchondral bone are placed in a rigid cylindrical chamber, where deformation can occur only in the direction of loading (Fig. 31.2). Load

Cartilage Subchondral bone

Porous filter

Fig. 31.2 Sectional view of a confined compression test fixture

Experimental Methods in Biological Tissue Testing

31.4 Experimental Methods in the Mechanical Testing of Articular Cartilage

n=0

(31.4)

where u(0, t) is the surface displacement, h is the specimen thickness, and F0 is the applied load. While a theoretical solution was documented some time ago for the relaxation test where step changes in displacement are applied to the specimen [31.53], few investigations have used this method. One reason may be the extremely high force response experienced in the test when the initial displacement input is applied rapidly to the specimen. This typically yields unsatisfactory results in the curve-fitting process.

31.4.4 Unconfined Compression Tests More typically, relaxation parameters are calculated from data obtained on cartilage using an unconfined compression test [31.54]. In this experiment cartilage discs are removed from the underlying subchondral bone and placed between two highly polished parallel plates (Fig. 31.3). A ramp compression is then applied to a prescribed level of strain (typically less than 20% of the tissue thickness) and held until an equilibrium load is reached. The linear biphasic solution for this problem has been given by Armstrong [31.55] as  (1 − vs )(1 − 2vs ) σ = E s εc 1 + (1 + vs )  α2 H k ∞  − n 2A 1 h e × , (1 − vs )2 αn2 − (1 − 2vs ) n=1

(31.5)

where εc is the applied strain, αn are the roots of the characteristic equation J1 (x) − (1 − vs )x J0 (x)/(1 − 2vs ) = 0, and J0 and J1 are Bessel functions. For the case of unconfined creep, the theoretical solution has also been given [31.56].

Load

h

Cartilage

Polished surfaces

Fig. 31.3 Schematic of an unconfined compression test fixture

Unconfined compression studies have documented a difficulty in using the linear biphasic model of cartilage for the fitting of experimental data [31.54]. The difficulty appears to arise from the inability of this model to adequately represent the lateral constraint generated by the collagen fibrils, which lie parallel to the tissue surface in the top layer. To account for their stiffening effect in the tissue under unconfined compression, transversely isotropic models of the solid phase have been proposed [31.57] and later disputed [31.58] because the models do not account for the tension– compression nonlinearity of the fibrils [31.59]. Such studies have led to more recent developments of computational models in which fibril reinforcements are added to simulate the effect of tension–compression nonlinearity in collagen fibrils [31.60, 61]. These more recently developed models adequately represent the response of articular cartilage in unconfined creep and relaxation compression tests [31.62].

31.4.5 Indentation Tests of Articular Cartilage Indentation tests have been used to characterize the compressive behavior of articular cartilage. A singlephase linear elastic model is often used when modeling either short-time response or the long-time equilibrium response of this tissue [31.63, 64]. In the Hayes et al. study, elastic solutions are given for the indentation of a rigid, flat or spherical indenter into a layer bonded to a rigid half-sphere. The solution for a flat indenter is  4Ga a P (31.6) = κ ,v , ω (1 − v) h where G is the shear modulus, v is the Poisson’s ratio, and κ is a correction factor that accounts for the finite layer effect. A nonlinear correction factor is used when there are deep penetrations into the tissue where nonlinear effects become more important [31.65]. In order to determine the shear modulus, Poisson’s ratio (v) must be either assumed a priori [31.66] or determined by other means [31.57, 67]. In the latter study, the authors

Part D 31.4

This is a uniaxial test. During loading, fluid escapes only from the top of the specimen through a porous platen. In the creep test a constant load is applied to the specimen [31.28]. Analysis of the steady-state stress– strain response provides the equilibrium compressive modulus HA , the aggregate modulus of the solid phase of the tissue. Other model parameters, such as permeability and Poisson’s ratio, are found by curve fitting the final 30% or so of the creep response using the solution according to the basic biphasic model for the cartilage [31.53], given as    2 2 2 ∞  e −(n+1/2) π HA kπ/h F0 u(0, t) 1−2 , = h HA (n + 1/2)2 π 2

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Force

δ δ(t) = 0.05 mm/s

t Tidemark

δ(t)

Cartilage thickness

Articular cartilage h

Time

Subchondral bone

Part D 31.5

Fig. 31.4 Typical results obtained from an indentation test to obtain cartilage thickness. The force trace shows the initial indentor contact with the cartilage surface (sudden resistance with increased displacement) and contact with subchondral bone (sudden change in slope)

used two flat indenters with different radii to determine G and v from the indentation tests on cartilage. In the former study, Poisson’s ratio was estimated, based on experimental test results, to be approximately 0.5 instantly to represent the incompressible nature of the tissue during the initial step of the relaxation test and v = 0.4 at equilibrium. Accurate knowledge of the tissue thickness at the site of indentation is essential for the extraction of material properties from this test. The methods for this measurement have included optical [31.68], needle probe [31.63, 69, 70], and ultrasonic techniques [31.69, 71–73]. In the needle probe method, a small-diameter

probe is displaced into the tissue until the zone of underlying calcified cartilage or subchondral bone are noted by an abrupt increase in reaction force (Fig. 31.4). It should be noted here that the shear modulus obtained from such in situ indentation tests is higher than that obtained from the in vitro unconfined or confined compression test, possibly as a consequence of indenter size [31.61]. The indentation creep experiment has been used extensively to determine the compressive behavior of articular cartilage [31.68, 69]. The mathematical solution for indentation creep (and relaxation) using a porous probe on articular cartilage modeled as a linear biphasic material has been described [31.74]. Early investigations with this model solution typically used a set of master curves to approximate the intrinsic parameters (HA , v, and k) of the cartilage. More recently the problem has been simulated computationally and the model parameters extracted using a least-squares fitting technique. Various experimental studies have since shown that, for an indentation creep test on articular cartilage with a porous probe and using the linear biphasic model, only about the final 30% of creep can be fitted for the extraction of HA , k, and v for the tissue [31.75, 76]. In some of the more recent models that include a fibril-reinforced network of collagen fibrils, the actual experimental curves are better fitted for the extraction of the intrinsic material parameters for the tissue [31.43, 77]. The same degree of fit can also be realized for these experimental curves by the inclusion of a viscoelastic matrix response in the model [31.72, 78, 79].

31.5 Bone Bone, like other biological tissue, is not homogeneous and isotropic and is a hierarchically organized composite material. It is important to have an appreciation of the microstructural organization of bone in order to understand the simplifying assumptions. A more in-depth treatment of bone structure and function may be found elsewhere [31.56]. There are two types of bone: cortical and cancellous. Cortical bone is the dense compact outer shell of what we casually call bone. Cancellous bone is a lattice of trabeculae surrounded by marrow, typically found at the ends of long bones. While the cortex of long bone thins with age, it is cancellous bone that is most profoundly affected by aging, particularly in postmenopausal women, and is associated with os-

teoporosis. Osteoporosis is the state of reduced bone density resulting from the bone-resorbing cells (osteoclasts) outpacing the bone-forming cells (osteoblasts).

31.5.1 Bone Specimen Preparation and Testing Considerations Gripping One of the great challenges in testing biological tissue is maintaining a firm grip at the tissue–fixture interface. When testing machined coupons, the gripping issue is typically resolved using standard engineering material grips. When testing involves whole bones, the varied geometry of the bone often poses a chal-

Experimental Methods in Biological Tissue Testing

lenge. Gripping can be achieved by inserting screws radially into the outer cortex [31.80], however, if the bone is osteoporotic, bone failure at the gripping site may occur and confound the results. As an alternative, bones may be potted in an acrylic using such products as Swiss Glas [31.81], Bondo [31.82], polymethylmethacrylate [31.83] or liquid metal [31.84].

Gripping Density Bone strength is typically reported to be a function of the square of the density, although there are reports that the strength can vary from a power of 1.3 to 3.0 [31.56]. Density varies geographically within a body. Metaphyseal bone is more susceptible to resorption than cortical bone. There are also bones which seem to be more affected by bone mineral loss than others. In particular, the proximal femur (hip), spine, and distal radius are common sites for fragility (osteoporotic) fractures. Bone mineral density can be established nondestructively by means of dual-energy x-ray absorptometry [31.86], quantitative computed tomography [31.87], and ultrasound [31.88]. Alternatively, density can be obtained destructively by removing (biopsy) a volume of bone, placing it in an oven to eliminate all moisture, and then measuring the resulting ash weight [31.89]. A less desirable method is to obtain a standard x-ray in the field of view of which is placed a step reference [31.90]. While all of the above methods are valuable for documenting density and provide an indication of specimen strength, density is not a good predictor of fracture strength. This is also true clinically. A patient with low bone mineral density is at risk of fracture, but density alone cannot predict with any certainty when or if the fracture will occur. Orientation Bone is a transversely orthotropic material. When measured along its longitudinal axis, bone exhibits an ultimate strength of approximately 133 MPa in tension, 193 MPa in compression, and 68 MPa in shear [31.91].

879

Femoral bone is strongest along its longitudinal axis and less strong in the transverse directions [31.91]. Similar trends are noted for modulus. Actually, there is substantial variation in the reported strength and modulus of bone [31.56]. The variation can often be attributed to a multitude of factors that affect bone material behavior. These factors include the location, temperature, orientation, hydration, gripping, and testing rate. Viscoelasticity Bone exhibits viscoelastic behavior; therefore, the rate at which tests are conducted can dramatically affect the measured response. Both the strength and stiffness increase as a function of increasing strain rate [31.92]. Storage Bone is best stored wrapped in saline-soaked gauze [31.93], double bagged, and frozen. Although it is preferred that tissue be tested as soon after harvesting as possible, it is not always practical to do so. Often investigators need to thaw, prepare, refreeze, thaw, and then test the specimens. Up to five cycles of freezing and thawing of cortical bone reportedly do not alter the compressive properties of bone [31.94]. For short-term storage, −20 ◦ C seems sufficient. If tissue is to be stored for the long term, −80 ◦ C freezer storage is preferred as it minimizes enzymatic activity [31.93, 95].

31.5.2 Whole Bone Whole-bone testing is used to measure the response of a particular bone under typical in vivo loads. It is often not possible, practical or ethical to acquire response parameters in vivo (particularly in humans), so cadaveric tissue is commonly used as a model. Understanding the mechanism of injury requires knowing the behavior of native tissue and how that behavior changes as a function of age, loading/deformation rate, healing, remodeling, drug use, and state of disease. Another need for whole-bone testing is to investigate the effect of stress concentrations on the structural response. In surgery, the placement of bone screws and pins [31.96], as well as defects left after tumor resection [31.97, 98], can significantly weaken bone until healing occurs [31.99]. Questions regarding the magnitude of the stress concentration and the duration of the concentration (unlike engineering materials, bone usually restores itself to nearly initial values) are important clinically [31.100]. Bones are normally not loaded in axial tension. Even long bones (femur, humerus, tibia), which are

Part D 31.5

Environment Bone is largely composed of water and its material properties change as a function of hydration. For longterm tests in which the bone may be exposed to the atmosphere, the bones must be hydrated [31.85]. Issues regarding hydration are similar to soft tissues. The reader is also referred elsewhere to obtain a review of general considerations of mechanical testing of bone such as pH balance, temperature, tonicity, and the use of antibiotics to prevent putrification during long tests.

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nominally cylindrical but in reality have some curvature, are difficult to grip and fixture such that tension is created uniformly across the cross section. For these same reasons, it is difficult to load to failure bones in tension without some bending superimposed by geometry or loading eccentricity. Torsion tests are typically conducted on whole bones to obtain shear properties [31.101, 102], but because of the natural curvature of long bones, it is often difficult to apply only torsion. Bones are naturally loaded in compression in vivo, so it is not surprising that bone is strongest in compression. Loading in compression to obtain in situ strain measurements is straightforward, but loading to failure results in fractures along the shear planes of the bone.

31.5.3 Constructs Construct testing is often used to compare various modalities of fracture fixation and thereby provide advice for orthopaedic surgeons on the fixation that is most stable or best satisfies biomechanical considerations (Fig. 31.5). Construct testing is often conducted to gather basic performance data for use in the Food and Drug Administration (FDA) approval process. Unfortunately, what constitutes an optimal construct is not obvious or well documented. Interpreting the results of a test and placing them in a clinically significant context is perhaps the most challenging. In the 1960s the watch words were rigidity of fixation. The prevailing wisdom was to fix fractures as rigidly as possible and to place screws in as many holes in the bone plates as were available. Surgeons quickly learned that, if the fixation was

excessively rigid, the plate stress shielded the fracture site. As a result the osteoblasts did not receive the appropriate mechanical signals for full healing, that is, the fracture site callus that formed did not ossify. Placing bone screws in all the available holes was also unnecessary [31.103]. The screws closest to and furthest from the fracture site are the ones that provide plate fixation. Surgeons often place a third screw on each side of the fracture as a safety in the event that one of the other screws fails. The goal of fixation is to keep fracture site motion to compressive strains below 2% [31.56]. Between 2% and 10%, a fibrocartilage callus forms and places the fracture at risk of nonunion. There is a broad range of construct stiffness that will yield the requisite fracture site strains for healing [31.104]. Similar healing criteria are not available by which to predict the efficacy of a particular construct based on the mechanical response of the construct. For example, in the spine, it is unknown how much motion spinal instrumentation should allow. If the instrumentation results in too stiff a construct, the patient’s range of motion may be impeded, degeneration of the disc proximal to the top of the instrumentation, or fracture of the proximal vertebral level may occur. If the instrumentation allows too much motion, the desired stabilization effect of the instrumentation may not occur. The difficulty is in determining the bounds of what constitutes optimal fixation. The determination of such criteria is further complicated by variation in patient activity level, bone health (density), and changes in mechanical demands placed on the instrumentation as healing occurs.

31.5.4 Testing Surrogates for Bone Synthetic bones that have material properties similar to native bone have been developed [31.105–108]. None of these surrogate bones match all of the material characteristics found in native bone [31.109, 110]. For example, while some surrogate bones may match the compressive modulus of normal bone, they may not match the shear modulus. Some may exhibit appropriate quasistatic behavior, but when loaded at high strain rates, the surrogate may not exhibit the appropriate viscoelasticity [31.111].

31.5.5 Outcome Measures Fig. 31.5 Displacement measured on a femoral stem prosthesis relative to the proximal femur. An acupuncture needle was inserted through the bone and points to a microscope calibration disc glued to the prosthesis

Kinematic Linear variable displacement transducers (LVDT)s have been mounted across the fracture site to measure fracture site motion, and if the initial fracture gap is known,

Experimental Methods in Biological Tissue Testing

A

PA D PB

P B

B

Fig. 31.6 Ghost point P is a virtual point defined on A and B. The virtual displacement of PB relative to PA is the true displacement at the fracture site between fragment A and B

The analysis systems typically report rigid-body rotations and translations relative to the origin of a given fragments reference system. If the origin coincides with the rigid body’s center of rotation, translation measurements can be decomposed into true translations and apparent translation due to rotation about the instantaneous center of rotation. Unfortunately, the center of rotation is not a fixed point for most fragment motion [31.113]. One could define a fixed reference, centered at a reproducible origin, i. e., center of mass, anatomical landmark, but such points are hard to identify in biological tissues or are not practical. One solution is to establish ghost or virtual points (Fig. 31.6). These are points of interest that are defined relative to both fragments’ reference frames before testing begins. For example, if we are interested in fracture gap motion, one could identify a point on the fracture line and define that point in both fragment coordinate systems. As the fragments move in rigid-body motion relative to each other, the virtual location of the ghost point can be determined in each fragment’s reference frame and then mapped to the global reference frame. The difference in the location of that ghost point, as predicted from the reference frame of each fragment, is the true translation of one fragment relative to the other at the location of the ghost point. Perhaps the most popular of the motion analysis systems use optical markers. These markers are either passive (reflective) markers or active light-emitting diodes (LEDs). The accuracy of the LED systems is reportedly [31.114] 0.3 mm in translation and 0.7◦ in rotation. For passive markers, accuracy of 0.1 mm and 0.2◦ has been reported [31.81]. Active markers have the advantage of being unambiguously recognized by the receiver. Because the LED markers emit light in a known sequence, the receiver can identify which marker is active at any given point in time, whereas passive markers have to be identified by their location in the temporal context. The accuracy of the systems is a function of the diagonal of the calibrated space, so it is important to calibrate only the volume required to conduct a given experiment, thereby maximizing the system accuracy [31.115]. LEDs, because they are hard-wired to the recording system, can be cumbersome because of the multiple wires running to the LEDs. Reflective markers are prone to contamination with blood and lose their reflectiveness. Both types of markers can be obscured by the testing apparatus and/or instrumentation. Further, optical systems using reflective markers are sensitive to errant reflections from liquid (hydrated tissue) and shiny

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calculate fracture site strain. Often the resulting fracture site motion, even under a simple load (i. e., uniaxial) is not one dimensional, making LVDT displacement difficult to interpret. Often the displacement sensed by the LVDT is a combination of translational and rotational and impossible to decompose. The problem has been overcome by mounting several LVDTs in complex arrangements to calculate three-dimensional (3-D) motion, but these methods are very cumbersome [31.112]. To overcome the shortcomings of LVDTs, motion analysis techniques have been developed. These technique usually consist of rigidly mounting some type of markers to the bone fragments and then placing the fragments into a calibrated space. The location of the markers is recorded by two or more receivers (cameras). The relative location and angle of the receivers is known (usually determined during calibration) so the 3-D coordinates of each marker location can be calculated by triangulation. If the motion to be measured is known to be planar, two-dimensional (2-D) coordinates can be calculated using only one camera. A major consideration of using the motion analysis systems is tracking motion at the site of interest.

31.5 Bone

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surfaces such as polished stainless-steel orthopaedic instrumentation (bone plates, screw heads) or the columns on the servohydraulic testing machine and fixturing. Instead of using infrared light as the conduit for obtaining position data, the flock of birds system (Ascension Technology Corporation, Burlington, VT) tracks the position and orientation of (slave) markers based on the magnetic field strength of the master sensor [31.116]. Acoustic sensors function on a similar basis to optical systems except that markers consist of spark emitters and the receivers are microphones [31.117]. Knowing the speed of sound and the relative position of the microphones, the transit time of the sound pulse emitted by a given spark is measured and the position of the emitter can be triangulated. While accurate, these systems are prone to interference from ambient noise and reflection off fixtures and test equipment. Gross strains, such as the strain across a fracture gap, can be calculated based on the displacements of points across the fracture divided by the initial gap. More typically we want measurements of strain in the cortex of the bone. In this case, strain gauges can be mounted directly to the cortex [31.118]. This requires stripping of the periosteum, which serves as a vapor barrier, so specimen dehydration must be considered. In trabecular bone, digital image correlation has been used to calculate strain [31.119]. Kinetic Gross applied force may be measured at the point of application using standard load cells. Because of the frequent use of baths and hydrated environment chambers, load cells are often mounted to the actuator rather than to the base of the testing apparatus, in order to prevent damage to the electronics from saline. For high-speed and cyclic tests, mounting the load cell to the actuator requires that any mass distal to the sensing beams be inertially compensated. We often want to know what forces are transmitted across fracture or osteotomy sites. Such measurements provide information about the reduction provided by various fixation devices. Changes in these measurements that occur with testing (acute or cyclic) inform us as to how the fixation is behaving, i. e., if the instrumentation is loosening and how it is sharing load with the bone.

Fig. 31.7 Washer load cell used to measure reduction

forces

A simple example of such a measurement is the use of a washer load cell to measure the reduction force provided by different types of lag screws for scaphoid fixation Fig. 31.7. A more elaborate example is the use of instrumented hardware and telemetry [31.120, 121]. While the data obtained from such devices is limited, it gives us considerable insight into the loads transmitted during healing, partial weight-bearing, and activities of daily living. The information is used to design rehabilitation programs, design new implants, and verify computer models. Some applications do not lend themselves to the use of a washer load cell to measure contact force as the washer may be too thick and thereby alter the load transmission path or pressure distribution. In the above example, pressure-sensitive film [31.122] could have been employed, but the film only provides maximum pressure readings, so loss of reduction force would have been missed. An alternative could be to use a pressure-sensitive polymeric film [31.123]. While this transducer provides real-time pressure distributions that can be integrated to calculate the total contact force, the transducers are temperamental, sensitive to temperature changes, need to be calibrated in the pressure range in which they will be used, and are easily damaged if kinked or exposed to sharp edges or to moisture, especially saline. The transducers are valuable for obtaining relative pressures and spatial distributions of pressure. They are less valuable for obtaining actual pressures accurately.

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31.6 Skin Testing 31.6.1 Background

31.6.2 In Vivo Testing In vivo testing avoids the issues of tissue environment and release of in vivo skin tension, and has the advantage of diagnosing skin dysfunction and pathology. There are several difficulties of measuring mechanical properties in vivo. Gripping the skin can be achieved by gluing grips to the epidermis using cyanoacrylate cement, but it is unknown how the surface traction is transmitted through the depth of the skin. Skin thickness may be measured using ultrasound or skin fold calipers, but both methods are prone to the inclusion of some measurement artifacts. Several attempts of uniaxial tests have been conducted by gluing grips to the skin surface. These tests are affected by the thickness of the skin, but this can be accounted for [31.130]. Of course uniaxial tests in vivo are not truly uniaxial as the lateral boundaries are not

31.6.3 In Vitro In vitro (ex vivo) testing reduces many of the challenges of in vivo testing, but replaces them with another set of challenges. In vitro testing of excised specimens permits the lateral boundaries to be traction free. The tests may also be started in a stress-free state, however, it must be noted that the specimen is under some in vivo tension that has been released during specimen excision. The investigator must document the in vivo specimen length so that, when it is excised and the specimen retracts, the investigator will know how to restore the original dimensions. One of the simplest ways of documenting in vivo dimension is to mark the skin with a 2-D grid using an ink stamp [31.139]. Uniaxial specimens obeying dimensions of ASTM D1708 can be stamped from skin specimens using a skin punch [31.140]. The mechanical response of the uniaxial skin specimen is less than would be expected if the specimen could be tested in vivo. Cutting uniaxial specimens from a field of skin severs the lateral boundary, damages the collagen network, and isolates the specimen from the reinforcing effect of the neighboring tissue.

Part D 31.6

Skin is the largest single organ in the body, accounting for about 16% of the total body weight and has a surface area of 1.5–2 m2 [31.2, 124]. Skin varies in thickness from 0.2 mm on the eyelid to 6.0 mm on the sole of the foot. Skin is composed of two layers: the epidermis which forms the superficial layer that serves as a barrier, and the dermis which provides structural integrity to the skin. As such, the dermis is the layer we are typically most interested in. The dermis consists mostly of collagen fibers (type I and III), elastin, and reticulin, surrounded by a hydrated matrix of ground substance. Also contained in the dermis are nerve endings, various ducts and glands, blood vessels, lymph vessels, and hair shafts and follicles. The dermis has an upper layer (papillary) of fine, randomly oriented collagen fibers [31.125] that connect the epidermis to the deep layer of the dermis, the reticular dermis. The reticular dermis contains layers of thick, densely packed collagen fibers that are organized in planes parallel to the surface of the skin [31.126, 127] with some fibers traversing between planes to limit interplanar shear. Within the planes, the collagen fibers appear to have a preferential orientation [31.128, 129] that governs the anisotropic behavior of skin. Surgeons are trained to cut along the dominant fiber orientation in order to minimize damage to the fibers and reduce tension in the healing skin incision.

traction free, nor is the interface of the dermis with the subdermis. Some have tested the skin in torsion by attaching a disk to the skin’s surface and applying a known twist or torque and measuring the response. The test was modified by gluing an outer ring to the skin such that only the annulus of skin between the ring and the disk was tested [31.131]. Saunders [31.132] estimated the modulus of elasticity from torsional tests. Wijn et al. [31.133] attempted to correlate torsional and uniaxial test measurements using the theory of elasticity of a homogeneous isotropic media, but did not recover comparable material constants. They concluded that skin cannot be treated as homogeneous and isotropic. Troubled by the inability to retrieve material constants, some investigators took a different approach [31.134–136]. They placed a type of suction cup on the skin, applied a vacuum, and measured the resulting dome height of the skin. A grid was applied to the skin before the test to track 2-D deformation and it was found that the resulting fields were inhomogeneous [31.137]. The investigators developed a technique to start the tests with the skin in a stress-free state [31.138].

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Lanir and Fung [31.141] developed a device to test membranous soft tissue biaxially. The device was capable of stretching in both directions, or one direction while holding the other dimension constant or stress free. Because skin is not a homogeneous isotropic material, there is concern over whether the standard dumbbell-shaped specimens satisfy the requirements

of being uniaxial with a uniform stress strain field in the gage area. In composite materials, the lengthto-width ratio can be many times greater than that needed for an isotropic material. In biological tissues, it is often not possible to obtain specimens with such aspect ratios and in these cases pure shear samples [31.142, 143] may be an attractive option.

References 31.1

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A. Race, A.A. Amis: The mechanical properties of the two bundles of the human posterior cruciate ligament, J. Biomech. 27, 13–24 (1994) R.C. Haut, A.C. Powlison: The effects of test environment and cyclic stretching on the failure properties of human patellar tendons, J. Orthop. Res. 8, 532–540 (1990) D.L. Butler, M.D. Kay, D.C. Stouffer: Comparison of material properties in fascicle-bone units from human patellar tendon and knee ligaments, J. Biomech. 19, 425–432 (1986) D.G. Ellis: Cross-sectional area measurements for tendon specimens: a comparison of several methods, J. Biomech. 2, 175–186 (1969) D.L. Buttler, D.A. Hulse, M.D. Kay, E.S. Grood, P.K. Shires, R. D’Ambrosia, H. Shoji: Biomechanics of cranial cruciate ligament reconstruction in the dog. II. Mechanical properties, Vet. Surg. 12, 113–118 (1983) B.N. Gupta, K.N. Subramanian, W.O. Brinker, A.N. Gupta: Tensile strength of canine cranial cruciate ligaments, Am. J. Vet. Res. 32, 183–190 (1971) F. Iaconis, R. Steindler, G. Marinozzi: Measurements of cross-sectional area of collagen structures (knee ligaments) by means of an optical method, J. Biomech. 20, 1003–1010 (1987) T.Q. Lee, S.L. Woo: A new method for determining cross-sectional shape and area of soft tissues, J. Biomech. Eng. 110, 110–114 (1988) S.L. Woo, M.I. Danto, K.J. Ohland, T.Q. Lee, P.O. Newton: The use of a laser micrometer system to determine the cross-sectional shape and area of ligaments: a comparative study with two existing methods, J. Biomech. Eng. 112, 426–431 (1990) D.K. Moon, S.D. Abramowitch, S.L. Woo: The development and validation of a charge-coupled device laser reflectance system to measure the complex cross-sectional shape and area of soft tissues, J. Biomech. 39, 3071–3075 (2006) D.L. Butler: Kappa delta award paper. Anterior cruciate ligament: its normal response and replacement, J. Orthop. Res. 7, 910–921 (1989) D.L. Bartel, J.L. Marshall, R.A. Schieck, J.B. Wang: Surgical repositioning of the medial collateral lig-

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the canine medical collateral ligament, J. Biomech. Eng. 103, 293–298 (1981) S.D. Abramowitch, S.L. Woo: An improved method to analyze the stress relaxation of ligaments following a finite ramp time based on the quasilinear viscoelastic theory, J. Biomech. Eng. 126, 92–97 (2004) R.C. Haut, R.W. Little: A constitutive equation for collagen fibers, J. Biomech. 5, 423–430 (1972) G.M. Thornton, A. Oliynyk, C.B. Frank, N.G. Shrive: Ligament creep cannot be predicted from stress relaxation at low stress: a biomechanical study of the rabbit medial collateral ligament, J. Orthop. Res. 15, 652–656 (1997) R.V. Hingorani, P.P. Provenzano, R.S. Lakes, A. Escarcega, R. Vanderby Jr.: Nonlinear viscoelasticity in rabbit medial collateral ligament, Ann. Biomed. Eng. 32, 306–312 (2004) W. Wilson, C.C. van Donkelaar, B. van Rietbergen, K. Ito, R. Huiskes: Stresses in the local collagen network of articular cartilage: a poroviscoelastic fibril-reinforced finite element study, J. Biomech. 37, 357–366 (2004) P. Bullough, J. Goodfellow: The significance of the fine structure of articular cartilage, J. Bone Joint Surg. Br. 50, 852–857 (1968) A. Benninghoff: Form und Bau der Gelenkknorpel in ihren Beziehungen zur Funktion. II., Zeitschrift für Zellforschung und Mikroskopische Anatomie 2, 793 (1925) S. Akizuki, V.C. Mow, F. Muller, J.C. Pita, D.S. Howell, D.H. Manicourt: Tensile properties of human knee joint cartilage: I. influence of ionic conditions, weight bearing, and fibrillation on the tensile modulus, J. Orthop. Res. 4, 379–392 (1986) M. Charlebois, M.D. McKee, M.D. Buschmann: Nonlinear tensile properties of bovine articular cartilage and their variation with age and depth, J. Biomech. Eng. 126, 129–137 (2004) S.L.Y. Woo, P. Lubock, M.A. Gomez, G.F. Jemmott, S.C. Kuei, W.H. Akeson: Large deformation nonhomogeneous and directional properties of articular cartilage in uniaxial tension, J. Biomech. 12, 437– 446 (1979) V. Roth, V.C. Mow: The intrinsic tensile behavior of the matrix of bovine articular cartilage and its variation with age, J. Bone Joint Surg. Am. 62, 1102– 1117 (1980) S.L.Y. Woo, P. Lubock, M.A. Gomez, G.F. Jemmott, S.C. Kuei, W.H. Akeson: Large deformation nonhomogeneous and directional properties of articular cartilage in uniaxial tension, J. Biomech. 12, 437– 446 (1979) N.O. Chahine, C.C.B. Wang, C.T. Hung, G.A. Ateshian: Anisotropic strain-dependent material properties of bovine articular cartilage in the transitional range from tension to compression, J. Biomech. 37, 1251–1261 (2004)

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891

Implantable 32. Implantable Biomedical Devices and Biologically Inspired Materials

Hugh Bruck

Mechanical forces play an important role not only in synthetic materials and systems, but also in the natural world. They have guided many aspects of the structural

32.1 Overview.............................................. 892 32.1.1 History ........................................ 892 32.1.2 Experimental Mechanics Challenges 893 32.2 Implantable Biomedical Devices............. 32.2.1 Applications ................................ 32.2.2 Brief Description of Examples ........ 32.2.3 Prosthetics .................................. 32.2.4 Biomechanical Fixation................. 32.2.5 Deployable Stents ........................ 32.3 Biologically Inspired Materials and Systems......................................... 32.3.1 Applications ................................ 32.3.2 Brief Description of Examples ........ 32.3.3 Functionally Graded Materials ....... 32.3.4 Self-Healing Polymers/ Polymer Composites ..................... 32.3.5 Active Materials and Systems ......... 32.3.6 Biologically Inspired Ceramics ........

899 899 899 900 904 907 909 909 910 911 915 917 921

32.4 Conclusions .......................................... 923 32.4.1 State-of-the-Art for Experimental Mechanics........... 923 32.4.2 Future Experimental Mechanics Research Issues ............................ 924 32.5 Further Reading ................................... 924 References .................................................. 924

and the fundamental scientific and technical insight that has been obtained into various aspects of processing/microstructure/property/structure/ performance relationships in these devices, materials, and systems will be reviewed.

development that is found in nature, primarily with respect to their effect on non-mechanical processes. For example, a plant must grow laterally in a manner that

Part D 32

Experimental mechanics is playing an important role in the development of new implantable biomedical devices through an advanced understanding of the microstructure/property relationship for biocompatible materials and their effect on the structure/performance of these devices. A similar understanding is also being applied to the development of new biologically inspired materials and systems that are analogs of biological counterparts. This chapter attempts to elucidate on the synergy between the research and development activities in these two areas through the application of experimental mechanics. Fundamental information is provided on the motivation for the science and technology required to develop these areas, and the associated contributions being made by the experimental mechanics community. The challenges that are encountered when investigating the unique mechanical behavior and properties of devices, materials, and systems are also presented. Specific examples are provided to illustrate these issues, and the application of experimental mechanics techniques, such as Photoelasticity, Digital Image Correlation, and Nanoindentation, to understand and characterize them at multiple length scales. It is the purpose of this chapter to describe the application of experimental mechanics in understanding the mechanics of implantable biomedical devices, as well as biologically inspired materials and systems. In particular, the experimental techniques used to develop this understanding,

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the volume reduces the stress due to bending from wind loading, but it must also distribute a sufficient amount of this volume axially in order to have sufficient surface area to gather sunlight necessary to support photosynthetic chemical processes within that volume. The need to understand the tradeoffs that natural systems make has provided an enormous opportunity to the experimental mechanics community to modify techniques normally used on synthetic materials to the study of these natural systems. This knowledge of the relationship between natural structure and functionality can then be used to guide the way natural materials are used in a diverse array of applications, ranging from implantable biomedical devices to new materials and systems that are referred to as Biomimetic or Biologically inspired/Biologically inspired depending on whether the concepts originating from nature used in their developed are either copied (biomimetic) or serve as inspiration (biologically inspired) [32.1].

One notable characteristic of natural materials that is of interest to the experimental mechanics community is the way in which the structure functions and adapts at a wide range of length scales. Much of the interaction that biological materials and structures experience occurs through mechanical contact. Therefore, to develop biologically inspired materials and systems it is necessary to process synthetic materials to adopt the attributes of biological materials and structures that affect mechanical behavior through similar microstructure/ property/structure/performance relationships. This has also been synergistic with the development of implantable biomedical devices, where it is essential to process materials with appropriate microstructure/ property relationships that are biocompatible (biomaterials), and deploy these biomaterials in devices with structure/ performance relationships that are appropriate for mechanically interacting with the body of an animal/human.

Part D 32.1

32.1 Overview 32.1.1 History Engineering can be defined as the effort by human beings to design and create materials and systems (i. e., technology) in order to improve their lives. Science can be defined as the effort by human beings to explain natural phenomena. To develop new technologies, Engineers employ scientific principles to manipulate raw materials that are found within their natural environment. The first engineered structures discovered by archaeologists were crude tools, utensils, and buildings consisting of natural materials such as stone, wood, and clay that were initially not refined [32.2]. Of these materials, only clay existed in both a liquid and solid phase. Therefore, it was the first material to be refined from its natural state by the addition of biological materials, such as straw, in order to form the first synthetic material: a natural composite. With the advent of the Bronze and Iron Ages, synthetic materials refined from inorganic natural materials became the focus for engineering many new technologies. Metallurgy became the foundation for the science and engineering of materials. Although stone, wood, and clay were still employed by engineers, they were often used without additional refinement. The need for a deeper understanding of these natural materials was obviated by the ability to utilize extra quantities of ma-

terial in order to create more conservative designs. This remained the case until the early 20th century, when the advent of synthetic polymers began to herald a new age in materials science and engineering: the age of plastics. The development of plastics coincided with scientific advances in the characterization of materials through the use of microscopes. Characterizing the microstructure of materials enabled engineers to develop new theories on the physical behavior of materials, and to use these theories to alter the way these materials were refined in order to improve their physical properties. Engineers returned to the concept of natural composite materials, and began to extend them to new synthetic composites formed from metals, ceramics, and plastics. The development of these composite materials was critical to the greatest engineering feat to date: landing a human being on the moon. Concurrently, these new materials were also having an impact on another important scientific/engineering feat that has transformed the lives of human beings: modern medicine. Much of the later half of the 20th century was focused on further development of advanced materials, with substantial motivation from the needs of modern medicine. One limitation of materials, which often restrict their applications in medicine, is the deterioration of their structural integrity due to damage accumula-

Implantable Biomedical Devices and Biologically Inspired Materials

Biomedical devices

Biological inspired materials & systems

Prosthetics

FGMs

Biomechanical fixation

893

Self-healing materials Active materials and structures

Deployable stents Bioinspired ceramics

Fig. 32.1 Relationship of implantable biomedical devices to biologically inspired materials and systems that are the current focus of applications for experimental mechanics

composites, active materials and structure, and biologically inspired ceramics. The scientific and technical relationship between these implantable biomedical devices and biologically inspired materials and devices that are the current focus for the application of experimental mechanics can be seen in Fig. 32.1. It is the relationship that will serve as a basis for discussing the experimental techniques that have been developed for addressing the challenges of applying experimental mechanics to understanding processing/microstructure/ property/structure/performance relationships in the implantable biomedical devices and biologically inspired materials and system to be reviewed in this chapter.

32.1.2 Experimental Mechanics Challenges Materials and Structural Characteristics For implantable biomedical devices, the applications are typically in vivo (i. e., within a living organism). As such, biocompatibility plays an important role in the use of materials for any implantable biomedical device. Consequently, biomaterials are often selected whose primary characteristic are chemical and mechanical properties that are compatible with the biological function of the organism. Unfortunately, biomaterials do not often possess microstructures with chemical or mechanical characteristics that are optimal for biomedical applications. Therefore, designing new materials with the constraint of biocompatibility is a challenge that requires understanding the microstructure/property relationships in biomaterials using novel experimental

Part D 32.1

tion. This problem, which exists to some extent in all synthetic materials, has resulted in the extensive development of advanced techniques for the measurement and characterization of damage that have evolved from Non-destructive Evaluation and Fracture Mechanics to quantify and understand damage mechanisms in biomaterials. On the other hand, natural materials and structures have evolved to actively minimize the impact of damage accumulation through processes such as self-healing and remodeling, which have been widely considered impractical in synthetic materials until only recently [32.3, 4]. Adapting these natural processes to synthetic materials has led to a new field of materials and systems research near the end of the 20th century known as biologically inspired materials and systems. The development of biologically inspired materials and systems has resulted from a new scientific and technical focus in materials research that has been placed on microtechnology and nanotechnology. This has been predicated by new characterization technologies, such as Atomic Force Microscopy that enabled visualization of natural phenomena down to the atomistic scale, as well as microtensile testers and nanoindenters for microscale and nanoscale mechanical characterization. At these size scales, new phenomena were revealed that presented scientists and engineers with a novel approach to engineering advanced materials and systems, such as biocompatible materials (biomaterials) for biomedical applications. Piehler has identified biomaterials as the dominant area of focus for materials research in the 21st century, one that will fuel significant economic expansion through the development of new implantable biomedical devices and biologically inspired materials and systems [32.5]. This new scientific and technical focus in materials research has provided a wonderful opportunity for experimental mechanics to play an important role in the understanding of processing/microstructure/property/structure/ performance relationships in implantable biomedical devices and biologically inspired materials and systems that are necessary for improving the development and deployment of both conventional synthetic materials and biomaterials. Although there are many opportunities for experimental mechanics in developing implantable biomedical devices, the majority of research has primarily focused on three technologies: prosthetics, biomechanical fixation, and deployable stents. Similarly, the development of biologically inspired materials and systems has focused on four technologies: functionally graded materials, self-healing polymer/polymer

32.1 Overview

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Part D 32.1

mechanics techniques, such as full-field deformation measurement at multiple length scales. In addition to biocompatibility of physical properties, it is also necessary for implantable biomedical devices to have structural compatibility for optimal performance, which requires geometric complexity. Therefore, new testing simulators have also been developed to accommodate these geometric complex structures in order to determine the structure/performance relationship for these devices. For example, For example, the rigidity of deployable stents will determine the forces that they apply to the plaque as they are being deployed, as well as the arterial walls with which they will have contact once they are fully deployed and the plaque removed from the arterial walls. Therefore, point or circular loads are applied to the stents to ascertain the diametrical response in rubber tubes that model the mechanical response of the arterial walls. Similar to biomaterials for implantable biomedical devices, biologically inspired materials and systems also require an understanding of microstructure/property relationships at multiple length scales in order to duplicate the function of natural counterparts. For example, the fracture toughness of the rigid biological ceramic composite microstructure found in mollusk shells (nacre) is superior to synthetic monolithic ceramic microstructures. Despite the inherently brittle nature of the ceramic phase (aragonite) that comprises 95% of the nacre, the microstructure formed when the aragonite is assembled in a mortar-andbrick composite microstructure using a natural polymer binder comprising the remaining 5% of the nacre enables the material to provide excellent resistance to crack initiation and propagation. The experimental mechanics community has been able to quantify the localized properties of the nacre microstructure and the transfer of load between the aragonite phases in order to understand the impact of the microstructure on crack growth mechanisms that has enabled the development of biologically inspired materials with similar improvements in fracture toughness. Natural materials and systems have also evolved to balance multiple functional needs. For example, the microstructures of bone and bamboo are not the same throughout the structure. Instead, it gradually varies depending on location in order to balance the distribution of load to resist failure, minimize the local mass to optimize the volume and reduce energy consumption, and to optimize transport and storage processes necessary to support growth and repair of the structure. Similar multifunctional benefits are being realized

by grading the microstructure in synthetic materials to create biologically inspired functionally graded materials. Therefore, the relationship between the gradient in microstructure/properties and various facets of the multifunctional performance has required the development of new testing and analysis techniques for experimental mechanics. Experimental Mechanics Issues Based on the Microstructure, Property, Structure, Performance Characteristics It should now be evident that implantable biomedical devices, as well as biologically inspired materials and systems, have posed a variety of challenges to the experimental mechanics community. Many of these challenges arise from understanding the mechanical behavior of these devices, materials, and systems in the context of their compatibility with and performance in biological environments (in vivo) rather than just external to these environments (in vitro). It is also clear that they are far more complex than the solid materials and structures that the mechanics community has conventionally focused on, since their physical properties are derived from structural interaction at multiple length scales through a wide variation in local physical properties in order to satisfy multiple functional requirements. Thus, the challenges that must be addressed can be generalized:

• • • •

wide range of physical properties, wide range of length scales (hierarchical), multifunctionality, environmental constraints (in vivo).

Each of these will now be discussed in more detail. Wide Range of Physical Properties. The development

of implantable biomedical devices involves working with materials as soft as an elastomer to as rigid as a ceramic. The design of implantable biomedical devices, such as implants using hard metals, ceramics, and composites for support and soft adhesives for fixation, often require understanding the impact of the differences of these material properties on the performance of the device, such as failure. To elucidate on the characteristics of these wide property variations, experimental mechanics approaches based on full-field deformation measurement techniques and micro/nanoscale indentation techniques are being developed. In nature, physical properties can also vary dramatically within the microstructure of a given material. For example, the seashell can have very hard arago-

Implantable Biomedical Devices and Biologically Inspired Materials

nite phases surrounded by a very compliant organic material [32.6]. Using standard experimental mechanics techniques, such as indentation, to characterize the wide range of properties, such as hardness, in these materials typically requires a transducer with high load resolution over a large load range. Additionally, understanding the deformation response of these types of materials can require techniques that have very high strain and displacement resolution over very large strain and displacement ranges, while having the additional capability of resolving discontinuities that result across the phase boundaries. These challenges are identical to those that are faced when engineering biologically inspired ceramics. Wide Range of Length Scales (Hierarchical). The

Nanoscale

Microscale

Collagen molecule

a nanovoid between the biopolymer ligaments whose growth is limited by a critical nanoscale crack tip opening displacement (CTOD) (Fig. 32.3) [32.6, 9]. Locally, on the nanoscale, the amount of resistance needs to be characterized by the surface energy and strength of the compliant organic material that are holding the aragonite phases together, as well as adhesive strength that will prevent a less energy intensive debonding process to occur. At the same time, the morphology of the aragonite phases forms a lamellar brick and mortar microstructure that distributes stress on the microscale that needs to be characterized to determine the relationship between the energy that can be released during formation of nanovoids and the direction of crack propagation in order to reconcile this behavior with the microscale distribution of resistance that is determined by the platelet morphology and binder thickness of the biopolymer. This behavior must then be reconciled with variations in the size of the phases on a mesoscale that can create further resistance to macroscale crack propagation through a gradient in resistance and a redistribution of stresses near the crack tip to guide energy away from the most favorable directions of crack propagation through the thickness towards the surface where a number of fine nanovoids are formed to dissipate energy. These spatial measurement challenges are identical to those faced when engineering biologically inspired FGMs. Multifunctionality. The hierarchical nature of natural

structures lead to an assembly of parts that make subsystems and collections of subsystems performing a variety of functions (i. e., multifunctional) [32.10]. Examples Mesoscale

Collagen fibres Haversian canal

286 nm

64 nm

Collagen fibril

Haversian Cement line osteon

Osteocyte lacuna

Lamella

260 nm

Canaliculi

Hydroxyapatite microcrystal 1 nm

100 nm

1 μm Size scale

10 μm

200 μm

Fig. 32.2 Hierarchical structure in bone [32.7]

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unique mechanical behavior of biological materials and systems and their potential biologically inspired counterparts can be attributed to microstructures that bridge many length scales (hierarchical) [32.8]. In calcified tissues, like bone, these length scales start with the collagen molecule at the nanoscale and organize into the haversian system at the microscale (Fig. 32.2) [32.7]. In bone tissues such as teeth, there are additional microscale structures defined by a dentin-enamel junction that serves as a natural barrier to cyclic fatigue crack growth. Using the seashell as a more detailed example of the relationship of these length scales to mechanical behavior, there are biopolymer phases at the nanoscale whose mechanical behavior is critical to providing a crack-bridging mechanism that substantially increase resistance to crack propagation through formation of

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Fig. 32.3 Wide range of properties and length scales associated with fracture processes in seashells

Nanoscale Aragonite Nanoscale CTOD

Formation of nanovoid between biopolymer ligaments

Aragonite Biopolymer

Fracture

Aragonite

Microscale Aragonite ”brick”

Crab shell 103 μm Wide range of properties

Propagating microcrack

Nanovoids

Biopolymer “mortar”

Mesoscale Gradient in size of phases

Part D 32.1

Wide range of length scales

of multifunctionality in natural structures can be seen in Table 32.1. The salient characteristics of these structures often dictate structural performance. For example, some natural objects are large and other objects are small, but the size of the structure of a particular animal does not necessarily have an influence on its speed. The smallest animal is not the fastest or slowest, and neither is the largest animal. The fastest mammal is the cheetah, whose size is somewhere in the middle. Having the right combination of skeletal structure for mechanical support and muscle structure for power are the largest contributors to the speed of an

animal. This is why the average ant, even though it has the same muscle strength as a human is able to carry anywhere from five to over twenty times its body weight because the muscle force scales with crosssectional area while the structural mass scales with the volume (i. e., relative lifting force = muscle strength × cross-sectional area/(density × volume × gravitational force)). The multifunctional structural principles found in nature can lead to new designs that are not just much more efficient, lightweight, or responsive than traditional designs. They can also lead to multifunctional

Table 32.1 Multifunctionality in natural structures

Biological material

Multifunctionality

Bone

Lightweight structural support for body Storage of inorganic salts Blood cell and bone formation Attachment for muscles Environmental protection Water barrier Light transmission Structural support Mechanical Anchoring Nutrient transport

Chitin-based exoskeleton in arthropods

Sea spicules Tree trunks and roots

Implantable Biomedical Devices and Biologically Inspired Materials

= 1240 × scaling factor × density1.8 ) that has also been correlated with hardness, while aiding in storage of inorganic salts and marrow for the formation of blood cells (Fig. 32.5) [32.12, 13]. The result is a lightweight microstructure with a balance of stiffness and strength for structural support of a body, and whose mechanical behavior creates low mechanical signals of 2000–3000 microstrain that can increase the rate of bone formation by 2.1-fold and the mineralizing surface by 2.4-fold in cancellous (porous) bone to strengthen it [32.14]. By using experimental mechanics to understand the mechanical principles of natural structures, new biologically inspired multifunctional materials and systems can be designed for optimum performance in a variety of applications. Environmental Constraints (In Vivo). The environ-

ments (e.g., the human body) under which implantable biomedical devices, as well as some biologically inspired materials and systems, operate present a unique challenge to the experimental mechanics community in developing testing techniques that can be conducted directly in these environments or in simulated environments for these materials and systems. For example, an implantable biomedical device for a soft collageneous tissue application, like tissue scaffolding, can be extremely permeable to water [32.15]. This degree of permeability will have an affect on the transport of oxy-

Mesoscale Microscale Nanoscale Secondary cell wall

Helicoidal transitions (cellulose microfibrils)

Primary cell wall

Glycoproteins Bundles of cellulose in amorphous matrix

Hemicellulose Microfibril

Pectins Macrofibrils in amorphous matrix

Tree

m

mm

Amorphous domain Crystalline domain

Parallel polymer chains

Macrofibril

Plant cell walls

μm

Microfibril

nm Scale

897

β-1,4-linked D-glucose

Å

Fig. 32.4 Hierarchical structure of

woode that leads to multifunctional behavior of tree trunk and root system [32.11]

Part D 32.1

structures that are capable of harvesting energy from the environment, supporting mechanical and thermal loads from the environment, transporting fluids, etc. For example, the hierarchical-structure of wood enables the trunk and roots of a tree to transport nutrients in solution and gather solar energy to support growth and repair processes while concurrently serving as the mechanical support (Fig. 32.4) [32.11]. For the trunk, this support allows for the distribution of mass through complex geometries during the growth of the tree that increases the surface area of the tree both vertically and horizontally needed for gathering solar energy used in the chemical process known as photosynthesis to drive the production of new organic matter required for the growth. For the roots, complex geometries increase surface area to allow for mechanical anchoring of the tree in soils while increasing the absorption of water and nutrients from the soil as well. Thus, new power generation materials and systems can be designed with hierarchical-structures for optimal power generation and external anchoring to structures. Another classic example of a multifunctional natural structure is bone. In a relative sense, bone can be dense (cortical, aka compact) and harder, primarily made up of collagen and calcium phosphate. However, bone can also contain pores (cancellous, aka spongy or trabecular) that reduce stiffness in a power law relationship with decreasing mass (e.g., Young’s modulus

32.1 Overview

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Cancellous bone (porous and ductile)

Cortical bone (dense and hard)

Femur

5 mm

Load (mN) 4 Cortical bone 3

Part D 32.1

2

Approaches for Addressing Issues To address the experimental challenges posed by implantable biomedical devices and biologically inspired materials and systems, there are several general approaches being employed.

Trabecular bone

1 0 –1 –100

0

100

200

300

Therefore, it is preferred for the experimentalist to conduct experiments on these materials in in vivo or simulated in vivo environments, rather than in vitro. The simulated in vivo environments are the easiest since they do not require surgical procedures and are more accessible to common experimental techniques employed by the experimental mechanics community; however they still present challenges when working with certain techniques, like optical methods whose noise level is very sensitive to the changes in optical characteristics presented by the aqueous and thermal conditions of these environments. For in vivo environments, it would be possible to utilize these techniques if accessibility issues can be overcome. When it is not possible to conduct experiments in either of these environments, it will be necessary to assess how the absence of an in vivo environment affects mechanical response in order to develop appropriate in vitro test methods.

400 500 600 Displacement (nm)

Fig. 32.5 Cross-section of a bone revealing porous and

ductile microstructure (cancellous) used for storage of inorganic salts and marrow for formation of red blood cells and dense and hard microstructure (cortical). Also seen are typical nanoindentation measurements indicating a 4-fold difference in stiffness [32.13]

gen and nutrients to the soft tissue interacting with the device. Removing water begins a degradation process in the interaction that can rapidly affect the functionality of the device. However, the biological processes are electrically-driven as well, so removing the device from its living organism will also begin a degradation process in the interaction, albeit the rate of decay is much slower than when water is removed. The degradation process can also be affected by different environmental factors associated with the applications of these devices, materials, and systems. For example, increasing the Ph and temperature associated with implanting implantable biomedical devices, such as polymer composites, in the human body can increase the rate of degradation for the implant materials [32.16].

1. Characterization of mechanical properties and performance using new/modified testing techniques and simulators (e.g., modified screw pullout tests, nano/microindentation). 2. Modeling of the devices, materials and systems to analyze experimental results using new mechanics principles (e.g., cohesive zone models for interfacial crack growth). 3. Development of model structures and experiments to ascertain mechanical performance of devices and structures (e.g., simulators for prosthetics, model arterial structures for deployable stents, model metal-ceramic composites for functionally graded materials). 4. Characterization of the processing/ microstructure/property/structure/ performance relationships (e.g., determining temperatures to process selfreinforced polymer composites for biomechanical fixation). These approaches represent application of the full range of testing techniques and mechanics principles employed by the experimental mechanics community. It is the intention of this chapter to provide details of the application of these approaches.

Implantable Biomedical Devices and Biologically Inspired Materials

32.2 Implantable Biomedical Devices

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32.2 Implantable Biomedical Devices prevent further plaque buildup using drug-eluting coatings to prevent re-stenosis. The joining of hard materials associated with prosthetics with soft biological tissues, such as bone, has also been enhanced through the development of new biocompatible adhesives, such as injectable biodegradable polymers for tissue scaffolding [32.17]. Most biocompatible adhesives have the traditional mechanics issues associated with their development involving interfacial properties and stresses associated with dissimilar materials. For new tissue scaffolding materials, the adhesion concept involves using porous microstructures that have features that are compatible with the growth characteristics of new cells to create adhesion. By providing a flexible pathway for cell growth, adhesion is achieved through rapid in vivo formation of a more compatible interface with interfacial properties that are a combination of the hard and soft materials leading to improved performance. For example, tissue scaffolding has been employed extensively in oral and maxillofacial surgery to aid in regeneration of tissue around dental implants.

32.2.1 Applications 32.2.2 Brief Description of Examples The primary application of implantable biomedical devices is for insertion into human beings or animals. For experimental mechanics, there are three primary classes of implantable biomedical devices of particular interest: 1. prosthetics, 2. biomechanical fixation, and 3. deployable stents. While the range of applications is limited, the ways they are utilized inside of the human being are varied. For example, prosthetics such as artificial bone and teeth can be used in place of decaying natural counterparts. The placement of the prosthetics can also require new biomechanical fixation technologies such as biocompatible adhesives. While it is sometimes necessary to replace decaying natural components of the human body, it is also possible to enhance their repair. For example, the removal of plaque buildups in arteries through surgical procedures such as angioplasty requires devices known as deployable stents that can be inserted into the narrowed artery and then expanded with pressures that are sufficient to remove the plaque buildup without damaging the arterial tissue. Furthermore, the stent can be left behind to

Prosthetics One of the most important applications of mechanics in bioengineering has been in understanding the mechanical response of materials and structures that are being developed for prostethetics. Prosthetics are artificial structures and devices that are used to replace natural structures and systems in living organisms. They can be used to replace damaged or missing body parts, or to enhance the performance of the living organism. They are often deployed as bone implants, with classic examples being ball and socket components for hip replacements and dental implants for tooth restoration. From a mechanics perspective, the development of prosthetics requires understanding the stresses that evolve within the natural body part due to the loading that that part will experience during use. This knowledge can be used to design and/or select biocompatible materials that will be sufficiently fracture-, fatigue-, and wear-resistant at the corresponding in vivo stress states and environment. One mechanics issue that has been extremely complex has been to develop biocompatible joining technologies for prosthetics that meet the aforementioned mechanical requirements.

Part D 32.2

With the advancement of medical knowledge, the development of implantable biomedical devices has also advanced. In some cases, advancements in the knowledge of the mechanics of materials and structures have led to the development of new medical procedures. For example, the ability to create metals with deployable geometries transformed cardiac surgery by enabling the developed of angioplasty for removing plaque buildup in arteries by insertion of a thin metallic wire that expands to break up the plaque and debond it from the arterial wall. Advances in materials and manufacturing technologies have also had a tremendous impact on replacing failed biological structures, such as prosthetics that have evolved from crude metal dentures that are removable, to more modern ceramic implants that are biocompatible and can be permanently fixated. Many applications of these implantable biomedical devices have required development and application of experimental mechanics techniques to facilitate their development, the details of which will now be discussed.

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Biomechanical Fixation A critical issue for implantable biomedical devices such as prosthetics is the ability to fixate the device to a natural structure through adhesion that is compatible with the natural structure (biomechanical fixation) in order for it to properly function. The mechanics issues involved with biomechanical fixation fall into the more traditional category of interfacial mechanics. For biocompatible fixation, the unique issue is balancing the biocompatibility characteristics of the fixation device with the mechanical characteristics necessary for optimizing the interfacial adhesion between natural and synthetic materials. A new biomechanical fixation device whose development is being guided through a more intimate understanding of mechanics is tissue scaffolding. To develop tissue scaffolding, there is a need to understand how the scaffolding can be designed to rapidly promote the growth of mechanically-sound tissue. The mechanical performance of this tissue is studied in an area of bioengineering known as tissue mechanics. The characterization of the evolution of the mechanical behavior of the tissue scaffold and new tissue has presented unique challenges for the experimental mechanics community to resolve. The porous nature of the scaffolding makes it difficult to measure the loads that are distributed within it to optimize its design, while it is also necessary to characterize in vivo the evolution of mechanical properties for the combined scaffold/new tissue system. Also, the adhesion of cells to the scaffold material, and its impact on the growth of new tissue, is also an area of active research for the development of implantable biomedical devices in general. Deployable Stents A very popular implantable biomedical device where mechanics has played an important role is the development of arterial stents. Arterial stents are small, deployable metal structures with complex geometries that enable them to expand after they are inserted into an artery clogged by plaque build-up in order to break up the plaque and restore the proper flow of blood. The stresses experienced by the stent when it is expanded must be characterized in order guide the design the geometry of the stent to mitigate stress concentrations, and to select appropriate biocompatible materials that can withstand the corresponding stress state. Originally, the pressures that induced the expansion of the metal stents were generated by a balloon in the center of the collapsed stent that could be pressurized after insertion. However, more advanced materials

have been recently investigated for deployable stents that are self-deployable. The mechanics of these materials, known as shape memory alloys (SMAs), are vastly different than the original metals. Upon heating, the SMAs can experience extremely large deformations associated with martensitic phase transformations. Because of these large thermally-induced deformations, SMAs are considered active materials, while the original metals are considered passive. Therefore, they just need to be designed to deploy and achieve the desired pressures within a very narrow range of temperatures that will not harm the arterial tissue in the human body.

32.2.3 Prosthetics Classification of Prosthetics Prosthetics generally fall into two categories:

1. Orthopaedic (bone and joints), 2. Dental (teeth). For orthopaedic applications, the structures are artificial load-bearing members designed to replace failed natural counterparts. A critical mechanics issue is the joining of these structures to the biological system, which often entails additional mechanics issues involving the joints themselves. In some cases, natural joints may fail and be replaced with artificial ones that may require modifying the natural structures that are being connected to the joint (i. e., biomechanical fixation), where pull-out forces associated with interfacial strength need to be characterized and will be discussed in more detail in terms of biomechanical fixation in Sect. 32.2.4. Finding materials and joining concepts that are amenable to the cyclic transfer of loads across the joints is usually the most important experimental mechanics consideration in developing orthopaedic prosthetics. Dental prosthetics differ from general orthopaedic prosthetics in that they employ a dental implant to serve as a root for a permanent tooth restoration known as the crown. Typically, these implants are designed to integrate with existing natural structure (e.g., jawbone), without need for a periodontal ligament. The interfacial mechanics issues associated with this design are typically the most important to understand, and will be discussed in more detail in terms of biomechanical fixation in Sect. 32.2.4. Additionally, the mechanical performance of the tooth restoration is much different than orthopaedic prostheses in that the surface is subjected to compressive point loads rather than distributed pressure loads. As such, the development of hard ma-

Implantable Biomedical Devices and Biologically Inspired Materials

32.2 Implantable Biomedical Devices

901

terials that can bear these loads without failing while being amenable to manufacturing technologies required to create complex shapes from these materials is the most important consideration. Types of Prosthetic Materials Prosthetic materials are generally made from:

1. 2. 3. 4.

metals, ceramics, polymers, and composites.

Experimental Mechanics Characterization Techniques There are three techniques generally used for characterizing prosthetics and their associated materials

1. Standard tension/compression testing. 2. Load frames with multiple degrees of freedom (DOF). 3. Micro and nanoindentation. Standard tension/compression testing can be conducted to elucidate on the properties of biomaterials, but they need to be modified to assess performance.

Fig. 32.6 Four DOF knee simulator from MTS (MTS Systems, Minneapolis) [32.18]

Load frames with multiple DOF are often employed to simulate the cyclic loading conditions that a prosthetic will be subjected to when it is deployed to determine its performance (Fig. 32.6) [32.18]. There have also been unique test fixtures designed to simulate natural boundary conditions, such as a wear simulator for shoulder prostheses, a new glenoid component test fixture for force controlled fatigue testing through the movement of the femoral head of shoulder prostheses, and a transverse plane shear test fixture for knee prosthetics [32.22–24]. To further elucidate on the details of the processing/structure/property relationship in biomaterials at the micro- and nanoscales, indentation and scratch tests can be employed.

Table 32.2 Classification of prosthetic materials

Class of prosthetic material

Type

Metal Ceramic

Ti6Al4V, CrNiMo stainless steel, NiTi, CoCrMo, gold Hydroxyapatite, alumina, zirconia, tricalcium phosphate, glass-ceramic, pyrolytic carbon Silicone, PMMA, polyurethane, polyethylene, acrylic, hydrogel, nylon, polypropylene, polylactic acid, polycaprolactone, polyglycolides, polydioxanone, trimethylene carbonate, polyorthoester Carbon fiber/epoxy, hydroxyapatite/PMMA, bioactive glass/PMMA, self-reinforced PMMA, self-reinforced PLA

Polymer

Composite

Part D 32.2

The primary materials used in prosthetic applications are as follows (Table 32.2) [32.19, 20]: Because of the time involved in approving new materials for biomedical applications, many of the materials in Table 32.2 have been used for decades and serve as the basis for much of the current research into the mechanics of not only prosthetics, but of implantable biomedical devices in general. The specific selection of an appropriate prosthetic material for a particular implant application will require a balance of mechanical properties, inert in vivo response, and availability [32.21].

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Strain gauges

Fig. 32.7 Strain gages used to determine strain distribution

cipal directions are along the axis of the bone, as in the case of stress shielding for a fixator plate on a femur [32.26]. For softer materials, video extensometry techniques like DIC can be employed [32.27]. Maximum pressure distributions associated with load transfer within articulating prosthetic joints are obtained using pressure-sensitive films with microbubbles that are designed to rupture and release liquids that former darker stains as the maximum pressure increases locally [32.28]. Photoelasticity has also been popular for characterizing the interfacial stresses generated at the interface of the prosthetic and the natural structure [32.29–31].

in femur after implantation of a hip prostheses [32.18]

Part D 32.2

Diagnostic techniques for mechanical testing of prosthetics also cover a wide range of experimental mechanics methods. For harder prosthetic materials, standard strain gage technology will be employed to understand the strain distribution in the prosthetic and/or the natural structure it is joined with, and can be used in vivo (Fig. 32.7) [32.25]. Uniaxial strain gages are also commonly used to characterize strains on the hard biological materials, such as bone, to which prosthetics are fixated, although triaxial rosettes are sometimes needed for more complete information unless the prina)

Microstructure/Property/Structure/ Performance Relationships Modal Response. The development of prosthetics using advanced materials, such as synthetic composites, requires modeling tools to relate that need experimental validation for microstructure/property/structure/performance relationships. It has been proposed that the modal response of the entire prosthetic structure can provide superior verification for these models over strain gage measurements since modal parameters (frequency, damping ratio, and mode shape) are intrinsic properties of the whole structure (Fig. 32.8) [32.32]. Modal analysis can also be used to characterize performance by detecting early loosening of prosthetics due to interfacial failure as the modal response transitions from linear to non-linear. Interfacial Damage. Since prostheses can be modu-

z y

x

b)

Magnitude (dB) 60 40 20 0 –20 –40 –60 –80 –100 –120 –140 0 200

400

600

800

1000

1200

1400

1600

Frequency (Hz)

Fig. 32.8 (a) Modal testing system for composite femur, and (b) comparison of experimental and numerical modal response in-

dicating good correlation [32.32]

lar (e.g., head-stem in hip implant), internal interface integrity plays an important role in its performance, the experimental characterization of the structure and properties of the interface are important for understanding failure [32.33]. In particular, characteristics of the modular interface, such as surface roughness, can change due to fretting and corrosion from wear at the interface causing inflammation [32.34]. As a result, the cyclic contact loading of modular orthopaedic prostheses can decrease the size of surface asperities after a few cycles, but corrosion can increase the size depending on the magnitude of the contact load. Using finite element analysis, significant residual stresses have been predicted at the trough of the asperities due to elastic interaction of plastically deforming peaks. Thus, a damage mechanism was proposed using the experimental and numerical results that explained the evolution of the measured surface roughness by incorporating the residual-stress devel-

Implantable Biomedical Devices and Biologically Inspired Materials

Fretting load-induced asperity deformation

903

Wrong method

P T

Surface roughness evolution

σres

32.2 Implantable Biomedical Devices

Residual stress distribution

σres

σres

Right method

Stress-assisted dissolution

Fig. 32.9 Experimental and numerically-derived damage mechanism at the interface of orthopaedic implants resulting from residual stresses due to contact loading and stress-assisted dissolution [32.33]

Fig. 32.10 Compression across fracture produced by rigid plate fixator [32.35]

Surface Damage. The performance of dental implants

also involves concentrated contact loading at the surface of the tooth restoration instead of distributed loading across the interface. This loading condition has been found to be accurately emulated using Hertzian indentation testing [32.36]. Using this testing technique, it has been determined that there are two failure modes of importance: 1. a fracture mode consisting of Hertzian cone cracks that develop outside of the contact zone due to brittle fracture in response to tensile stresses, and 2. a quasi-plastic deformation mode consisting of microdamage beneath the contact zone in response to shear stresses. There is a progressive transition from the first to the second mode depending on the microstructural heterogeneity, and is manifested through non-linearity in the indentation stress-strain curve. Threshold loads for the initiation of fracture and deformation modes have been related to indenter radius in order to assess prospective tooth restoration materials with different microstructure/property relationships for various masticatory conditions.

namic compression plate, the mechanics of the device are designed to produce compressive stress on a fracture to promote healing through membranous bone repair (Fig. 32.10). For these prosthetics to work properly, the fundamental principle is to keep the two pieces of fractured bone from moving relative to each other through rigid fixation. However, there are degrees of this motion that need to be constrained less than others, in particular the axial direction where the rigidity can cause stress shielding during the healing process that reduces the density of the remodeled bone. For example, a thin Ti6Al4V plate that has low stiffness in the axial and bending directions is not adequate, but a thin tubular plate with low stiffness in the axial direction but moderate stiffness in the bending and torsional directions produce superior mechanical and structural properties that do not degrade even after the application is prolonged for 9 months [32.35]. Composite materials would seem to possess the ideal microstructure to tailor the directional properties for this application. While promising, conventional carbon-fiber/polymer composites have been found to be brittle and degrade rapidly when exposed to body fluids [32.16, 37]. Alternatively, the use of bioresorbable screws with rigid plates have been found to potentially take advantage of the degradation behavior in reducing axial stiffness as the bone heals, while elastic inserts inside of the screw slots have also been found to structurally produce an axially flexible plate through the geometry of the slot and the stiffness of the elastic insert [32.38, 39].

Part D 32.2

Axial Flexibility. For temporary prosthetics, like a dy-

opment governed by the yield stress of the implant material normalized by the contact stress, along with the stress-assisted dissolution of the implant material (Fig. 32.9).

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32.2.4 Biomechanical Fixation Types of Biomechanical Fixation A critical issue for successful deployment of prosthetics has been the development and characterization of fixation concepts between the prosthetic and the natural structure with which it functions, typically bone. These concepts generally fall into two categories:

1. biocompatible adhesives, 2. mechanical fixation (screws, roots).

Part D 32.2

The biocompatible adhesives can take many forms, and are most advantageous since they do not require any modification of natural structures. For example, the most common biocompatible adhesives are epoxies, such as DePuy CMW or Zimmer bone cements, which not only possess excellent biocompatibility, but can also develop strong adhesion with metal, ceramic, and biological materials [32.40]. However, they are inherently weak and can be exothermic when they set causing bone necrosis. Alternatively, soft tissue scaffolding materials have been developed to promote cell growth at the interface between prosthetics and natural structures, and are bioresorbable so they disappear once the desired growth has been achieved. While adhesive materials can be ideal for most joining applications, it is not always possible to achieve the desired performance. Therefore, more conventional mechanical joining techniques are often employed. These include the use of screws as fixators, as well as rivets, staples, and other fastening devices. Experimental Mechanics Characterization Techniques The most common technique for characterizing biomechanical fixation performance is pull-out testing. In pull-out testing, the need is to identify the critical force required to exceed the interfacial strength and pull out a screw, nail or root that is being used to anchor a prostheses. The resulting pull-out response is a combination of the interfacial properties, in particular the shear strength, and the geometry of the interface. Alternatively, lap-shear testing can be used to isolate the shear strength component in the pull-out response. Novel microscale testing techniques, such as microfabricated arrays of needles, are also being developed to better understand how the adhesion mechanics evolve between new biomechanical fixation devices, such as tissue scaffolding, and the biological materials, such as individual cells and tissues. However, the more conventional techniques of pull-out and lap-shear tests are still

needed to assess the interfacial properties and performance. Processing/Microstructure/Property/Structure/ Performance Relationships Cell–Substrate Interaction. Because biomechanical fixation is unique in that it involves bonding synthetic biomaterials with biological materials, such as individual cells and tissue, it is essential to develop an understanding of the fundamental physics governing the microstructure/property/structure/performance relationship involved in biomechanical fixation. In order to develop this relationship, novel experimental mechanics techniques have been developed, such as a microfabricated array of elastomeric microneedles that isolate mechanical force on deforming cells to study the mechanics of cell-substrate interaction and the effects of the topology, flexibility, and surface chemistry of the substrate on this interaction to establish the role that cells play in the interfacial strength (Fig. 32.11) [32.41]. By measuring the deflection of the posts δ the contractile forces F in cells can be quantified and related to the cell morphology and focal adhesions using the following standard bending equation   3E I (32.1) δ, F= L3

where E, I , and L are the Young’s modulus, moment of inertia and length of the posts. Characterization of the scaffold itself usually involves compression testing, and analyzing the results for initial stiffness, collapse stress, and densification using a standard cellular solid model with an open-cell tetrakaidecahedron microstructure to describe the scaffold mechanics [32.42]. The collapse stress σcl depends on the initial relative density of the scaffold ρ∗ /ρs and the yield stress of the scaffold material σy as follows:  ∗ 3 ρ 2 σcl = 0.3 . σy ρs

(32.2)

The densification of the scaffold microstructure after the collapse stress is exceeded provides insight into the permeability changes that will occur according to the formula [32.43]  3 ρ∗ 2 , (32.3) k = A d 2 1 − ρs where k is the permeability, A is a dimensionless system constant, and d is the scaffold mean pore size, and the relative density changes are obtained from the

Implantable Biomedical Devices and Biologically Inspired Materials

a)

b)

Before

Before

e)

f)

32.2 Implantable Biomedical Devices

c)

d)

After BDM

g)

905

After cyto D

h)

Force (nN) 80

60

40

0

50 nN

50 nN

0

4

8

12

FA area/post (μm2)

Fig. 32.11a–h Measurement of contractile forces in cells using a microfabricated array of elastomeric microneedles. The plot in the lower left (e) is the plot of the force generated on each post as a function of the total area of focal adhesion staining per post [32.41]

compression test. Changes in permeability are especially important since they affect scaffold performance by controlling the transport of nutrients and waste products within the structure that affect processes like cell adhesion and degradation [32.44]. Alternatively, the effects of scaffold properties and microstructural characteristics on cell adhesion can be qualitatively assessed by counting the number of cells that adhere to a scaffold, where more cells imply stronger interfacial adhesion [32.45]. From these studies, it was concluded that cell attachment and viability are primarily influenced by the specific surface area over a limited range of pore sizes (95.9–150.5 μm). As a general rule, it has been determined that adhesion rarely occurs when pore sizes are less than 20 μm or greater than 120 μm [32.46]. Pull-Out Strength and Failure Mechanisms. More

traditional experimental mechanics testing techniques, such as pull-out tests, have been employed to characterize the interfacial strength and failure mechanisms

for conventional prosthetic biomechanical fixation, such as bone fasteners. Using this test, it has been possible to characterize the ultimate pull-out forces for commercial orthopaedic self-tapping screws and machine screws [32.47]. Three failure mechanisms were identified depending on the level of pull-out force. 1. Bone-thread shear (low); 2. Bone splitting (intermediate); 3. Bone fragmentation (high). As expected, the ultimate pull-out force was maximum at the mid-length of a bone and minimum at the distal end. With machine screws, the ultimate shear stress ranged from 3.5 to 5.8 ksi depending on the size of the screw and interference fit, which produced a maximum pull-out force at 50% difference between the diameter of the screw and the diameter of the drilled hole. Self-tapping screws produced about 50% greater shear strengths than machine screws. More advanced techniques are also being developed for characterizing the mechanics associated with

Part D 32.2

20

906

Part D

Applications

biocompatible interfaces. Fiber bragg gratings (FBGs) combined with strain gages have been employed to characterize the strains associated with a fixator platebone interface [32.48]. The plates are amenable to strain gages, but the optical FBGs were determined to be superior to electrical resistance strain gages because it is less intrusive for biological materials and smaller. From tests on a synthetic femur specimen, It was determined that strain shielding is more pronounced in the distal region of the plate, which is 130% higher than for the a)

Load

Load Peak load Peak load

deg

deg

Displacement

Part D 32.2

Load

Displacement Load

Time

Time

b) Stiffness (N/mm) 2000

**

*

* Stress Relaxation Standard

1000

0

1

5

25

Rate (mm/min) Peak load (N) 4000

* *

3000 Stress Relaxation Standard

2000 1000 0

1

5

25

Rate (mm/min)

Fig. 32.12 (a) Comparison of new testing method for screw pullout with the standard testing method, and (b) comparison of stiffness and peak load measurements obtained from the two methods from lumbar spines [32.49]

intact femur and is 640% higher at the medial-distal side of the plated femur. Another complication in assessing the pull-out strength of screws is the complex mechanical behavior of bone. For example, the viscoelastic properties of bone can result in stress relaxation that can affect the pullout strength of orthopaedic screws (Fig. 32.12) [32.49]. The viscoelastic behavior is more likely to be present in vivo, and will significantly reduce the pullout strength and stiffness compared to in vitro tests where the behavior is more elastic. Thus, in vitro tests will overpredict mechanical performance due to the viscoelastic properties of bone in which the screws are implanted. To quantify these differences, a new screw pullout test was developed were the load was periodically held in order to quantify the relaxation effects and obtain the stiffness from the peaks of steps. Stress at Geometrically Complex Interface. In contrast

to the biomechanical fixation obtained with screws, the fixation of dental prosthetics requires understanding the stresses and displacements associated with a wedgelike root-periodontium interface that is geometrically complex [32.30]. In this investigation, a photoelastic specimen was prepared, and loading applied according to a model of the loading condition on a tooth associated with orthodontic appliances (Fig. 32.13). Although a 5◦ angle is more typical of the interface, a 15◦ angle was employed for a valid photoelastic test. The variation of interfacial stress was correlated to theoretical solutions obtained from elementary elasticity theory in Timoshenko and Goodier. Although these results are for a simplified 2-D model assuming elastic behavior, they can still be used to gain insight into the critical loads and displacements associated with the biomechanical fixation of a dental prosthetic. Recently, more detailed nonlinear two-dimensional interface element for finite element analyses of the bone-dental prosthetic interface have also been developed and validated using photoelasticity [32.31]. Processing/Microstructure/Property Relationship at Nanoscale. While the structure/performance rela-

tionship for biomechanical fixation can be characterized using novel microscale and conventional macroscale testing techniques, more advanced techniques, such as nanoindentation, have been used, to extend this characterization to the understanding the processing/microstructure/property relationship at the nanoscale. Using nanoindentation across the surface of a degradable polymer, polylactic acid (PLA),

Implantable Biomedical Devices and Biologically Inspired Materials

a)

32.2 Implantable Biomedical Devices

907

Residual scratch depth (nm) 950 900 850

2Fo

800

2Do

750 700 650 600

so

Parallel Orthogonal Neat PMMA

550

Mo No

No

Δ

Mo so

?

Fig. 32.13 (a) Model of loading conditions on tooth from orthodontic appliance, and (b) experimental photoelastic model used to measure the root-periodontium interfacial stresses and displacements and resulting fringe field [32.30]

which is being used for orthopaedic screws to provide temporary biomechanical fixation, it was possible to characterize that mechanical changes occurred sooner at the nanoscale than at the macroscale in a phosphate buffered saline solution after 12 weeks [32.50]. There were also statistical differences between the hard-

450 100

110

120

130

140

150

160

Processing temperature (°C)

Fig. 32.14 Variation of the plastic deformation with processing temperature in self-reinforced PMMA both parallel and transverse to the fiber direction indicating increasing resistance to plastic deformation at decreasing processing temperatures due to molecular orientation [32.51]

ness at the surface and interior of the specimen. It was concluded that nanoindentation can be used to discern changes in mechanical properties sooner than macroscale tests. The effects of processing conditions on the mechanical behavior of polymer composites for prosthetic applications have also been investigated using nanoindentation [32.51]. A self-reinforced PMMA composite was processed at a range of temperatures by laying fibers in a mold and heating for twelve minutes. At low processing temperatures, the fibers retain their orientation, and do not form a strong matrix bond. At higher processing temperatures, the outer surfaces of the fibers are able to bond better, at the expense of retained molecular orientation in the fibers. The nanoindenter was used to characterize these mechanical changes by scratching parallel and perpendicular to the fiber direction (Fig. 32.14). Scratching parallel produced similar amounts of plastic deformation as control specimens. However, scratching perpendicular to the fibers resulted in increased resistance to plastic deformation at low processing temperatures. It was concluded that the molecular orientation in the fibers can be indirectly ascertained from the deformation induced by scratching.

32.2.5 Deployable Stents Experimental Mechanics Characterization Techniques Deployable stents are geometrically complex tubular structures that are designed to undergo very large de-

Part D 32.2

o

b)

500

908

Part D

Applications

a) Relative change of dimension

a)

1.8 1.6 1.4 1.2

2.5"

Length – wire diam 0.18 mm Length – wire diam 0.2 mm Radius – wire diam 0.18 mm Radius – wire diam 0.2 mm

1 0.8 0.6

1"

0.4 0.2 0

b) Alternating strain range 0

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b) Relative change of dimension

3.5 4 4.5 Axial force (N)

1.4 1.2

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Part D 32.2

Length – rising 25 mm Length – rising 28 mm Radius – rising 25 mm Radius – rising 28 mm

1 0.8 0.6

0.6 0.4

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0

0

Mean strain = 2.4% Mean strain = 3.4% Mean strain = 4.8%

0

0.5

1

1.5

2

2.5 3 Axial force (N)

Fig. 32.15a,b Variation of rigidity for SMA stents with (a) wire diameter and (b) rise of the windings indicating

more sensitivity to wire diameter [32.53]

formations in order to expand and unblock clogged arteries, or reinforce weakened arteries. They are capable of being expanded up to 4 times their original diameter. Mechanical testing of these stents usually consist of applying point loads or circular loads to obtain qualitative comparisons of different stent designs through their diametrical response [32.52]. However, this mechanical characterization is not adequate for providing the quantitative pressure-diameter relationships necessary to analyze of the coupled artery-stent design problem because it lacks the external resistance provided by the clogged artery. To gain insight into these relationships, specialized test methods have been developed where the stents are inserted into rubber tubes that are used to simulate the arterial wall [32.55]. When external pressure applied, the resulting change in diameter can be measured and related to the pressure. A more direct measure can be obtained by placing a stent within a collar with tabs that are pulled apart to contract the collar and apply ra-

0

100 M

400 M 700 M Number of cycles to failure

Fig. 32.16 (a) Fatigue test specimen design for deployable stent, and (b) fatigue performance data [32.54]

dial pressure to the stint [32.56]. The resulting in load N with respect to stent diameter D used to determine the stiffness of the stent K lows:   dN 2 , K= bD0 dD

change can be as fol-

(32.4)

where b is the width of the collar. A test method has also been developed to obtain pressure-diameter relationships by simulating the in vivo mechanical response of stents using wrinkled polyethylene bags that are capable of expanding without stretching [32.57]. The bags could be placed internally or externally to generate pressure, such that the resistance from expansion comes only from the stent itself and not the polyethylene film. The strains associated with the change in diameter were measured and related to the pressure and axial strain. From this experimental method, it was determined that the stiffness of the stents changed dramatically as they expanded, and could cause excessive wall stress in an artery if it is overexpanded. Using an analogous model of a beam on an elastic foun-

Implantable Biomedical Devices and Biologically Inspired Materials

dation and a thin-wall tube under pressure, the coupled response of the stent and aorta could be modeled based on the experimental data [32.58]. By formally stating the coupled problem, the specific need has been established for an appropriate pressure-diameter relationship for the artery. Property/Structure/Performance Relationship Recently, interest in stent technology has shifted towards shape memory alloys (SMAs). Mechanical testing of these stents has focused on how the geometric characteristics of the stents affect their rigidity [32.53]. These tests have determined that the diameter of the wire plays a more important role in rigidity than the rise of the windings (Fig. 32.15). The rigidity also increases with increasing temperature due to the transformation from Martensite to a stiffer Austenite phase. However, the temperature at which this occurs increases as the

32.3 Biologically Inspired Materials and Systems

pressure on the stent increases, so that transformation may be suppressed if too much force is required to expand the stent in the artery. Not only will the static performance of deployable stents be important, but the fatigue performance will also dictate the life expectancy of the stent, where a pulse of 75 beats per minute over a 10 year period will produce roughly 400 million cycles of service [32.54]. Therefore, a fatigue testing protocol has been developed to determine safety factors for stents made from materials such as SMAs. Specimens are designed that reflected the areas where there would be high strain, so that the local testing conditions in the specimen and the stent were identical (Fig. 32.16). Tests conducted to assess the effects of mean strain on the fatigue life of the stent material indicated that the mid- to highcycle fatigue life of the material appears to increase with increasing mean strain in the range of 2–4%.

sizing materials, while mechanics can be used to interpret the manner in which the properties and functionality associated with the structure of natural materials have enabled them to adapt to environmental stimuli.

32.3.1 Applications The use of biologically inspired materials and systems is quite varied, ranging from new adhesive materials to specialized robotic applications. The primary mechanical benefit in these applications often involves more durable and reliable materials, structures, and systems. For example, nacre-like ceramics are being developed that can provide superior fracture resistance in applications such as ceramic armors and thermal barrier coatings (TBCs). In the case of ceramic armors, the constituents may consistent of synthetic ceramics and polymers instead of natural. For the TBCs, it may even require using a higher temperature metal in place of the polymer, in essence resulting in a bio-inspired ceramic matrix composite (CMC). Not only can the properties of materials be improved, but the joining of dissimilar materials can also be enhanced. For example, it is possible to create new adhesion concepts that mimic the behavior of gecko feet by micromachining pillars on a flexible surface to provide new mechanically-based adhesion mechanisms

Part D 32.3

32.3 Biologically Inspired Materials and Systems The science and engineering of synthetic materials has been traditionally separated into classes of structures, length scales, and functionality that are used to differentiate disciplines such as experimental mechanics and materials science from each other. However, biological materials do not conform to disciplinary boundaries, since they possess structures that span across a full range of length scales in order to react to a variety of environmental stimuli with optimal functionality. A number of technological breakthroughs may be achieved through mimicry of the multi-scale optimization of structure and functionality in natural materials, such as material systems with morphogenesis capabilities and biomorphic explorer robots with versatile mobility [32.59]. The translation of this multiscale optimization of structure and functionality to the science of advanced materials is a part of the new field where science and engineering are joined together: biomimetics. Biomimetics refers to humanmade processes, substances, devices, or systems that imitate nature, and has led to the development of new biologically inspired materials based on biological analogs. The research in this area can either be focused on the investigation of natural materials, or on the processes that optimize the structure of materials in a manner similar to that occurring in nature. Therefore, biology can become the basis for developing new processes required for synthe-

909

910

Part D

Applications

Part D 32.3

based on the pillar radius and aspect ratio for improved joining to chemically inert materials, such as ceramics [32.60]. Furthermore, the geometry of the interface can be altered to provide mesoscale and macroscale structural features that enhance joining, like the features associated with the growth of tree trunks into soil and around obstacles. Structurally, the mechanical behavior of a component can be enhanced through variations in the distribution of materials. For example, the microstructure of a metal-ceramic composite can be gradually varied from a strong outer surface, resistant to the high stresses associated with loading events such as cutting, to a more ductile inner surface, resistant to the cracks that may grow from the global deformations associated with loading events such as flexural. This is similar in principle to the way that bamboo distributions strong fiber reinforcement from high concentrations at the outer surface of a piece of culm to low concentrations at the inner surface. In addition to the improved mechanical behavior that can be obtained from biological materials, it is also possible to couple the mechanical response with others, such as thermal or chemical, in order to create new multifunctional structures that are more compact, reliable, efficient, and autonomous by embedding sensors, controls, actuators, or power storage elements in polymers and polymer composites for applications such as morphing structures or robotics. For example, active materials such as shape memory alloys (SMAs) can undergo very large deformations through thermally-activated martensitic phase transformations and generate very high thermomechanical contractile forces when embedded in the form of a wire inside a polymer or polymer composite, much like the chemomechanical contractile forces produced by muscle tissues embedded within a body. This can enable the structural support in a morphing or robotic structure to change its shape for reduced aerodynamic resistance or for locomotion to mimic the performance of natural counterparts. Materials can also be designed with entirely new transduction principles as well. For example, microencapsulated monomers can be distributed in polymers that are impregnated with a catalyst. When a crack begins to grow in the polymer, the microencapsulation is breached, releasing the monomer and initiating a polymerization reaction that self heals the surfaces of the cracks, reversing the damage process. This is similar in principle to the self-healing of natural structures through processes like bleeding.

32.3.2 Brief Description of Examples Functionally Graded Material (FGMs) In nature, many natural structures exhibit very complex microstructural distributions. For example, the connectivity and size of porosity will change from the outer surface to inner volume of a bone. Similarly, the concentration of fibers can change within the cross section of plants like bamboo. The experimental mechanics community has attempted to quantify and model the effects of these microstructures in order to gain inspiration for developing new materials, known as functionally graded materials (FGMs), for optimal structural performance. In particular, these materials provide an optimal distribution of phases at the microstructural level in order to minimize the amount of material and weight needed to fabricate the structure. Furthermore, the microstructural distributions can be tailored to minimize stress distributions around stress concentrations, which nature has done with holes in bones, known as foramen, and which bioengineers are now doing with the interfaces of prosthetics implanted in bone. Active Materials Conventional materials are characterized by mechanical behavior that is fairly inert or passive (i. e., they only move when a load is applied to them). Many biological materials and systems do not display this behavior, with even single cells capable of locomotion. This has led to a desire to engineer active materials, which are synthetic materials capable of responding to their environment. For example, one active material known as a Shape Memory Alloy, can undergo a martensitic phase transformation when heated that enables it to undergo substantial structural reconfiguration not associated with conventional thermal expansion. Active materials have also been synthesized from polymers (electroactive), and ceramics (piezoelectric). Combinations of materials, such as ferromagnetic particles and a liquid, can be used to create magnetorheologic fluids. For the experimental mechanics community, there has been a great deal of interest in quantifying the mechanical limitations of these active materials and overcoming these limitations through composite structures. Self-Healing Polymers/Polymer Composites Just as many biological materials and systems are active, they are also capable of self-healing. This behavior makes it very difficult to permanently damage them. It also enables them to be reconfigured, as is the case with

Implantable Biomedical Devices and Biologically Inspired Materials

bone tissue remodeling. Developing similar concepts in synthetic materials has provided the experimental mechanics community with a unique opportunity to contribute to the engineering of these biologically inspired materials. In particular, efforts have focused on quantifying the mechanics of the interactions of cracks with microencapsulated monomers that are the healing agent for the material in order to design an appropriate size and thickness for the encapsulation in order for the crack to penetrate the microcapsule and release the healing agent [32.61].

cent advances in the selection of materials over the past few decades have provided engineers with the new opportunities to engineer materials using FGM concepts. Currently, most structures are engineered by using a large number of uniform materials that are selected based on functional requirements that vary with location. For example, hard and wear-resistant materials are used to keep the edge of a knife sharp for cutting purposes, but to improve durability it is necessary to use strong and tough materials for the body of the blade. Abrupt transitions in material properties within a structure that result from these functional requirements lead to undesirable concentrations of stress capable of compromising structural performance by promoting crack growth along the interface. In nature, these stresses are controlled by gradually varying the material behavior through a structure, resulting in a FGM. In a variety of biological structures, from insect wings to bamboo, evidence can be found that functionally graded materials have been selected through natural evolution to optimize structural performance through the unique coupling of material and stress distribution [32.64, 65]. FGM concepts have also been inadvertently exploited for years in synthetic structural material systems as common as dual-hard steels, through the use of surface heat treatment processes [32.61]. However, the advent of new materials and manufacturing processes now permit approaches for developing engineered materials with tailored functionally graded architectures, such as the inverse design procedure [32.66]. Thus, component design and fabrication have been synergistically combined, not just for the manufacturing of FGMs, but for the establishment of an entirely new approach to engineering structures.

Microscale

Macroscale x

32.3.3 Functionally Graded Materials Classes of Functionally Graded Materials Biological materials have served as inspiration for a new class of materials that has become of significant interest to the mechanics and materials communities: functionally graded materials (FGMs). FGMs are defined as materials featuring engineered gradual transitions in microstructure and/or composition, the presence of which is motivated by functional requirements that vary with location within the component [32.63]. Re-

• Microstructure (microscale) • Layering (mesoscale) • Interlayer thickness (macroscale)

Mesoscale

f=0

t Continuously graded

f=1

f = (x/t) p f=0

Discretely layered

f=1

Fig. 32.17 Functionally gradient architectures and various length scales related to the architectural features involving microstructure, layering, and interlayer thickness [32.1]

911

Part D 32.3

Biologically Inspired Ceramics One of the most unique materials found in nature are seashells. Seashells are made from calcium carbonate, which makes it chemically the same as chalk. However, the mechanical behavior of shells and chalk vary tremendously. Shells are very hard and difficult to break, while chalk tends to be much softer and more brittle. Thus, shells make very good protective coatings for living organisms, while chalk is good to write with on hard surfaces. The difference in the mechanical behavior of chalk and shells is attributed to the microstructures of the two materials. The calcium carbonate atoms in chalk tend to arrange in more of a granular, porous microstructure, exhibiting the stochastic fracture behavior observed in conventional ceramics and governed by the size of the pores. In the nacre of mollusk shells, these same atoms tend to configure into micron-sized, lamellar blocks known as aragonite, which are held together with a nanometerthick compliant organic material in a brick-and-motor microstructure. It has been of tremendous importance to the experimental mechanics community to understand the effect of this microstructure on crack initiation and growth processes in order to quantify the benefits of creating similar, biologically inspired microstructures in conventional ceramics with processing techniques similar to biomineralization [32.62].

32.3 Biologically Inspired Materials and Systems

912

Part D

Applications

The end result is the capability of engineering parts that correspond to designer-prescribed properties; e.g., parts with negative Poisson’s ratios [32.67]. This approach has already being used to accurately design FGMs for aerospace applications [32.68]. FGMs have also been successfully developed for bone replacement in prosthetic applications [32.69]. Property Distributions The gradual material variation results in a functionally graded architecture described with a continuously graded or discretely layered interlayer that has several relevant length scales, seen in Fig. 32.17, and a variation in properties in the interlayer, typically denoted by one of the following models [32.70].

f (x) = [x/t] p

(power law) ,

f (x) = a0 + a1 x + a2 x

Part D 32.3

f (x) = a0 + a1 eδx

2

(polynomial) ,

(exponential) ,

(32.5) (32.6) (32.7)

where f (x) is the value of the property at a microscale location x in the interlayer, t is the thickness of the interlayer on the macroscale, and a0 , a1 , a2 and δ are constants describing the magnitude and rate of change in properties. When properties vary through discrete layering, the resulting gradient architecture will no longer have property variation on the microscale but rather possess mesoscale variations that are controlled by the number and relative thicknesses of the discrete layers. Experimental Mechanics Characterization Techniques A variety of experimental mechanics diagnostic techniques are being employed to understand FGMs. These include

1. 2. 3. 4. 5. 6. 7.

To further advance the development of FGMs, it will be absolutely essential to use experimental mechanics to characterize the coupling between material and stress distributions. In particular, fullfield deformation measurement techniques, such as Digital Image Correlation, combined with advanced microscopy techniques, such as Electron Microscopy and Atomic Force Microscopy, and localized property characterization techniques, such as microtensile testing and nanoindentation, are being developed to elucidate on the unconventional structure/property/stress relationship produced by this coupling. Non-traditional testing methods, such as hybrid numerical-experimental techniques, are also being developed because of the inherent inhomogeneous behavior of these materials.

x-ray diffraction, neutron diffraction, laser fluorescence spectroscopy, digital image correlation, coherent gradient sensing, photoelasticity, moiré interferometry.

These techniques are being employed on a fullfield or array basis to understand the thermomechanical residual strain and stress distributions that form at the graded interface, as well as at the tip of stationary and propagating cracks.

Failure Mechanics Fracture Resistant Materials. Recently, material and

mechanical characterization of synthetic and natural FGMs has become more extensive. Most of the work on synthetic materials has focused on controlling stress and crack growth in metal/ceramic composites for applications such as prosthetics, while the natural materials have been limited to understanding the relationship between stress and microstructural distributions in bone, bamboo, and shells [32.71–80]. Rabin et al. successfully employed FGMs to minimize the peak axial thermal residual stress that develops in metal-ceramic joints during powder processing [32.81–83] (Fig. 32.18). New theoretical concepts are also being developed and experimentally characterized in order to understand the fracture mechanics of FGMs. Eischen has shown that conventional theoretical fracture mechanics concepts, such as the stress intensity factor, can be used to describe the stress fields around a crack tip in a FGM, as [32.84]   K I −1/2 f 0 (θ) + C1 f 1 (θ) + O(r 1/2 ) , r σx + σ y = 2π (32.8)

where K I is the mode-I stress intensity factor, and the functions of angle θ are identical to the homogeneous case. The J-integral concept has also been modified in order to preserve the path independence in the presence of the inhomogeneities occurring in the interlayer [32.85]. At the microscale, Dao et al. have focused on a physically based computational micromechanics model to study the effects of random and discrete microstructures on the development of resid-

Implantable Biomedical Devices and Biologically Inspired Materials

32.3 Biologically Inspired Materials and Systems

913

Stress (TvPa) 800

0 % Ni Alumina

400

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Fig. 32.18 Schematic of design approach for minimization of peak axial stresses in powder-processed metal ceramic FGM joints through an optimal gradient

 Ke =

BT (x, y)C(x, y)B(x, y)dV ,

(32.9)

Ve

where Ke is the element stiffness matrix, B(x, y) is the matrix of shape function derivatives, and C(x, y) is the compliance matrix derived from the property gradient, E(x, y) = E 0 e(αx+β y) . FGMs have provided a plethora of unique challenges that theoretical, experimental, and computational mechanics have had to overcome in order to characterize and model the behavior of these biologically inspired materials. Dynamic Mechanical Response. Gradient architectures

have also been developed to optimize the energyabsorbing capabilities of armors, inspired by the superior impact resistance provided by wood, bone, and the shells of sea and land animals [32.88]. For these applications, it has been important to understand the dynamic failure of gradient microstructures caused by impact loading events. This failure is governed by the response of stress waves generated by the impact loading as they interact with the gradient microstrucure [32.89–94]. For example, it was determined that a stress wave with magnitude f i will reflect with magnitude f r at time t from

Part D 32.3

ual stresses in functionally graded materials [32.86]. New finite element methods have also been developed to provide insight into the complex stress distributions that can develop within and near the graded interlayer as [32.87]

a graded interface as [32.89], x

1 fr = fi 2

d

(κ¯ − 1)nτ n−1 [1 + (κ¯ − 1)τ n ]−1 dτ

0

 x n

1  , = ln 1 + (κ¯ − 1) 2 d x d tc0 = 2 [1 + (κ¯ − 1)τ n ]−1 dτ , d

(32.10)

(32.11)

0

where d is the interlayer thickness, c0 is the wave speed in base material 1, κ¯ is the acoustic impedance of base material 2 normalized by the acoustic impedance of base material 1. From (32.11), it was determined that the gradient architecture creates a time delay effect that allows for peak stresses to form more slowly and for more energy to be absorbed before crack initiation when a gradient microstructure is present versus a sharp interface. Once a crack initiates due to stress wave loading and begins to grow, the mechanics governing crack propagation differ substantially from static loading. Higher order asymptotic analyses are required to incorporate the effects of gradient property distributions into the localized stress fields around a dynamically growing crack tip [32.95]. The results of this analysis for various values of the material non-homogeneity parameter, α = δ cos ϕ, where δ describes the gradient in material properties in (32.7) and ϕ is the property gradation direction, can be seen in Fig. 32.19. The

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Applications

a) y/t 1

b) y/t

Normalized crack velocity (C/Cs(x))

1

a

a:α = –0.5 b:α = 0 c:α = 0.5

b c

0.5

a:c/cs = 0.1 b:c/cs = 0.3 c:c/cs = 0.5

c b

0.5

0.6

a 0

0

– 0.5

– 0.5

0.5 0.4 0.3

–1 –1

– 0.5

0

0.5

1

–1 –1

– 0.5

0

0.5

x/t

c) y/t 1

d) y/t c

Part D 32.3

0

0

–0.5

–0.5

– 0.5

0

0.5

1

x/t

–1 –1

a:dc/dt = 106 b:dc/dt = 107 c:dc/dt = 108

a b c

0.5

a

0.2 FGM Polyester

0.1

1

a:dK10/dt = 0 b:dK10/dt = 105 c:dK10/dt = 106

b

0.5

–1 –1

1

x/t

0 0.8

1

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Normalized stress intensity factor (KID /KIC(x))

Fig. 32.20 Experimentally determined dynamic constitutive fracture relationship between crack velocity and dynamic stress intensity factor in FGMs that is distinctly different in nature from homogeneous counterparts [32.98] – 0.5

0

0.5

1

x/t

Fig. 32.19a–d Maximum shear stress contours around crack tip at (0, 0) with crack lying along the negative x-axis for (a) various material non-homogeneity parameters, (b) for different crack speeds at a non-homogeneity parameter of 0.57, (c) effect of rate of change of dynamic stress intensity factor of 1 MPa m1/2 for crack velocity of 650 m/s, (d) various levels of crack acceleration [32.95]

data in Fig. 32.19a illustrates that for a homogeneous material, the contours tilt backward due to inertial effects, but the material gradient can compensate and tilt the contours forward. In Fig. 32.19b–d, it can be seen that they will tilt forward more and decrease in size as the crack propagates slowly, will tilt backward more and increase in size as the stress-intensity factor changes, and tilt forward more and decrease in size again as the crack accelerates. Numerical methods using cohesive-volumetric finite elements have also been employed to model dynamic crack propagation in FGMS, and parametric studies of dynamic fracture in Ti/TiB FGM specimens have shown a great sensitivity of crack motion to the gradient of bilinear cohesive failure parameters, σmax (the maximum stress in the CVFE) and Δnc (the critical normal displacement component in the CVFE) [32.96]. Cohesive failure parameters have been measured using digital image correlation near slowspeed cracks exhibiting stable growth in a poly(ethylene carbon monoxide) copolymer, ECO, with a gradient in

ductile-to-brittle transition controlled through UV exposure [32.97]. A dynamic constitutive fracture relationship between the crack velocity and dynamic SIF has also been established experimentally using photoelasticity on polyester-based FGMs (Fig. 32.20) [32.98]. This relationship has three regions: 1. a stem, which is nearly vertical and indicates that the velocity is independent of the stress intensity factor, 2. a transition region, and 3. a plateau where the crack reaches a terminal velocity (usually around 0.6 of the shear wave speed) that does not change substantially with stress intensity factor. Comparing the results of an FGM with a homogeneous counterpart in Fig. 32.20 indicates that at a dynamic stress intensity factor approximately 10% greater than the fracture toughness the crack speed in the FGM increases to almost 2-fold of the speed in the homogeneous counterpart. Experimental results from a variety of experiments have indicated that this relationship is unique in nature for FGMs, and is distinctly different from homogeneous counterparts [32.99]. The dynamic stress intensity factor has also been experimentally measured in compositionally graded glass-filled epoxies at low impact velocities using the technique of coherent gradient sensing [32.100]. Using the crack tip stress field analyses from [32.84] and [32.95], it was

Implantable Biomedical Devices and Biologically Inspired Materials

observed that the dynamic stress intensity factor continuously increases with crack growth when the filler volume fraction is monotonically increasing, and continuously decreases when the gradient is of the opposite sense similar to a homogeneous material (Fig. 32.21). Geometrically Complex Interfaces. Another area where

KID/(KiD)i FGM: E2/E1 > 1 FGM: E2/E1 < 1 Homogeneous

2

1

0

1

2

3 Δa/a

Fig. 32.21 Experimentally measured dynamic stress intensity factor versus crack length in FGMs indicating increases with increasing filler volume fraction (E 2 > E 1 ) in and decreases when the gradient is in the opposite sense, similar to a homogeneous material [32.100]

related to variations in the elastic modulus to determine an optimal gradient for reducing the stress concentration. Assuming a power law relationship between density of the material ρ and the elastic modulus and yield strength, E : ρb and σy : ρβ , the optimal gradient was found to depend on the ratio of the power law exponents, β/b. At some point (β/b ≈ 0.75), when strength does not increase fast enough with density compared to the modulus, then it pays to divert the stresses away from the hole, as is the case of a foramen in bone (β/b ≈ 0.5). A variation on the use of FGMs at geometrically complex interfaces has been the engineering of the geometric complexity to increase the fracture resistance of the interface by redistributing the stress and strain at the interface. For example, development of a new multi-material multi-stage molding technology enabled the geometric features of the interface to be precisely engineered at the milliscale [32.104]. Experimental measurements of deformation fields associated with circular and square features using DIC determined that the strength of the interface could be enhanced by 40% using the features to convert the failure mode from the traditional K-dominant fracture to ligament failure. It was also determined that mechanical interlocking mechanisms attributed to the geometric features enabled the interface to retain at least 30% of its strength in the absence of chemical bonding. Another investigation conducted into free-edge stress singularities associated with bio-inspired designs for dissimilar material joints was conducted based on the growth patterns when trees grow around an obstruction [32.105]. Using photoelasticity, model polycarbonate-aluminum specimens were prepared with varying degrees of joint angles associated with the free-edge of the interface between the dissimilar materials. The convexity associated with these angles was determined to eliminate the free-edge stress singularity, increasing the tensile load capacity by 81% and reducing the material volume by 15% over traditional butt-joint specimens with sever free-edge stress singularities.

32.3.4 Self-Healing Polymers/ Polymer Composites Microstructure/Property/ Performance Relationship One of the most unique characteristics of biological systems has been their ability to heal. Recently, advancements have been made in the formulation of

915

Part D 32.3

functionally graded materials have been making an impact is to enhance the toughness of interfaces with more geometrically complex features. For example, one of the simplest geometrically complex interfaces is a circular inclusion. A classic circular inclusion is a hole, which is conventionally characterized as an inclusion with no mechanical properties. For a two-dimensional elastic material, this case produces the classic stress concentration of three times the far field stress applied to the structure. In nature, holes exist within structures, such as bones, in order to provide internal access for arteries and veins. One such hole is known as a foramen, and the study of the graded material property distribution around the foramen has become the basis for a new concept in stress concentration reduction for holes in synthetic structures, like rivet holes in the skin of aircraft structures [32.101–103]. Using the fullfield deformation technique of Moire inerferometry, the strains around the foramen were characterized and were

32.3 Biologically Inspired Materials and Systems

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Table 32.3 Summary of biological sources of inspiration for self-healing and potential approaches using composite/polymer engineering [32.106]

Biological source of inspiration

Composite/polymer engineering

Molecular cross-linking Bleeding

Remendable polymers Microcapsules, hollow microfibers Nanoparticles, healing resin Short microfibers

Blood clotting Bone reparation

Part D 32.3

polymers and polymer composites to impart this ability to synthetic materials [32.106]. Material heal thyself has now become a reality. Most of the work conducted on self-healing polymers/polymer composites has focused on the impregnation of polymers with a Rhuthenium-based catalyst known as Gibbs’ catalyst, that will polymerize monomers that are microencapsulated and dispersed in the polymer. When a crack propagates through the polymer, it breaks open the microcapsules, and then enables the monomer to polymerize and heal the fracture surfaces in a process inspired by bleeding. The size and concentration of the microcapsules has a significant effect on the healing efficiency. Variations on this bleeding inspired concept include vascular networks consisting of hollow microfibers instead of hollow spheres, and use of microfibers or nanoparticles to mimic the behavior of bone reparation or blood clotting [32.4, 106–108]. A summary of biological sources of inspiration for selfhealing processes and some potential approaches to developing them using composite/polymer engineering can be seen in Table 32.3 [32.106]. Remendable Polymers. Another attractive self-healing

approach for pure polymers is to use mending crosslinked molecular chains activated by a thermal input to remend the fracture surfaces [32.109]. Remendable polymers have been engineered at the molecular level to exhibit this self healing behavior. Using nuclear magnetic resonance spectroscopy, the disconnecting and reconnecting of intermonomer linkages have been tracked during heating and cooling of the material. These polymers are engineered by forming a macromolecular network formed in its entirety by reversible cross-linking covalent bonds, where the degree of crosslinking is so high that they are relatively stiff and

strong. Healing efficiencies of over 50% were reported for these polymers. Photochemical healing has also been achieved via re-photocycloaddition of cinnamoyl groups that form during crack growth in a polymer engineered using a tricinnamate monomer [32.110]. Microencapsulated Healing Agents. One characteris-

tic of natural materials that differs substantially from synthetic is their ability to heal after small amounts of damage. This ability is conveyed through the growth of new tissue by a collagen-based DNA-controlled mechanism. While the building blocks of natural materials has been inherently designed for this capability, there is no such mechanism available in synthetic material systems. A bio-inspired alternative was developed in the late 90s employing the technique of microencapsulation to embed healing agents inside of polymers and polymer composites that would be released whenever a crack compromised the encapsulation, creating an auGrubbs Ru catalyst Crack

Microcapsule

DCPD healing agent

Polymerized healing agent

Load (N) 250 200 Injected DCPD + catalyst

150 Neat epoxy Virgin

100 Selfhealed

50

0

0

200

400

600 800 1000 Displacement (μm)

Fig. 32.22 Fundamental concept for microencapsulated

healing agents and healing efficiency [32.61]

Implantable Biomedical Devices and Biologically Inspired Materials

tonomic healing response with efficiencies greater than 70% (Fig. 32.22) [32.3, 61, 111]. This novel approach raised a number of mechanics issues to optimize the design of this bio-inspired material that are being addressed by the experimental mechanics community that will now be discussed. For the design of microencapsulated healing agents there are two fundamental mechanics issues: 1. how can they be compromised by propagating cracks, and 2. what is their healing efficiency. The first is a microscale issue while the second is macroscale. For the first issue, there are a number of design characteristics for the microencapsulation that can be optimally engineered: thickness of encapsulation, properties of encapsulation, size of encapsulation, and the dispersion of the microscapsules.

An additional issue that is being investigated in order to enhance the commercial appeal of these bio-inspired materials is replacement of expensive ingredients, such as the Ruthenium-based Grubbs’ catalyst. Characterization of Fracture Mechanics To determine the optimal design characteristics, the fundamental interaction between microcapsules and propagating cracks must be investigated. Currently, computational experiments are being performed to understand this interaction at a very basic level. However, the development of microscale and nanoscale mechanical and microstructural characterization techniques, such as microscale DIC, are now enabling the assumptions in these computational experiments to be verified by direct measurement of mechanical properties on and near microcapsules, by determining path of crack tips near microcapsules, and by measuring the local deformation fields near the crack tip and the microcapsules. For the healing efficiency, very fundamental mechanical characterization experiments have been performed using unconventional fracture specimens with tapered geometries to arrest the development of cracks in more brittle polymers [32.111]. The use of brittle polymers is dictated by the interaction of the crack with the microcapsules, where more ductile or rubbery polymers permit cracks to deflect between microcapsules rather than through them. The fracture resistance of ductile or rubbery polymers also preclude the neces-

sity of healing agents at this time, unless an embrittling mechanism is present such as an oxidizing atmosphere like ozone or an aging mechanism like chain decomposition. Cracks can be grown either monotonically or fatigue, and then the change in load-bearing capacity with respect to healing time can be assessed. For optimal healing efficiency, almost 100% of the load bearing capacity can be recovered. It has also been demonstrated that the microencapsulation can be designed such that the fracture toughness and strength of the pure polymer is not compromised. In addition to the healing efficiency, it is also desirable to increase the speed of the healing process. For general fatigue fracture, this is generally not an issue. However, for overloads, significant crack propagation can be reversed if the healing time is faster than the time between overloads. Currently, healing agents can achieve near optimal efficiency as quickly as a few hours. It is envisioned that healing times of minutes, similar to the cure time of fast-curing epoxies, will be achieved in the near future using cheaper ingredients then are currently employed.

32.3.5 Active Materials and Systems Classes of Active Materials and Structures Active Materials and Structures (aka, smart or intelligent materials and structures) are defined by their ability to mechanically deform in response to an external stimulus such as an electrical signal, similar to muscle tissue. They basically fall into three categories.

1. SMA and SMA composites. 2. Piezoceramics. 3. Electroactive polymers (EAPs). Each class is essentially defined by the characteristic of the main active material ingredient (i. e., metal, ceramic, polymer). The active response of each class is summarized in Table 32.4 and compared with muscle tissue. Table 32.4 Comparison of active response for each class

of active material in comparison to natural muscle tissue Class of active material

Active strain (%)

Active stress (MPa)

Frequency response (Hz)

EAP SMA Piezo Muscle

300 10 1 30

10 200 100 1

1000 100 100 000 1000

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Part D 32.3

1. 2. 3. 4.

32.3 Biologically Inspired Materials and Systems

918

Part D

Applications

Part D 32.3

SMA and SMA Composites In many experimental investigations of biologically inspired materials and systems, characterization has focused on relating the distribution of material properties, such as hardness, to the microstructure, independent of the stress distribution in the structure. For advanced structural systems, known as smart structures, the material properties can actually depend on this stress distribution. In particular, smart structures can be fabricated from materials, such as shape memory alloys (SMAs). SMAs experience martensite-austenite phase transformations that enable them to fully recover inelastic deformations due to martensite detwinning when they are heated above their austenite finish temperature (known as the shape memory effect), and will exhibit pseudoelastic behavior when deformed above these temperatures (Fig. 32.23). This novel thermomechanical behavior has led to extensive use of these materials in biologically inspired and biomedical applications, such as deployable stents [32.53]. They can also be embedded within other materials, such as composites where functionally grading the distribution of SMA wire reinforcement has been demonstrated to theoretically increase buckling strength by controlling the distribution of recovery stresses that are generated when the wires are heated [32.112]. By thermomechanically treating SMAs, they can be trained to deform to two different states upon heating and cooling, an effect known as the two-way shape memory effect (TWSME), which is a deformation be-

havior similar to the chemomechanical constriction of muscle tissue. Functionally graded smart materials are now being fabricated on the microscale from SMA thin films with compositional gradients for microelectromechanical systems (MEMS) applications [32.113]. By grading these films, a unique coupling develops between the microstructure and residual stresses that enable these materials to exhibit thermomechanical behavior as fabricated that would not otherwise be possible, such as repeatable actuation behavior (i. e., the TWSME). Using the biologically inspired actuation behavior of these graded films, new biomimetic concepts can be developed for novel structures fabricated from these materials, such as micropumps for MEMS applications. A similar coupling has also been achieved by functionally grading the distribution of SMA wires exhibiting the one-way SME in polyurethanes to produce an equivalent TWSME that mimics the actuation behavior of biological structures such as the fibrous muscle tissue found in a heart or a bird’s wings (Fig. 32.24) [32.114]. The equivalent TWSME for an SMA wire can be determined for wires embedded in a beam structure from the bending curvature k2 and elongation e of the beam associated with the location of the SMA wire d the geometric properties of the beam, and the properties of the wire and beam material as [32.115] k2 =

 t αβdγ (γβ − γ − β) ε + αa (T − T0 ) , 2 (γ + β)γ − αβd (γβ − γ − β) (32.12)

T < As σ

Detwinned martensite

Stress

e=

T > Af

(γ + β)γ

− αβd 2 (γβ − γ

− β)

εt + αa (T − T0 ) , (32.13)

Detwinning

Twinned martensite

where T − T0 is the temperature change in the SMA wire, εt is the stress-free transformation strain, αa is the

ε

Cooling

Austenite



γ 2β

Heating recovery

Detwinned martensite

Strain

T

Shape memory effect

Pseudoelasticity

Flying bird

373 K 293 K

Two way shape memory effect

373 K 273 K 373 K 10 cm

Fig. 32.23 Novel thermomechanical behavior of Shape Memory

Alloys exploited for numerous biologically inspired and biomedical actuation applications. related to the deformation and transformation of martensite and austenite phases at different temperatures

Morphing wing

Fig. 32.24 Biologically inspired morphing wing structure using graded one-way SMA wire distribution in polyurethane that reversibly deform when heated

Implantable Biomedical Devices and Biologically Inspired Materials

coefficient of thermal expansion, γ is the fraction of the recovery strain transferred to the matrix, and α ≡ A/I2 , β ≡ E a Aa /(E A) ,

(32.14)

where A is the cross-sectional area of the beam, Aa is the cross-sectional area of the wire, E is the Young’s modulus for the beam, E a is the Young’s modulus of the beam, and I2 is the moment of inertia of the beam cross section. The experimental mechanics technique known as digital image correlation has been used to quantify the transfer of the recovery strain between the SMA wire and the matrix for comparison with 3-D thermomechanical finite element analysis (Fig. 32.25) [32.116].

limited to primarily vibration control in the form of patches which are attached externally to a passive structure [32.118]. They have also been popular as strain and force transducers, where they exhibit greater sensitivity and signal-to-noise ratios than conventional strain gages or load cells [32.119]. In order to overcome some of these limitations for applying piezoceramics, there has been a great deal of historical work by the experimental mechanics community to design mechanical amplifiers based on flextensional principles for piezoceramic structures to enhance the deformation of stacks for low-frequency high-power applications such as sonar [32.120]. Applying these amplification principles at the microscale, microamplifiers for piezoceramics have been fabricated using LIGA with a mean static amplification factor of 5.48 and frequency response of 82 kHz [32.121]. In addition to the electromechanical strain limitations, piezomaterials tend to be very brittle, with the fracture toughness of the most common piezoceramic (PZT) being below 1 MPa m1/2 . Their fracture behavior is also very sensitive to applied electric fields, such that a positive electric field increases the apparent fracture toughness for cracks propagating parallel to the poling direction, while a negative electric field has an opposite effect which can be modeled using a work energy based criterion for domain switching and a Reuss-type approximation for poly-domain piezoelectrics [32.122]. Thus, there is interest in the experimental mechan-

DIC deformation measurements

SMA wire

z y

x

Finite element analysis

Fig. 32.25 Quantification of the transfer of recovery strain between an SMA wire and matrix material using the experi-

mental mechanics technique of digital image correlation and comparison with 3-D thermomechanical FEA [32.116]

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Part D 32.3

Piezoceramics Another active material system that has seen limited use in biologically inspired structural applications has been piezoceramics. Piezoceramics produce deformation by coupling electric fields with mechanical and thermal loadings (piezoelectric effect). In reverse, they can produce electrical signals upon mechanical loading similar to act as a force sensor, similar to bone [32.117]. The deformations associated with these fields are very fast (>10 kHz) and very small (0.1 Hz), have difficulty holding their strain, and produce relatively low actuation forces. The forms of ionic EAPs include ionic polymer gel (IPG), ionomeric polymer-metal composite (IPMC), conductive polymer (CP), and carbon nanotube (CNT). EAPs can exhibit mechanical properties and actuation behavior that are very close to muscle tissue. The simplest design of an EAP is to sandwich a polymer with a low elastic stiffness and high dielectric constant

b) Sensor voltage 4.6

Microamplifier static amplification experiment

4.4 Amplifier sensor

4.2

4

Least squares (y =A+Bx) amplification factor = 2.74

3.8

3.6 –120

PZT sensor

–100

– 80

– 60

– 40

–20 0 PZT voltage

Fig. 32.26 (a) Microamplifier fabricated using LIGA sandwiched between piezoceramics, and (b) the static amplifi-

Fig. 32.27 Strains associated with extensional activation of

cation of the piezoceramic displacement [32.119]

a synthetic muscle formed from EAPs [32.125]

Implantable Biomedical Devices and Biologically Inspired Materials

in between two electrodes, creating an electric EAP which is known as an electrostatically stricted polymer (ESSP). By applying a high voltage V to electrodes separated by a thickness t of a polymer with dielectric constant ε and area A an actuation force F will be produced as

32.3.6 Biologically Inspired Ceramics Ceramics are a wonderful engineering material. Extremely hard, stiff, fairly light, chemically inert, and thermally-resistant, they have found many uses, especially in harsh environments and protective applications. However, they suffer from a major drawback: they are very brittle and easily fracture. This is due to the bonding and microstructure of ceramics. Ceramics are ionically bonded, which is a very strong and highly directional bonding. Thus, it takes a great deal of local force to rearrange the bonding and to create defects, which is necessary to generate plasticity through the traditional mechanisms of dislocation motion in a crystal. However, the lack of defects creates grain morphologies that result in significant amounts of porosity in the microstructure. This porosity becomes sites for initiating crack growth, and the strength of the material is ultimately determined by the loading necessary to generate cracks from these pores rather than to propagate dislocations through a crystal structure. To overcome the drawback of brittleness manifested through low fracture toughness, high-performance ceramics are being engineered with reduced grain sizes that reduce the size and amount of pores, as well as to create more tortuous crack paths for intergranular crack growth mechanisms that lead to more crack bifurcation and diffuse microcracking to consume more energy through the formation of greater crack surface area. While this significantly strengthens the ceramic, it

921

still does not permit the loads or generates the defects that are necessary to initiate the motion of dislocations to consume significantly greater fracture energy. Nacre Inspired Ceramics The inherent limitation on dislocation motion due to the type of bonding present in ceramic has significantly limited progress with the current ceramic engineering approach Composites have been an attractive alternative, where very ductile, relatively strong metals, such as aluminum, can significantly enhance fracture toughness of ceramics, such as silicon carbide, at the expense of thermal resistance, while ceramic-ceramic composites, such as carbon–carbon, can create mesoscopic interfaces to provide additional surface area at another length scale to consume more fracture energy. On the other hand, rigid biological ceramic composites, such as the nacre of mollusk shells, possess fracture mechanisms that increase the absorption of fracture energy through the use of very thin layers of an biopolymer (proteinaceous) phase between platelets of calcium carbonate known as aragonite that generate crack bridging and tunneling mechanisms at the nanoscale through the molecular strength of the molecules in the biopolymer [32.5, 128]. Digital image correlation has been a)

b)

100 μm

20 μm

c) Flexural stress (M) 1000 900 800 700 600 500 400 300 200 100 0

Silica rod Spicule

0

0.5

1

1.5

2

2.5

3

3.5

Strain (%)

Fig. 32.28a–c Micrographs of (a) amorphous silica rod and (b) a sponge spicule, along with the associated four-point bend data (c) indicating a 4-fold increase in strength and a 3-fold increase in

failure strain [32.127]

Part D 32.3

εε0 AV 2 , (32.15) 2t 2 where ε0 is the dielectric constant of a vacuum. The resulting deformation will depend on the constitutive behavior of the polymer. The response of these electroactive polymers is fast (ms levels), and the response can be held for long periods of time under DC activation. However, the required voltage levels are high (≈150 MV/m), and the glass transition temperature is inadequate for low temperature applications. Recently, 3-D digital imaging techniques have been employed to look at the deformations associated with synthetic muscles formed from EAPs (Fig. 32.27) [32.125]. F=

32.3 Biologically Inspired Materials and Systems

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Applications

Part D 32.3

Actual bone

3D model created by scanning

Replica of bone

Mold

Fig. 32.29 Replamineform inspired bone structures (RIBS) fabricated using a new multistage molding technology to create geometric complex features on the milliscale and advanced ceramic gelcasting technology with carbon fugitives that form replamineform inspired microstructures [32.130]

used at the microscale to determine that sliding of the platelets during crack growth can give rise to a rising crack resistance curve characteristic of a material that can stabilize and even arrest cracks [32.129]. The geometric features of the ceramic platelets alone can substantially alter the fracture resistance, as evidenced by a 4-fold increase in strength and 3-fold increase in failure strain when comparing a sponge specula made of silica with a standard amorphous silica rod (Fig. 32.28) [32.127]. At the nanoscale, the characteristics of nacre inspired ceramics leads to an insensitivity of fracture processes to flaws that are present in the biopolymer phase, which is governed by the aspect ratio of the platelets ρ∗ and the shear strength of the biopolymer τpf as

1 αE m γ ∗ (32.16) ρ = f , h τp where E m is the Young’s modulus of the platelet, γ is the surface energy, h is the thickness of the platelet, and α depends on the crack geometry [32.9]. Nacre also exhibit a much higher fracture toughness in water, where a ten-fold increase can occur through the activation of

more plastic work at the platelet interfaces normal to the crack propagation, and are extremely temperature dependent [32.6, 131]. Synthetic methods for processing these lamellar microstructures have been developed using Portland-cement to create a self-aligned calcium carbonate plates or prismatic morphologies, however they have yet to be mechanically characterized in a manner similar to Nacre [32.132]. Replamineform Ceramics Another natural ceramic that possesses unique microstructural characteristics is coral. It has been a source of inspiration in bone healing and remodeling since the 1970s due to the interconnected porosity that develops during the biomineralization process [32.133]. Pores with the appropriate diameter (>100 μm) allow the in-growth of vascularized bone tissue for prosthetics and prosthetic fixation [32.134]. The replamineform process was developed to duplicate the interconnected, controlled pore size microstructure of particular coral species (such as Porites and Goniopora) [32.135,136]. Processes have also been developed to fabricate biologically inspired ceramics that are potential prosthetic materials or structures using hydroxyapatite particles and taking advantage of the

Implantable Biomedical Devices and Biologically Inspired Materials

physics of ice formation [32.137]. These materials are four times stronger than conventional prosthetic materials. Also, replamineform inspired bone structures (RIBS) have been formed by combining a new multi-piece molding technology for creating geometrically complex milliscale features with an advanced ceramic gelcasting technology using carbon fugitives to create replamineform inspired microstructures (Fig. 32.29) [32.130]. Experimental mechanics investigations on RIBS have indicated that 35% total porosity in bone repli-

32.4 Conclusions

923

cas can decrease strength 86%, decrease stiffness 75%, and reduce total deformation 30% in comparison to bone replicas with 13% total porosity fabricated without carbon. The reduction in stiffness was consistent with a model developed from microscale finite element analysis of overlapping solid spheres that produce microstructures similar to those observed in these specimens. Furthermore, the increased porosity of the material permits axial cracks to grow and bifurcate more easily at substantially reduced deformation and load levels.

32.4 Conclusions 32.4.1 State-of-the-Art for Experimental Mechanics

Part D 32.4

Experimental mechanics is being applied to the development of new implantable biomedical devices through understanding the processing/microstructure/property relationship in biocompatible materials and the structure/performance of the devices that are being made from them. It is also being used to provide insight into the fundamental processing/microstructure/property/ structure/performance relationship at multiple length scales for new biologically inspired materials and systems that are analogs of biological counterparts. To elucidate on the synergy between the research and development activities in these areas through the application of experimental mechanics can be seen through specific examples that are motivated by the following challenges that are faced in understanding the mechanics of these unique materials and systems: Wide range of properties, wide range of length scales, thermal and chemical constraints (in vivo), and multifunctionality. The application of experimental mechanics to implantable biomedical devices has accelerated their development by understanding fundamental performance issues. Prosthetics have benefited from a detailed understanding of the microstructure/property relationship in composites from which they are made, as well as the development of simulators and testing techniques for characterizing the structure/performance relationship when they are deployed. Characterizing the processing/microstructure/property relationship in biomaterials using micro/nanoidentation have facilitated the development of biodegradable polymer composites for biomechanical fixation. The design

of deployable stents has been improved by characterizing their performance using new test methods and relating it to the materials, geometry of the stents, and the mechanical interaction of system components. The novel use of experimental mechanics has also led to the development of new implantable biomedical devices, such as SMA stents that can selfexpand through heating instead of requiring a balloon as the original passive metal stents used in angioplasty. All of the experimental mechanics knowledge obtained from implantable biomedical devices and the biological systems they are used in is now being applied to developing new biologically inspired materials and systems. Graded natural structures formed from adaptation to changing environmental conditions and defects, such as foramen holes, have provided inspiration for creating Functionally Graded Materials out of composites formed from polymers, metals, and ceramics that provide greater fracture resistance and tougher geometrically complex interfaces. The actuation behavior of soft tissues such as muscle is providing inspiration to create active structures using Shape Memory Alloy metals, piezoceramics, and Electroactive Polymers that provide more compact, reliable, efficient, and autonomous replacements for conventional actuation systems. The self-healing behavior of biological materials are also inspiring the development of similar attributes in polymers and polymer composites using engineered polymer molecules and microencapsulated healing agents that mimic the bleeding process. Finally, the microstructures of bone and seashells are serving as templates for creating more fracture resistant and biocompatible ceramics materials and structures.

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Applications

32.4.2 Future Experimental Mechanics Research Issues It is clear from the experimental mechanics research that has been conducted to date on implantable biomedical devices and biologically inspired materials and systems, that the following research issues need to be addressed to further the development of these devices, materials, and systems. 1. Further development of 3-D measurement techniques are required to understand the mechanics associated with the geometric complexity of biologically inspired systems and advanced implantable biomedical devices. 2. Further development of non-destructive and noninvasive sub-surface measurement techniques are

required to elucidate more directly on the interfacial interactions between natural structures and implantable biomedical devices both in vitro and in vivo. 3. Advancement of multiscale measurement techniques to elucidate on the link between nanoscale, microscale, and mesoscale mechanisms involved in the transfer of load and resulting failure in biologically inspired materials. 4. More dynamic measurement techniques to resolve time-dependent processes involved in polymerbased implantable biomedical devices and biologically inspired materials and systems. 5. More advanced loading systems and test methods are required for improved experimental in vitro simulation of the performance of implantable biomedical devices.

Part D 32

32.5 Further Reading • • • •

Overview of experimental mechanics testing methods for prosthetics [32.18]. Overview of tissue scaffolding [32.15, 17, 44]. Overview of biomaterials [32.8, 20]. Overview of hierarchically structured materials [32.6, 10, 11].

• • • •

Overview of mechanics and fundamental principles in biologically inspired materials and systems [32.1, 2, 21]. Overview of processing, characterization, and modeling of FGMs [32.63, 70]. Overview of dynamic failure in FGMs [32.98]. Perspective on self-healing polymers/polymer composites [32.106].

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929

High Rates an 33. High Rates and Impact Experiments

Kaliat T. Ramesh

From a mechanics viewpoint, the consequences of an impact are threefold. First, stress waves or shock waves are propagated inside the impacted bodies, and the propagation of these waves must be understood. Second, large inelastic deformations may be developed, typically at high rates of deformation. Third, the entire impacted structure may be excited by the impact, leading to structural dynamics and vibration problems, typically at long times. This chapter explicitly ignores the last of these consequences. Further, if the impact velocity is sufficiently small, all of the stress waves propagated inside the impacted bodies will be elastic. The measurement of elastic wave propagation is discussed in other chapters, notably those on ultrasonics and photoacoustic characterization. Thus, our focus is on the experimental techniques associated with the propagation of nonelastic waves, and with the measurement of high-strain-rate behavior. Our emphasis is therefore on the measurement of the phenomena that are

33.1 High Strain Rate Experiments ................ 33.1.1 Split-Hopkinson or Kolsky Bars ...... 33.1.2 Extensions and Modifications of Kolsky Bars .............................. 33.1.3 The Miniaturized Kolsky Bar........... 33.1.4 High Strain Rate Pressure-Shear Plate Impact ................................ 33.2 Wave Propagation Experiments ............. 33.2.1 Plate Impact Experiments.............. 33.3 Taylor Impact Experiments .................... 33.4 Dynamic Failure Experiments ................. 33.4.1 Void Growth and Spallation Experiments........... 33.4.2 Shear Band Experiments ............... 33.4.3 Expanding Ring Experiments ......... 33.4.4 Dynamic Fracture Experiments ....... 33.4.5 Charpy Impact Testing................... 33.5 Further Reading ................................... References ..................................................

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developed during the early times after a relatively high velocity impact. It is useful to develop a sense of the range of strain rates developed during typical impact problems (note that the strain rates developed as a result of an impact are always functions of time, and therefore we focus here on the peak strain rates developed). Thus during an asteroid impact on the Earth, the peak strain rates that are developed are likely to be of the order of 108 s−1 (this results from hypervelocity impact, i. e. impact velocities above 5 km/s). For impacts corresponding to typical velocities from defense-related terminal ballistics (≈ 1–2 km/s), the peak strain rates developed are of the order of 105 s−1 to 106 s−1 . All strain rates below the peak strain rate are likely to be developed during the event at sufficiently long times, and their significance to the problem must be determined on a case-by-case basis. In both the planetary impact and ballistic impact cases, for example, substantial parts of

Part D 33

Experimental techniques for high-strain-rate measurements and for the study of impactrelated problems are described. An approach to classifying these experimental techniques is presented, and the state-of-the-art is briefly described. An in-depth description of the basis for high-strain-rate experiments is presented, with an emphasis on the development of a range of strain rates and a range of stress states. The issues associated with testing metals, ceramics and soft materials are reviewed. Next, experimental techniques that focus on studying the propagation of waves are considered, including plate impact and shock experiments. Experiments that focus on the development of dynamic failure processes are separately reviewed, including experiments for studying spallation and dynamic fracture.

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Part D

Applications

the deformation occur at lower strain rates (perhaps as low as 102 s−1 ). Further, significant parts of the damage that occur as a result of these events might be a result of the lower strain rate deformations (depending on the damage growth kinetics). This is because the high-strain-rates are typically sustained for much longer times than the ultra-high-strain-rates, and it takes time for many damage mechanisms (for example, the nucleation and growth of voids) to develop. Experimental techniques that are colloquially defined as impact experiments can have dramatically different objectives. Since the design of the experimental technique is critically influenced by the final objective of the experiment, it is important to first decide what information one wishes to extract from the experiment. Typically, what are commonly called impact experiments fall into one of four different categories in order of increasing complexity in terms of the dynamics.

• •

• •

High-strain-rate experiments, that is, experiments designed to measure the high-strain-rate properties of a material. Wave-propagation experiments, that is, experiments designed to understand the characteristics of wave propagation within the material or structure; these may develop high strain rates as well, but the high rate deformations vary in both space and time. Dynamic failure experiments, that is, experiments designed to understand the processes of dynamic failure within a material or structure. Direct impact experiments, that is, experiments designed to understand or discover broad impact phenomena (such as cratering efficiency experiments or vehicle crash experiments).

We examine each of these types of experiments in detail.

33.1 High Strain Rate Experiments

10–6

10– 4

10–2

100

102

Pressure-shear plate impact

Miniaturized Kolsky bar

Specialized machines

Conventional Kolsky bars

high strain rates. Conventionally, strain rates at or below 10−3 s−1 are considered to represent quasistatic deformations, and strain rates below 10−6 s−1 are considered to be in the creep domain. Creep experiments are typically performed at relatively high temperatures, and a variety of specialized machines exist for these kinds of loading; dead loads are often of particular interest. Quasi static experiments are typically accomplished through a variety of servohydraulic machines, and ASTM standards exist for

Servohydraulic machines

Part D 33.1

One of the defining features of impacts that occur at velocities sufficiently large to cause inelastic (and particularly plastic) deformations is that most of these deformations occur at high strain-rates. These deformations may also lead to large strains and high temperatures. Unfortunately, we do not understand the high-strain-rate behavior of many materials (often defined as the dependence σf (ε, ε˙ , T ) of the flow stress on the strain, strain rate and temperature), and this is particularly true at high strains and high temperatures. A number of experimental techniques have therefore been developed to measure the properties of materials at high strain-rates. In this section we focus on those experimental techniques that develop controlled high rates of deformation in the bulk of the specimen, rather than those in which high-strain-rates are developed just behind a propagating wave front. The primary experimental techniques associated with the measurement of the rate dependent properties of materials are described in Fig. 33.1 (note that the stress states developed within the various techniques are not necessarily identical). An excellent and relatively recent review of these methods is presented by Field et al. [33.1]. For the purposes of this discussion, strain rates above 102 s−1 are classified as high strain rates, strain rates above 104 s−1 are called very high strain rates, and strain rates above 106 s−1 are called ultra

104 106 108 Strain rate (s–1)

Fig. 33.1 Experimental techniques used for the devel-

opment of controlled high strain rate deformations in materials

High Rates and Impact Experiments

most of these experiments. Most servohydraulic machines are unable to develop strain rates larger than 100 s−1 repeatably, but some specialized servohydraulic machines can achieve strain rates of 101 s−1 . Finally, the strain rates in the intermediate rate domain (between 100 s−1 and 102 s−1 ) are extremely difficult to study, since this is a domain in which wave-propagation is relevant and cannot be easily accounted for (however, the strain rates in this domain are indeed of interest to a number of machining problems). The primary approach to testing in this strain rate range uses drop towers or dropweight machines [33.2], and great care must be exercised in interpreting the data because of the coupling between impact-induced wave propagation and machine vibrations [33.3–7]. There are existing ASTM standards for some drop-weight tests, including ASTM E208-95a (2000) for measuring the ductility transition temperature of steels, ASTM E43603 for performing drop-weight tear testing of steels and ASTM E680-79 (1999) for measuring the impact sensitivity of solid phase hazardous materials. In this section we focus on experimental techniques for the higher strain rates (greater than 102 s−1 ), including the high (102 –104 s−1 ), very high (104 –106 s−1 ) and ultra-high-strain-rate (> 106 s−1 ) domains.

The now-classical experimental technique in the highstrain-rate domain is the Kolsky bar or Split-Hopkinson pressure bar (SHPB) experiment for determining the mechanical properties of various materials (e.g. [33.8– 11] in metals, ceramics and polymers respectively) in the strain rate range of 10+2 –8 × 10+3 s−1 . The terms Split-Hopkinson pressure bar and Kolsky bar are often used interchangeably. However, one should note that the term Split-Hopkinson pressure bar implies the performance of compression experiments, whereas the term Kolsky bar is more general and may include compression, tension, torsion or combinations of all of these. Since the fundamental concept involved in this technique – that of determining the dynamic properties of materials using two long bars as transducers, with the specimen size much smaller than the pulse length – was developed by Kolsky [33.12], we will use the term Kolsky bar in the rest of this chapter. We begin by describing the compression Kolsky bar (an extensive description is presented in [33.13]). A schematic of the typical experimental apparatus is shown in Fig. 33.2. The device consists of two long bars (called the input and output bars) that are de-

Input bar

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Interface 2

Specimen

Output bar

V

Fig. 33.2 Schematic of the compression Kolsky bar. The input and output bars are designed to remain elastic, as is the pacting projectile

signed to remain elastic throughout the test. These bars sandwich a small specimen (usually cylindrical), which is expected to develop inelastic deformations. The bars are typically made of high-strength steels such as maraging steel, with a very high yield strength and substantial toughness. Other bar materials that have been used include 7075-T6 aluminum, magnesium alloys and PMMA (for testing very soft materials) and tungsten carbide (for testing ceramics). One end of the input bar is impacted by a projectile made of a material identical to that of the bars; the resulting compressive pulse propagates down the input bar to the specimen. The wave propagation is shown in the Lagrangian wave propagation diagram of Fig. 33.3, where time t = 0 corresponds to the instant of projectile impact. Several reverberations of the loading wave occur within the specimen; a transmitted pulse is sent into the output bar and a reflected pulse is sent back into the input bar. Typically, resistance strain gages are placed on the input and output bars for measuring 1. the incident pulse generated by the impacting projectile, 2. the reflected pulse from the input bar/specimen interface and 3. the transmitted pulse through the specimen to the output bar. Note that the strain gauge locations are ideally such that the incident and reflected pulses do not overlap, as shown in Fig. 33.3. The strain gage signals are typically measured using high-speed digital oscilloscopes with at least 10-bit accuracy and preferably with differential inputs to reduce noise. Let the strain in the incident pulse be denoted by εI , that in the reflected pulse by εR , and that in the transmitted pulse by εT (these are bar strains as measured by the strain gages). Examples of the raw signals measured on the input and output bars are presented in Fig. 33.4

Part D 33.1

33.1.1 Split-Hopkinson or Kolsky Bars

Interface 1

33.1 High Strain Rate Experiments

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Part D

Applications

by t

e˙ s = εT

εR

v1 − v2 cb = (εI − εR − εT ) , l0 l0

(33.3)

where l0 is the initial specimen length. Since the strains in the elastic bars are known, we can compute the bar stresses and thence the normal forces at the two specimen/bar interfaces as

εI

x

P1 = E b (εI + εR ) Ab

(33.4)

P2 = E b (εT ) Ab

(33.5)

12

Fig. 33.3 Wave propagation (also called time-distance or Langrangian) diagram showing the stress waves progagating through space and time in the Kolsky bar arrangement. The strain gauge locations are also shown. Speciman length is exaggerated to show the wave reverberation in the specimen

for a test on vanadium. Then the characteristic relations associated with one dimensional elastic wave propagation in the bar tell us that the particle velocity at the specimen/input-bar interface is given by   v1 (t) = cb εI − εR ,

(33.1)

Part D 33.1

and that at the specimen/output-bar interface is given by v2 (t) = cb εT ,

(33.2)

√ where cb = E b /ρb is the bar wave speed, with E b the Young’s modulus and ρb the density of the bar material. The mean axial strain rate in the specimen is then given Volts 0.08 0.06 0.04 0.02 0 –0.02 Input bar signal Output bar signal

–0.04 –0.06 –0.08

0

0.1

0.2

0.3

0.4 Time (ms)

Fig. 33.4 Examples of strain gauge signals obtained from the input and output bars for a high strain rate test on pure vanadium tested at 600 ◦ C

and

at the specimen/input-bar and specimen/output-bar interfaces respectively. The mean axial stress s¯s in the specimen is then given by s¯s (t) =

E b Ab P1 + P2 1 = (εI + εR + εT ) 2 As0 2 As0 (33.6)

where As0 is the initial cross-section area of the specimen. It is instructive to examine the wave propagation within the specimen in some detail (Fig. 33.3). When the incident pulse arrives at the specimen a compressive loading wave is generated inside the specimen. The compressive loading wave propagates through the specimen and arrives at the specimen output-bar interface. By design, the specimen’s impedance is smaller than the impedance of the bars surrounding the specimen on both sides. Since the output bar has a higher impedance than the specimen itself, the wave that reflects from the specimen/output-bar interface remains a loading wave, resulting in an even higher compressive stress. This wave now arrives at the specimen/input-bar interface, again sees a higher impedance and again reflects as a loading wave, resulting in a further increase in the compressive stress. This process continues until the stress within the specimen reaches a value that is sufficiently high to generate inelastic strains, resulting in finite plastic flow of the specimen under the compressive loading. Once substantial plastic flow of the specimen material has commenced, we may neglect further wave propagation within the specimen, since the amplitude of the subsequent wavefronts will be very small. Thus at these later times the stress within the specimen is essentially uniform; the stress is said to have equilibrated. If the boundary conditions are frictionless, the specimen stress is also uniaxial.

High Rates and Impact Experiments

Once an equilibrium condition has been achieved, we have P1 = P2 , and (33.4) and (33.5) then imply that εI + εR = εT .

(33.7)

This condition, together with (33.1), (33.2), (33.4) and (33.5), substantially simplifies equations (33.3) and (33.6) for the specimen strain rate and specimen stress. That is, by assuming stress equilibrium, uniaxial stress conditions in the specimen and 1-D elastic stress wave propagation without dispersion in the bars (discussed et seq.), the nominal strain rate e˙ s , nominal strain es and nominal stress ss (all in the specimen) can be estimated using 2cb e˙ s (t) = − εR (t) , (33.8) l0 t es (t) = e˙ s (τ) dτ , (33.9) 0

and

True stress (MPa) 300 250 200 150 100 50 0

Vanadium (600 °C) at 4500 s–1

0

0.05

0.1

0.15

0.2 0.25 True strain

Fig. 33.5 True stress versus true strain curve obtained using the signals presented in Fig. 33.4. The material is vanadium tested at 600 ◦ C at a strain rate of 4500 s−1

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specimen are then given by [33.14] εs (t) = − ln [1 − es (t)] ,

(33.11)

and ε˙ s (t) =

e˙ s (t) . 1 − es (t)

(33.12)

Note that compression is defined as positive. Assuming plastic incompressibility, the true stress σs in the specimen is obtained as σs (t) = ss (t) [1 − es (t)] .

(33.13)

From (33.9) and (33.10), a stress-strain curve is obtained at a strain rate defined by an average taken over the strain rate history obtained in (33.8); an example of such a stress-strain curve is presented in Fig. 33.5 (obtained on vanadium using the pulses presented in Fig. 33.4). We note here that the correction to a true strain rate represented by (33.12) has often not been used by researchers in the field, resulting in the peculiar situation that papers will present true stress versus true strain curves obtained at an engineering (rather than true) strain-rate. The magnitude of this correction can be considerable if the final strain is of the order of 20%. The Design of Kolsky Bar Experiments The design of a specific Kolsky bar experiment essentially comes down to the design of the specimen (i. e., the choice of specimen length l0 and diameter d0 ) and the choice of the projectile impact velocity so as to attain a specific strain rate in the specimen. Other design parameters such as the bar length L and the bar diameter D are more relevant to the design of the Kolsky bar system rather than the specific experiment. The most important design parameters are the three ratios: L/D, D/d0 and l0 /d0 . Typically, L/D is of order 100, D/d0 is of order 2 to 4, and l0 /d0 is 0.6 to 1. The choice of bar diameter itself depends on the material being tested, the typical specimen sizes that are available, and the degree of wave dispersion that is allowable. Although using a large value of D/d0 provides a large area mismatch and thus allows for the testing of much harder specimen materials, the expectation that the specimen-bar interface can be approximated as a plane places limits on the value of this ratio (that is, one needs to avoid the punch problem). The majority of Kolsky bars in common use in the world are between 7 mm and 13 mm in diameter, although very large bar diameters have been used for experiments on concrete (e.g., [33.15]; very large bar diameters have a specific disadvantage in that launching the projectile may require substantial gun facilities).

Part D 33.1

E b Ab εT (t) , (33.10) As where As and l0 are the initial cross-sectional area and length of the specimen respectively. Note the negative sign in (33.8). This arises because the strain in the reflected pulse has the opposite side of the strain in the incident pulse; the latter is compressive and is conventionally considered positive in these experiments. Thus the specimen strain rate expressed by equation (33.8) is a positive strain-rate, i. e., in this convention, a compressive strain-rate. The true strain and true strain rate in the ss (t) =

33.1 High Strain Rate Experiments

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Part D

Applications

A systematic way to perform experimental design is presented below. The strain in the incident pulse is given by V , (33.14) 2cb where V is the projectile velocity. Using this and the equilibrium condition (33.7) in (33.10), we can write the specimen stress entirely in terms of the reflected strain and the impact velocity: εl =

E b Ab E b Ab εT (t) = (εI + εR ) As As   E b Ab V = + εR . As 2cb

ss (t) =

(33.15)

However, (33.8) allows us to write the strain in the reflected pulse in terms of the specimen strain rate l0 e˙ s . (33.16) 2cb Using this in (33.15), we obtain the relation between the specimen stress and the specimen strain rate that must be satisfied during any given test using a specific Kolsky bar, a specific specimen, and a particular choice of impact velocity    Ab V ss (t) = E b A 2c s   b  Ab l0 (33.17) e˙ s . − Eb As 2cb εR = −

Part D 33.1

This last result represents what we call a test characteristic response: that is, the constraints of the

experimental configuration force any observed stresses and strain rates in the specimen to satisfy (33.17). The result is shown as the straight line in Fig. 33.6, which is a plot of the stress versus the strain-rate (hereafter called a response diagram). Note that in this figure compressive stresses are presented along the positive Y-axis, following our convention. The straight line in Fig. 33.6 represents the locus of all points in stress versus strainrate space that can be reached using a compression Kolsky bar, a specific specimen geometry, and a specific projectile impact velocity. Figure 33.6 also shows the curve corresponding to the constitutive response of the specimen material (typically the flow stress increases with strain-rate). The stresses and strain rates actually observed in a given test correspond to the intersection of the test response line with the constitutive curve, as shown in Fig. 33.6. Given that the constitutive response of the material is not controllable, a specific strain-rate can only be achieved by moving the test response line; that is, in order to sample other points along the constitutive curve, we would need to move the straight line corresponding to the experimental parameters. Equation (33.17) indicates that the maximum strain rate that can be observed in a Kolsky bar test is given by e˙ s |max =

V , l0

(33.18)

while the maximum compressive stress that can be observed in such a test is given by    Ab V (33.19) ss |max = E b . As 2cb

Specimen stress Increasing area ratio A V ss|max = Eb ⎛ b⎛ ⎛ ⎛ ⎝As⎝ ⎝2cb ⎝

Test response line

Observed spec. stress

Increasing V

Specimen material response curve

Decreasing l0

Observed spec. e·

V e·s|max = l0

Specimen strain rate

Fig. 33.6 Response diagramm for the compression Kolsky bar, showing the intersection of the material response curve (dashed) with the test response line

High Rates and Impact Experiments

Requirements for Valid Kolsky Bar Experiments Several basic requirements must be satisfied in order to ensure that the measured response in a Kolsky bar experiment is truly the constitutive behavior of the specimen material. We discuss each of these requirements briefly below. Equilibrated Stresses. The nature of the Kolsky bar

experiment is such that the loading arrives first at one side of the specimen. Because stress waves propagate in the specimen at finite speeds, equilibration of the stresses takes a non-zero time. Equilibration of the stresses is required to ensure that the stress measured from the output bar side of the specimen represents the average stress in the specimen. In terms of experimental design, equilibration of the stresses will occur if the loading time of interest is long in comparison with the specimen characteristic time (τs = csp /l0 , the time for the compressive wave to traverse the length of the specimen). Davies and Hunter estimated that three reverberations of the loading wave in the specimen are required for stress equilibration [33.17] for ductile metal specimens. The ability to check that the axial stress has in fact equilibrated at the time of inter-

935

est is critical for the successful use of any Kolsky bar technique. The typical approach to checking equilibrium involves a comparison of the force histories on the two sides of the specimen [33.13]. Note that the axial stress distributions in the specimen cannot be measured in Kolsky bar experiments, since only the forces at the two ends of the specimen are experimentally accessible. For some situations (such as an experiment on a soft specimen material or at very high strain rates) even the force at the input-bar/specimen interface is difficult to obtain accurately, because the measure of this force involves a subtraction of two pulses with similar magnitudes that have suffered wave dispersion. For the more complex experiments, involving very soft specimens or extremely high-strain-rates, it is sometimes necessary to perform a full computational analysis of the wave propagation within the specimen to obtain an estimate of the time required for equilibration [33.16]. Friction Effects. Friction at the specimen/bar interfaces causes the state of stress to deviate from the uniaxial stress condition and leads to spuriously stiff results [33.18, 19]. As Fig. 33.5 shows, decreasing the specimen length can greatly increase the accessible strain rate range for a given experiment. However, as the l0 /d0 ratio of the specimen becomes smaller, the effects of friction at the interfaces become substantial if the coefficient of friction at the specimen/bar interfaces is not sufficiently low, leading to inaccurate measures of the stress [33.19]. For quasi-static compression tests, specimen l0 /d0 ratios of 1.5 to 3.0 are recommended in various ASTM standards [33.20], with proper use of lubricant. In contrast, for conventional compression Kolsky bar tests, l0 /d0 ratios of 0.5–1.0 are widely used, following Bertholf and Karnes [33.19]. The interfacial friction depends on the smoothness of the end surfaces, the bar and specimen materials and lubricant used, impact velocity and temperature. Commonly used lubricants in Kolsky bar experiments include MoS2 lubricants and lithium greases. Several researchers have measured the friction of lubricants at various strain rates using ring specimens (e.g. [33.21]). These measurements have generally shown that the friction decreases at high strain rates. Dispersion Effects. Longitudinal waves in elastic bars

suffer from geometric dispersion, so that the incident, reflected and transmitted pulses change as they propagate along the input and output bars; however, (33.1–33.13) essentially ignore dispersion. Wave dispersion affects the measured strain pulses, which in

Part D 33.1

Figure 33.6 shows that the maximum strain rate can only be approached an actual Kolsky bar test if the material response is exceedingly soft, so that the observed specimen stress is very close to zero. Conversely, the maximum specimen stress can only be observed in the test if the specimen strain rate is very close to zero. This is a mathematical manifestation of the well-known result that a given test procedure will generate lower strain rates on strong materials, while very soft materials can be deformed at very high strain rates in an identical test. As Fig. 33.6 shows, it is apparent that the easiest way to get a higher maximum strain rate would be to increase the impact velocity or to reduce the specimen length. Similarly, an increase in the maximum observable stress (e.g. when studying very strong materials) can be obtained by increasing the impact velocity or by increasing the area mismatch Ab /As . However, these parameters cannot be varied arbitrarily; restrictions are usually imposed by the requirements for test validity (see, for example, Jia and Ramesh [33.16], outlined next). Other restrictions arise from such issues as the yielding of the bar material (which limits impact velocity) and the sensitivity of the strain gages and signal-to-noise ratio of instrumentation (which determines the smallest stresses that can be effectively measured).

33.1 High Strain Rate Experiments

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Applications

turn affect the measured stress-strain response. Dispersion effects can be rigorously included through elastic wave propagation computations (e.g. [33.22]), but two consequences of dispersion remain important. First, dispersion inevitably induces superimposed oscillations in the loading of the specimen. Second, shorter stress pulses with sharper rise times cause greater dispersion because of the greater high-frequency content and the wider range. It is now possible to routinely correct for the effects of dispersion using elastic-wave propagation calculations, and indeed this must be done for some experiments where information that is present early in the stress-time signal is important. One approach to minimizing the effects of dispersion is to use so-called buffer materials or shaper materials that are placed between the impacting projectile and input bar [33.23]. This results in the shaping of the incident wave with a longer rise time and thus lower dispersion. Inertial Effects. The stresses associated with axial and

Part D 33.1

radial inertia should be small compared with the flow stress of the material under investigation. The stress wave loading that is used in high strain rate experiments causes inertia to have an influence on the measured properties, particularly at very high strain rates [33.24]. The magnitude of the inertial contribution to the apparent stress also depends on the density and size of the specimen [33.16]. Gorham [33.24] (following [33.25]) developed an approach to estimate the inertial effect in Kolsky bar experiments that can be rewritten to emphasize specimen size and l/d ratio effects as follows:     1 1 l0 2 2 2 + σs,2 − σy = ρd ε˙ 64 6 d0     ρl v˙ 1 1 l0 2 2 − − , − ρd ε¨ 32 6 d0 2

second and the third terms, the relative error (r) in flow stress measurement due to inertia is     σs,2 − σy ρd 2 1 1 l0 2 2 ε˙ . =r = + σy σy 64 6 d0 (33.21)

It is apparent that (for a given strain rate) the error drops significantly when the specimen size (length or diameter, given l0 /d0 ) is reduced (this is one driver for miniaturizing the Kolsky bar arrangement). It is also apparent that tests on materials with high flow stress and low density are less prone to such inertial errors, whereas very high density materials and very soft materials will be problematic. A useful way to consider inertial effects is to examine the limiting strain rate that can be attained while keeping the inertial error below some chosen value. For a given value of the relative error r in stress and a given ratio of l0 /d0 , the limiting strain rate can be expressed as a function of the specimen length and the specific strength σy /ρ as ⎛ ⎞

1

σy ⎝ 192 ⎠   , (33.22) ε˙ lim =  r l0 ρ 32 + β32 where β = l0 /d0 . This gives us, for example, ε˙ lim = 1/l0 [(48r)/11]1/2 [(σy )/(ρ)]1/2 when β = l0 /d0 is assumed to be 0.5. Figure 33.7 shows this latter strain rate limit (due to inertia) Limiting strain rate (s–1) 20 000 Copper

Alphatitanium

15 000 L = 1 mm

(33.20)

where σs,2 and σy are the flow stress measured from the output bar and the actual yield stress of the material respectively, ρ is the density of the specimen material and v is the velocity of the interface between the specimen and the output bar. This rearrangement is useful because the l/d ratio is constrained to remain within a fairly narrow range (0.5 ≤ l/d ≤ 1), and so (33.20) allows us to focus on the effect of specimen size through the specimen diameter. In any experiment with a nearly constant strain rate, the magnitude of the second term is much smaller than that of the first term, and the magnitude of the third term is generally small for materials which do not exhibit a very high strain hardening. Ignoring the

Al alloy 6061-T651

10 000

L = 2 mm 5000 L = 4 mm 0 0

5 000

10 000

15 000

20 000

Specific strength (m2/s2)

Fig. 33.7 Limiting strain rates in compression Kolsky bar experiments due to inertial errors in flow stress measurement

High Rates and Impact Experiments

as a function of the specific strength for a value of error r of 3% (variations smaller than 3% are typically not observable in Kolsky bar experiments). Three typical specimen sizes and three commonly used materials are also shown in the figure [33.16]. This is a useful approach to estimating the limiting strain rate in very-high-strainrate experiments for a given specimen size and a known specific strength of the material. This is also helpful for clarifying the sometimes dramatic reported changes of flow stresses in the range of strain rates of ≈ 1 × 103 – ≈ 5 × 104 s−1 , by helping to distinguish valid from invalid tests.

33.1.2 Extensions and Modifications of Kolsky Bars A very large number of extensions and modifications of the traditional Kolsky bar system have been developed over the last five decades. Most of these are listed in Table I of the review by Field et al. [33.1], who include an exhaustive literature set. We discuss some of the key modifications in the rest of this section, focusing on some particular challenges.

or a pulse-shaper material at the projectile/impact-bar interface (this material is typically a soft metal, for example copper) [33.28]. The question of when equilibration occurs remains a matter of some debate. A rule of thumb that is commonly used is that at least five reverberations of the elastic wave should occur before the difference in the stress in the two sides of the ceramic specimen becomes sufficiently small to be ignored. This also generates limits on the effective strain rates that can be developed for valid Kolsky bar testing [33.29]. Note that for the harder hot-pressed ceramics, the specimen strain is extraordinarily difficult to measure because of indentation of the ceramics into the platens. The specimen design also has significant impact on the results obtained in Kolsky bar experiments on ceramics. In a sense, it is possible to view experiments in ceramics in one of two modes. One wishes to measure the properties of the ceramic itself – this is often the original intent. However, one may instead be examining the properties of the ceramic specimen as a structure, since the stress concentrations at the corners and the edges can have such a strong influence on the failure process. In the quasi static mechanical testing of ceramics, researchers have gone to great lengths to optimize the specimen design. This degree of optimization has not yet been accomplished in the dynamic testing of ceramics. Several specimen designs have been proposed and used, e.g. [33.30]. It appears that substantial gains in understanding can be obtained by juxtaposing high-speed photography (Fig. 33.8) of specimen deformation during loading with the recorded stress-time curves [33.27]. The figure demonstrates that, in this polycrystalline ceramic: 1. axial splitting is not the mode of compressive failure and 2. the interaction of flaws is of great importance in dynamic brittle failure under compression. Modifications for Soft Materials The testing of very soft materials (such as polymers and soft tissues) in the Kolsky bar represents the alternate extreme of difficulty in comparison to testing ceramics [33.31]. The two primary difficulties here are

1. the transmitted stress is very small, and so can be measured only with great difficulty, and 2. equilibration of the stress in the specimen can take a very long time [33.32]. There have been substantial improvements in the testing of soft materials over the last two decades, primarily

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Part D 33.1

Modifications for Ceramics There are three major difficulties associated with testing ceramics in the conventional compression Kolsky bar. First, these materials are extremely hard, and thus are likely to cause damage to the bar ends. Second, these materials do not develop substantial plastic strain, so that failure will be caused essentially in the elastic domain and specimen strain can be very difficult to measure. Third, since these materials are extremely brittle, the specimen design can easily introduce defects that will cause premature failure, raising the question of whether the material properties can be measured. An excellent summary of these issues is presented in the ASM Handbook article by Subhash and Ravichandran [33.26]. The first of these difficulties can be somewhat mitigated by using special inserts (called platens) at the end of the bars that are made of even harder ceramics, e.g. [33.27]. The second and third difficulties are much harder to address. The fact that failure occurs immediately after the elastic domain means that it becomes very important to ensure that stress equilibration has occurred in the specimen, and this cannot be done with the traditional trapezoidal pulse. The best way to resolve this issue is to use a controlled rise time for the incident pulse, and to ensure that equilibration occurs well before failure develops. The incident pulse shape is controlled [33.23] by using a buffer material

33.1 High Strain Rate Experiments

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Applications

1

5.2 mm

2

3

4

6

7

8

2.3 mm

5

Fig. 33.8 High speed photographs of the development of microcracks in a transparent polycrystalline ceramic (AION)

under dynamic compression along the horizontal axis [33.27]. The interframe time is 1 μs and each exposure time is 20 ns

Part D 33.1

involving the use of polymer or magnesium bars with small elastic moduli, so that small stresses can still generate large enough strains to measure [33.33]. The use of semiconductor strain gages instead of foil gages has also provided significant improvement [33.33]. The equilibration question is addressed sometimes by the direct measurement of force using piezoelectric quartz gages [33.34]. All of these approaches require great care with both experimental setup and data analysis [33.35]. One particularly important question that must be explicitly addressed for each material is that of material incompressibility. Modifications for High Temperature Testing The most common approach to running high temperature Kolsky bar experiments is to simply heat the specimen while it is in contact with the input and output bars (typically inside a tube furnace), allowing a strong temperature gradient to develop within the bars. If an estimate of the temperature gradient can be made, its effect can be compensated for by making the proper adjustments to the wave-speeds due to the temperaturedependent elastic modulus of the bar material [33.36]. Such techniques have limitations on the attainable specimen temperature that are imposed by heat conduction in the bars and by the properties of the bar materials. One solution to this problem has been to provide an insulating layer between the specimen and bars in the form of a short length of impedance-matched ceramic bar [33.37]. A variant of this technique has been used to perform high temperature torsion tests [33.38]. One method of providing rapid heating in such an

arrangement (for conductive specimens) is induction heating [33.37]. A second, more modern approach to running high temperature experiments in the Kolsky bar is to heat the specimen alone, and then to bring the bars into contact with the specimen just before impact [33.39]. This setup has the advantage of reducing temperature gradients in the bars, and the added benefit of requiring relatively small energy inputs to heat the specimen. The major design issue in such a system is the cold contact time – the duration of time over which the input and output bars are in contact with the specimen before the arrival of the incident wave. During this contact time, thermal gradients develop within the specimen, and the overall specimen temperature drops by an amount that is typically not recorded. Various arrangements have been used to reduce this cold contact time, e.g. [33.39] and [33.40]. Approaches to heating the specimen include RF heating and radiative heating [33.39]. Conductive heating and pulse heating approaches have also been used to develop very large heating rates in metallic specimens [33.41]. Tension Kolsky Bars A version of the Kolsky bar system that is effective in tension was developed in the 1960s [33.42]. The basic principle of this system is identical to the compression system, except that a method for generating a tensile wave must be used, and special tension grips are required for the specimen. There are two basic approaches to generate a tensile wave in the input bar. The first approach, known as an impact-tension approach, con-

High Rates and Impact Experiments

pulse provides the strain rate history (which can then be integrated over time to obtain the specimen strain). Multiple strain gauge stations may be needed in order to separate out the incident and reflected pulses. The equations are identical to those used for the compression Kolsky bar, with the specimen length replaced by a gauge length on the tensile specimen. A full tensile stress-strain curve is obtained at fairly high strain rates (on the order of 102 –103 s−1 using this system). However, the traditional approach to obtaining the specimen strain in the tension Kolsky bar provides an inaccurate measure of strain, in general. This is because it measures the relative displacement of the two specimen-bar interfaces, and converting this to specimen strain requires an assumption of a specimen gauge length (this is very well known in compression, but is not well defined in tension). Since fillets are required in tensile specimens in order to avoid failure due to stress concentrations at the specimen flanges, the non-uniform deformation developed at the flanges contributes to the total deformation of the specimen. It is therefore difficult and at times impossible to determine the effective gauge length to be used in the calculation of specimen axial strains [33.42]. Ideally one should have a strain-gage mounted directly on the specimen to provide measures of strain throughout the experiment. One alternative approach is to use a high-speed camera to directly measure the specimen diameter and length during an experiment. A second alternative is to use the Laser Occlusive Radius Detector (LORD) technique to directly determine the L0

D1-20-UNF

D0 R L2

L1

L2

Strain gauge stations

Axial actuator

Input bar

Clamp

1

Specimen

2 3048

Output bar

3 1524

Fig. 33.9 Schematic of the direct-tension Kolsky bar. The inset shows an example of a threaded specimen

939

Part D 33.1

sists of firing a tubular projectile at a flange at the end of a bar, thereby generating a tensile wave within the bar. A variant of this approach uses a reaction mass and causes the reflection of a compressive wave as a tensile wave [33.43]. The second, and less common, approach consists of storing a large tensile strain within a section of a bar and then suddenly releasing it is so as to generate a tensile wave (this is known as the direct-tension approach) [33.44]. The gripping of tensile specimens requires some care, because otherwise the spurious wave reflections from the grips would make it difficult to interpret the experiments. Commonly used grips include threaded ends and adhesively bonded ends. A general rule of thumb is that if the gripped ends are made of the same material as the bar and are no longer than the bar diameter, then the effect on the wave propagation will be minimal. An example of the direct-tension Kolsky bar is shown in Fig. 33.9 [33.45]. A tensile force is applied to a section of the input bar using the hydraulic actuator system, while the rest of the input bar is restrained using a friction clamp. When the friction clamp is suddenly released, a tensile pulse propagates down the input bar and loads the specimen in tension; part of this incident pulse is transmitted into the output bar, and part is reflected back to the input bar. Strain gages are mounted on the input and output bars to measure elastic strain in the bars. Using these strain gages, the incident, reflected and transmitted pulses in the bars can be measured. As before, the transmitted pulse provides the stress history of the specimen, while the reflected

33.1 High Strain Rate Experiments

940

Part D

Applications

local specimen strains during high-strain-rate experiments [33.45]. This approach requires the assumption of incompressibility in the specimen in order to estimate the axial specimen strain (an assumption that is typically not valid once necking begins). However, it is capable of measuring fairly large plastic strains, which are generally not measurable using standard onspecimen strain gauges because they debond during the deformation.

Part D 33.1

Torsional Kolsky Bars The torsional Kolsky bar is an adaptation of the compression Kolsky bar that is used to generate high strain rate shearing deformations within typically tubular specimens [33.46]. An excellent summary of the technique is presented by Hartley et al. [33.47]. A schematic of such a device is presented in Fig. 33.10. The principle of operation of this bar is identical to that at the tension Kolsky bar, except that a torsional wave is now propagated down the input bar towards the specimen. One advantage of the torsional Kolsky bar is that primarymode torsional waves are not dispersive in the bar, so that dispersion corrections are not necessary. A second advantage of this system is that (in principle) very large shearing strains can be sustained and so material behavior can be studied over very large strain regimes. That is, the difficulties associated with maintaining friction during large deformations (in the compression bar) and avoiding necking (in the tension bar) are avoided in the torsional bar system. Further, radial expansion and contraction are not significant in torsion of isotropic materials, so that many of the inertial and end effects present in the tension and compression tests are absent in the torsional Kolsky bar experiment. However, shear bands may develop during torsional deformations: in Torquing device

Clamp

Input bar

fact this is one of the most common applications of the torsional bar system [33.48]. A significant experimental concern in the setup of the torsional Kolsky bar system is the avoidance of bending waves. Flexural waves in a long bar may have signal content propagating at speeds similar to that of the torsional wave, resulting in some difficulties in wave analysis. It is particularly important that the friction clamp design not generate flexural waves within the bars – note in particular that the use of heavy friction clamps is not an advantage. Careful placement of the supports and specific initial measurement of the propagation of bending waves within the bar is warranted. The difficulties associated with specimen gripping are also an issue with the torsional Kolsky bar design. Typical approaches to gripping the specimen include the use of adhesives, asymmetric flanges, and various types of compression grips. Specimen design is critical, with the usual questions associated with the definition of specimen gauge length and definition of fillet radii. It is recommended that a full finite deformation analysis of the plastic deformation of a specimen be performed before using the given specimen design with an experiment. The equations associated with the use of the torsional Kolsky bar system are different in detail, but not in character, from those used for the compression Kolsky bar system. A full description of the relevant equations is presented by Hartley et al. [33.47]. The primary distinctions arise because of the torsional mode: one must consider torques and angular velocities rather than axial forces and translational velocities. We note that a response diagram very similar to Fig. 33.6 may also be constructed for the torsional Kolsky bar, and may be used in a very similar way to accomplish

Specimen

Output bar

Fig. 33.10 Schematic of the torsional Kolsky bar. Note the location of the friction clamp with respect to the supports

High Rates and Impact Experiments

experimental design. Examples of shear stress versus shear-strain curves obtained using the torsional Kolsky bar are presented in Fig. 33.15 for the aluminum alloy 6061-T6. The torsional Kolsky bar system has also been modified to perform high-temperature testing at high shear strain rates [33.38]. In this approach the necessity of using grips on the specimen can be turned into an advantage, with the grips acting as thermal insulators and reducing the heating of the bars themselves.

33.1.3 The Miniaturized Kolsky Bar

1. assuming the projectile is effectively rigid [33.49] and 2. measuring the velocity of the back surface of the projectile [33.50].

Fig. 33.11 Photograph of a miniaturized Kolsky bar (total

length of 30  )

Gorham [33.51] obtained strain rates of 4 × 104 s−1 in direct impact using a high-speed camera for strain measurement, and even higher rates were obtained in a similar configuration by Safford [33.52]. Gorham et al. [33.53] and Shioiri et al. [33.54] obtained the strain in direct impact experiments by assuming that stress equilibrium is satisfied throughout the deformation; strain rates of 2–4 × 104 s−1 are claimed. Kamler et al. [33.55] developed a very small (1.5 mm dia.) direct impact Kolsky bar and performed experiments on copper at very high strain rates of 6 × 103 to 4 × 105 s−1 ; again, stress equilibrium was assumed. In the third approach, several fully miniaturized versions of the compression Kolsky bar have been developed (e.g., see Jia and Ramesh [33.16]). In the Jia and Ramesh version (see Fig. 33.11), the bars are on the order of 3 mm in diameter and 30 cm long, and may be made of maraging steel or tungsten carbide. Sample sizes are cubes or cylinders on the order of 1 mm on a side; cube specimens are used when the failure mode must be imaged using a high-speed camera, or when the amount of material is so small that only cuboidal specimens can be cut. Very high strain rates (up to 5 × 104 s−1 ) can be attained in miniaturized systems, while retaining the ability to study materials at strain rates as low as 1.0 × 103 s−1 (the maximum achievable strain-rate is limited by an inertial correction and varies with the material being tested). Both computational and experimental results have shown that this extended capability can be attained without violating the requirements for valid high-rate testing, and indeed while improving the quality of the experimental results in terms of precision and accuracy [33.16]. The technique is simple, and the entire system can be designed to fit on a desktop – the technique is therefore sometimes referred to as the desktop Kolsky bar. Equilibration of stresses can be rigorously checked from both sides of the specimen in the experiment even during the highest rate deformations that can be attained. Both experiments and simulations show that rapid equilibration of the stress can be achieved even at very high strain rates. However, effective lubrication remains critical. The miniaturized system makes it possible to reach a very high strain rate in a small sample with a relatively high l/d ratio, thus minimizing the influence of friction. Inertial errors in stress measurement are significantly reduced with the smaller specimens. A comparison of the response measured by this miniaturized technique with the results of the traditional compression Kolsky bar system is presented in

941

Part D 33.1

Attempts to push the Kolsky bar arrangement to higher strain rates have led to three modifications: decrease of the specimen length, direct impact on the specimen (thus increasing effective velocity), and miniaturization of the entire system. The first approach, with smaller specimen sizes, is typically limited by frictional effects. The second approach is the so-called direct impact technique: a projectile directly impacts a specimen placed in front of an elastic bar (which operates like the output bar in the regular version of the Kolsky bar). Since there is no input bar, the impact velocity can be very high. However, because of the absence of an input bar, there is no reflected signal from which the strain rate and strain in the specimen can be extracted. Strain measurement in the direct impact approach has been addressed in several ways, including

33.1 High Strain Rate Experiments

942

Part D

Applications

Flyer

Flow stress (MPa)

Anvil

600 500

Desktop Kolsky bar (l /d = 0.55) Desktop Kolsky bar (l /d = 1.1) Conventional Kolsky bar (l /d = 0.55)

V0

400

u0

υ0

TDI Incident beam NVI

300

TDI

200

Specimen

100 0 103

104

105

Strain rate (s–1)

Fig. 33.12 Comparison of results obtained using the minia-

turized and traditional compression Kolsky bars. The material is 6061-T6 aluminium alloy

Part D 33.1

Fig. 33.12 for 6061-T6 aluminum alloy. This figure shows the rate-sensitivity over an extended range of strain rates that is not attainable with the traditional technique. Even higher-strain-rates can be attained using the pressure-shear plate impact experiment, described et seq. The miniaturized Kolsky bar or desktop Kolsky bar experimental technique is particularly effective when it comes to understanding the properties of nanostructured materials, or of materials with very fine grain sizes. Note, however, that the small specimen sizes can make it difficult to use this approach with materials (such as solders) that have large grain sizes or a large microstructural length scale.

33.1.4 High Strain Rate Pressure-Shear Plate Impact The high strain rate pressure-shear (HSRPS) plate impact technique was developed to study the shearing behavior of materials undergoing homogeneous shearing deformations at extremely high shear rates of 104 –106 s−1 , while under superimposed hydrostatic pressures of several GPa. A detailed description of this technique is provided by Klopp and Clifton [33.56]. The experiment involves the impact of plates that are flat and parallel but inclined relative to their direction of approach (Fig. 33.13). The specimen consists of a very thin (of the order of 100 μm thick), and very flat plate of the material under investigation. This thin specimen is bonded to a hard plate (the flyer), itself carried

Fig. 33.13 Schematic of the high-strain-rate pressure-shear plate impact experiment. The specimen thickness is graetly exaggerated for clarity

on a projectile which is launched down the barrel of a gas gun towards a stationary target or anvil plate. The projectile velocity (V0 ) at impact may be decomposed into a normal component u 0 = V0 cos θ and a transverse component v0 = V0 sin θ, where θ is the angle between the normal to the plates and the direction of flight of the projectile. The target is positioned in a special fixture (known as the target-holder) within an evacuated chamber. The flyer and the anvil plates are aligned before impact using an optical technique described by Klopp and Clifton [33.56]. Rotation of the projectile is prevented by a Teflon key in the projectile, which glides within a matching keyway machined in the barrel. The misalignment (tilt) between the two impacting plates is obtained by measuring the times at which the flyer makes contact with four pins that are mounted flush with the impact face of the target. Tilts better than t

Thin specimen

Flyer

Anvil TDI window NVI window

x

Fig. 33.14 Wave propagation diagram for high-strain-rate pressure-shear plate impact, with the specimen carried on the flyer

High Rates and Impact Experiments

1 milliRadian are routinely achieved in these experiments. At impact, plane longitudinal (compressive) and transverse (shear) waves are generated in the specimen and the target plate (Fig. 33.14) propagating at the longitudinal wave speed cl and the shear wave speed cs . These waves reverberate within the specimen, resulting in a buildup of the normal stress and of the shear stress within the specimen material. Information on the stress levels sustained by the specimen material is carried by the longitudinal (normal) and transverse waves propagating into the target plate. Since the target remains elastic, there is a linear relationship between the stresses and the particle velocities in the target plate. Thus it is sufficient to measure the normal and transverse particle velocities in the target plate to deduce the stress state and deformation state within the specimen. The entire experiment is completed before any unloading waves from the periphery of the plates arrives at the point of observation, so that only plane waves are ina) Normal stress σ

Flyer states

ρ cl ρ cl

volved and a one dimensional analysis is both sufficient and rigorously correct. Like most plate impact experiments, this is a uniaxial strain experiment in that no transverse normal strains can occur during the time of interest. Measurements of the particle velocities at the free surface of the target plate are made using laser interferometry off a diffraction grating that is photodeposited onto the rear surface. The normal velocity and the transverse displacement at the center of the rear surface of the target are measured using a normal velocity interferometer (NVI) and a transverse displacement interferometer (TDI), e.g. [33.57]. The laser used is typically either an ion laser or a diode-pumped, frequency doubled, Nd:YAG laser. The interferometric signals are captured by high speed silicon photodiodes (with rise times < 1 ns) and recorded on high-speed digitizing oscilloscopes, using typical sampling rates of 1 ns/point and bandwidths of 500 MHz – 1 GHz. The longitudinal and transverse waves generated on impact (Fig. 33.14) reverberate within the specimen, remaining loading waves since the impedance of the material of the flyer and target plates is (by design) higher than that of the specimen materials. The development of the stress in the specimen is most easily seen in the characteristics diagram of Fig. 33.15. The characteristic relations (σ ± ρcl u = const. or τ ± ρcs v = const., with σ and τ the normal stress and shear stress and u and v the normal and transverse particle velocities respectively; ρ is the density of the plate material) in the linear elastic plates give us −σ + ρcl u = ρcl u 0 , τ + ρcs v = ρcs v0 ,

u0

b) Shear stress τ

in the flyer and −σ + ρcl u = 0 , τ + ρcs v = 0 ,

Anvil states

ρ cs

Flyer states

τf

υA

υfs

υF

υ0

Fig. 33.15a,b Characteristic diagram for high-strain-rate pressure-shear plate impact. (a) Development of normal stress and (b) shear stresses in the specimen

(33.23)

(33.24)

in the anvil. Thus all states in the flyer will lie on the solid line passing through u 0 (for the normal stress, Fig. 33.15a) or v0 (for the shear stress, Fig. 33.15b), while all states on the anvil will lie on the solid line passing through the origin. In the case of the normal stress (Fig. 33.15a), the normal stress in the specimen must always lie along the characteristic lines shown by the dashed lines. The state of the specimen at any given time is given by the intersection of the characteristic line of specimen with either the characteristic line of the anvil or the characteristic line of the flyer. If there is a difference in the normal velocities on the two sides

943

Part D 33.1

Anvil states

33.1 High Strain Rate Experiments

944

Part D

Applications

of the specimen (the flyer side and the anvil side), then the specimen will continue to be compressed in the normal direction. However, the plate impact experiment is a uniaxial strain experiment rather than a uniaxial stress experiment like the Kolsky bar. The continued compression along the normal direction thus implies a change in volume. Due to the finite compressibility of the specimen, the normal stress in the specimen attains an equilibrium value (corresponding to the intersection of the flyer and anvil characteristics) given by 1 σeqm ∼ = ρcl V0 cos θ , 2

(33.25)

Part D 33.1

where ρcl is the acoustic impedance of the steel, V0 is the projectile velocity and θ is the skew angle. At equilibrium, the hydrostatic pressure in the specimen can be approximated to the normal compressive stress, differing from it by at most the strength of the specimen material. Note that the difference in normal velocities on the two sides of the specimen continuously decreases. Thus the normal strain-rate in the high-strain-rate pressure-shear plate impact experiment is initially high but tends towards zero, actually approximating zero when normal stress equilibrium has been achieved. This is again a result of the uniaxial strain condition. The shear stress in the specimen also increases with each reverberation of the shear wave within the specimen (Fig. 33.15b), until the specimen starts flowing at a stress level τf (the flow stress). Thereafter, a finite difference in the transverse velocity can be maintained across the two surfaces of the specimen (unlike the normal velocity situation, it is possible to sustain a difference in transverse velocities across the specimen because the uniaxial strain condition does not present a constraint in this case). The velocity difference vF − vA also corresponds to (see Fig. 33.15b) the velocity difference v0 − v fs , where v fs is the transverse free-surface velocity measured at the rear surface of the target using the TDI. The nominal shear strain rate in the specimen is then given by V0 sin θ − v fs , γ˙ = h

(33.26)

t γ˙ (τ) dτ . 0

1 τ(t) = ρcs vA = ρcs v fs (t) , 2

(33.27)

(33.28)

where ρcs is the shear impedance of the anvil plate. The shear stress and the shear strain in the specimen can now be cross-correlated to give the shear stress-shear strain curve (Fig. 33.16) for the material tested at an essentially constant shear rate corresponding to (33.26). Examination of (33.25)–(33.28) shows that the critical measurements that must be made during the experiment are of the projectile impact velocity V0 and the transverse free-surface particle velocity vfs . The latter particle velocity is measured using the transversedisplacement-interferometer [33.57]. The projectile impact velocity can be measured in a number of different ways. These include measuring the times when the projectile comes in contact with specific pins placed a known distance apart; measuring the times when the projectile crosses two magnetic or radiofrequency sensors or laser beams that are spaced a known distance apart; or using a Laser Line Velocity Sensor [33.58]. Finally, the misalignment (tilt) between the flyer platespecimen combination and the anvil plate must be measured for every test, both for purposes of diagnostics and to remove small components of the normal velocity that may show up within the transverse velocity signal due to the misalignment. Shear stress (MPa) 400 350

High-strain-rate pressure-shear plate impact

300 250 200

Torsional Kolsky bar

150

650 s–1 850 s–1 9.5×104 s–1 2.3×105 s–1

100

where h is the specimen thickness. The nominal shear strain rate in the specimen can be integrated to give the nominal shear strain history in the specimen: γ (t) =

The shear stress history in the specimen is obtained from the transverse particle velocity history using the elastic characteristics of the target plate (Fig. 33.14b)

50 0

Al 6061-T6: Shear

0

0.05

0.1

0.15

0.2 Shear strain

Fig. 33.16 Shear stress versus shear strain curves for 6061-T6 aluminium alloy obtained using high-strain-rate pressure-shear plate impact, compared with those obtained usinf the torsional Kolsky bar

High Rates and Impact Experiments

The high strain rate pressure-shear plate impact technique is capable of achieving shear rates of 8 × 104 to 106 s−1 , depending on the specimen thickness, [33.57–59]. A version of this experiment that is designed to allow recovery of the specimen (for microstructural examination) after a single high-strain-rate shear loading has also been developed [33.60]. The primary equations (33.25)–(33.28) can be rewritten to provide a locus of test response, as presented for the Kolsky bar in Fig. 33.6. Here we find that the test requires that the shear stress and the shear strain rate obey the relationship τ = 0.5ρcs (V0 sin θ − γ˙ h), representing the test response line. The maximum shear strain rate that can be attained is γ˙max = V0 sin θ/h (and corresponds to τ = 0), while the maximum shear stress that can be attained is τmax = 0.5ρcs V0 sin θ (and corresponds to the apex of the characteristic triangle in Fig. 33.15). The specific shear stress and shear rate

33.2 Wave Propagation Experiments

945

obtained in a given test is the intersection of the material response curve with the test response line, as in Fig. 33.6. The superimposed hydrostatic pressures that can be exerted during the high-strain-rate pressure-shear plate impact experiment may be as high as 10 GPa, depending on the impedances of the flyer and target plates and the projectile velocity. The superimposed hydrostatic pressures must always be remembered when comparing high-strain-rate pressure-shear plate impact data with data obtained using the other techniques shown in Fig. 33.1, since all of the other techniques can generate essentially uniaxial stress states, typically corresponding to low hydrostatic pressures. In particular, while the effect of pressure on the flow stress of most metals is negligible in comparison with the effect of strain-rate, the effect of pressure on the strength of polymers and amorphous materials may be substantial (even in comparison with the effect of strain-rate).

33.2 Wave Propagation Experiments a fundamental feature of large amplitude wave propagation in materials: in the timescales associated with the wave propagation, it is typically not possible to observe the far-field stress state, and so locally one is typically exploring the uniaxial strain condition. In other words, some local confinement is an inherent characteristic of large amplitude wave propagation in materials as a result of impact. This can make it difficult to compare results obtained at ultra-high strain rates (typically obtained with uniaxial-strain experiments) with results obtained at high and very high strain rates (typically obtained with uniaxial stress experiments), particularly if the material has pressure dependent properties. The strain rates developed in large-amplitude wave propagation experiments (where shocks are developed) can be of the order of 106 –108 s−1 , but exist only for a short time behind a propagating wave front, and because of inelastic dissipation (as well as reflections from surfaces), the strain rates will vary with position within an impacted plate. The temperatures behind the wave front may be substantial, and must be accounted for as well. Comparisons of material properties estimated using wave propagation experiments and high-strain-rate experiments (the distinction made in this chapter) can therefore require careful parsing of experimental conditions.

Part D 33.2

Experiments designed to study the propagation of large amplitude stress waves within materials constitute a very broad class of impact experiments. Note that we do not include in this category those experiments that are designed to study the propagation of waves within structures - such experiments are better considered in discussions of structural dynamics or of elastic wave guides (see photoacoustic characterization chapter). Our interest here is in experiments that examine the interactions of waves with materials, particularly exciting inelastic modes such as plasticity, cracking or other kinds of damage. In contrast to the previous section, the experiments in this section all generate strain rates and stress states that vary in both space and time, and the wave propagation is fundamentally dispersive because of material behavior. In broad terms, wave-propagation experiments of this type fall into the same two categories considered in the previous section: bar wave experiments [33.61– 66] and plate impact experiments, or more specifically, uniaxial-stress wave propagation experiments and uniaxial-strain wave propagation experiments. The plate impact experiments are far more commonly used, since they can explore a wider range of the phenomena that arise in impact events, and so we focus on such experiments here. It is perhaps worth pointing out

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33.2.1 Plate Impact Experiments

Part D 33.2

The basic concept of the plate-impact experiment has been touched on in the description of the high-strainrate pressure-shear plate impact experiment above. A flyer plate is launched down a gun barrel towards a stationary target plate, typically using a gas gun for launch. The flyer plate is normally carried on a projectile known as a sabot, and is launched at velocities ranging from a few tens of meters per second up to several kilometers per second using a variety of gas and powder guns. The flyer plate may impact the target plate at normal incidence, resulting in what is called a normal plate impact experiment. An oblique plate impact experiment consists of an impact that occurs at an angle as described in the pressure-shear plate impact experiment. The vast majority of plate impact experiments that are performed today are normal plate impact experiments, largely because oblique plate impact experiments typically require a keyed gun barrel to prevent rotation of the projectile during flight. We consider only normal plate impact experiments in the remainder of this section. A schematic of the simplest form of the normal plate impact experiment is shown in Fig. 33.17. The minimum diagnostics that are typically obtained consist of measurements of the projectile impact velocity and the rear surface particle velocity. Once the impact occurs, normal uniaxial-strain compressive waves will be propagated in both the flyer and the target. The case where these waves are elastic is not interesting. Typically the target plate is expected to deform inelastically during the experiment, and in most experiments both flyer and target will do so. If the impact velocity is sufficiently high, one or both of the propagating waves within the plates will be inelastic. At velocities just above the velocity required to cause yield of the target material, both elastic and plastic waves will propagate into the target, with the elastic precursor propagating at a higher wave Projectile velocity υ

Target plate

Interferometry

Sabot

Flyer

Fig. 33.17 Schematic of the normal phase impact experiment, including the sarbot

speed. Significantly higher impact velocities will result in the propagation of shock waves into the plates. The propagation of plastic waves within materials is a topic of some interest; however, the majority of plate impact experiments are conducted at velocities sufficiently high to generate shock waves within the plates, and so we will describe shock wave experiments in some detail. It should be noted that the strain state in normal plate impact experiments is the uniaxial strain state only for a finite time. Once release waves from the boundary of the plates arrive at the point of interest, the strain state is no longer uniaxial-strain, and much more complex analyses are necessary. However, before the release waves arrive at the point of interest (for instance, the location at which particle surface velocities and displacements are being measured using interferometry), the strain state is rigorously one-dimensional and can be analyzed much more easily. Longer experimental times in the uniaxial-strain condition can be obtained by using larger diameter plates, but the increased cost of performing such experiments with large diameter plates is substantial. Typical experiments are performed with plate diameters on the order of 50 mm. Shock Wave Experiments In any uniaxial strain wave, there is always a difference between the normal stress σa (the stress along the direction of wave propagation) and the transverse stress σt (the stress transverse to the direction of wave propagation, which arises because of the uniaxial-strain constraint). The difference between these two stresses can be considered to be a shear stress and is of the order of magnitude of the deviatoric strength (yield strength for metals) of the material. The difference in the amplitude σa of the stress wave and the propagating pressure ( p = (σa + 2σt )/3, all stresses defined as positive in compression) is (2/3)(σa − σt ). Since this latter quantity is of the order of magnitude of the flow stress, the difference between the longitudinal stress and the pressure is of the order of the strength of the material. When the amplitude of the stress wave is extremely large in comparison to the strength of the material, the stress wave may be approximated as a shock wave, and treated essentially as a problem in hydrodynamics. A shock wave is a sharp discontinuity in pressure, temperature energy and density (more specifically, a shock wave is a traveling wave front across which a discontinuous adiabatic jump in state variables occurs). There are several excellent reviews of shock wave propagation ([33.67, 68]). The application of the balance of mass, balance of momentum and balance of energy conditions

High Rates and Impact Experiments

to the shock wave gives us the so-called Rankine– Hugoniot jump conditions across the shock [33.67]. These are: ρ0 Us = ρ1 (Us − u p ) , P1 − P0 = ρ0 Us u p , e1 − e0 = 1/2(P1 + P0 ) × (v0 − v1 ) .

Mass : Momentum : Energy :

(33.29) (33.30)

(33.31)

Here the initial state is denoted by the subscript 0 and the final state is denoted by the subscript 1; v represents the specific volume (the reciprocal of the density). In addition, an equation of state must be prescribed for each material that connects the specific internal energy of the material to the pressure, temperature and density. A shock wave generated during a plate-impact experiment propagates at a shock speed Us that varies with the particle velocity u p , and it is commonly observed that these two variables are related linearly or nearly so: Us = U0 + su p ,

(33.32)

Pressure Hugoniot curve

PB = ρ0B U0B u pB + ρ0B sB u 2pB

Final state

P

from the biannual meetings of the American Physical Society Topical Group on Shock Compression of Condensed Matter published by the American Institute of Physics [33.70–74] and a series on the Shock Compression of Solids published by Springer [33.75–79]. Experimental details are often only presented in these conference proceedings, and so the reader is advised to examine these books carefully. The intent of most shock wave experiments is to measure a so-called Hugoniot curve for a material. A Hugoniot curve (Fig. 33.18) is the locus of all possible final states (e.g., in a pressure-specific volume space) attainable by a single shock from a given initial state. We note that such a curve couples the material response with the thermodynamics, and does not represent a true thermodynamic path. The shock speed Us is given by the slope of the Rayleigh line shown in Fig. 33.18, and is thus clearly dependent on the particular final state. As an example, consider the shock wave analysis of the normal impact of two identical plates in the schematic experiment shown in Fig. 33.17. We assume that the impact velocity is V0 and that there is no unloading from the edges during the time of interest. At the moment of impact shock waves are generated in both the target and the flyer plates. What are the pressure and velocity states in the two plates after the shock? The initial conditions are that the target plate is initially at rest, while the flyer plate is initially moving at the impact velocity (Fig. 33.19). The interface condition is one of traction continuity and velocity continuity at the A-B interface between the two plates. Applying (33.30) and (33.32) on each side of the interface, we obtain (33.33)

in the target and PA = −ρ0A (U0A + sA u pA + V0 )(u pA − V0 ) (33.34)

Rayleigh line

a) Flyer

Target

b)

t

Initial state

P0

A B υ

υ0

u0 =V0

Specific volume

Fig. 33.18 Schematic of a Hugoniot curve from a shock

experiment showing how the initial and final states are connencted by a Rayleigh line

u0 = 0

x

Fig. 33.19 (a) Schematic of symmetric impact shock wave experiment. (b) First phase of shock wave propagation in impacted plates

947

Part D 33.2

where U0 and s are material-specific parameters (the first is essentially the sound wave speed in the material). Large numbers of experiments have been performed to determine these parameters in various materials (e.g., a summary of such data is presented by Meyers [33.68] Table 4.1). Another useful reference is that by Gray [33.69]. The shock wave propagation literature is extensive, including a large number of conference proceedings

33.2 Wave Propagation Experiments

948

Part D

Applications

in the flyer, with the subscripts A or B used to indicate variables in the target or flyer respectively. Thus the states at A and B (the pressure and velocity states immediately after the shock has been generated) are given by the solutions of (33.33) and (33.34). The solution is shown graphically in Fig. 33.20; note the similarity to the characteristics diagram in Fig. 33.15 for the highstrain-rate pressure-shear plate impact experiment. This approach is sometimes called the impedance matching method for determining the solution. Problems involving asymmetric impact can be solved in exactly the same way. We note that the release waves (reflections from the free surfaces) are somewhat more complicated, since release does not occur along the Rayleigh line but along the isentrope instead. Detailed approaches to the solution of specific shockwave problems are presented in the books by Drumheller [33.67], Meyers [33.68] and Zukas [33.80]. The major experimental issues associated with shockwave plate impact experiments are

Part D 33.2

1. the development of gun launching facilities at the appropriate velocities; 2. the accurate measurement of projectile velocity; 3. the measurement of the stress state within the specimen; and 4. the measurement of the particle velocities in the targets. We do not discuss the first of these issues in any detail: such gun-launch facilities are typically extremely specialized facilities made by a small number of companies and national laboratories, and extraordinary precautions must be taken to ensure the safety of all laboratory perPressure

A/B Target states (33)

Flyer states (34)

A or B

υ0 Particle velocity

Fig. 33.20 Solving for the initial state in the symmetric

impact shockwave experiment

sonnel. Most of these facilities are either gas guns, light gas guns or powder guns, and for the higher velocities multistage guns are typically required. Since kinetic energy increases with the square of the velocity, reaching higher velocities typically requires the use of lowermass sabots and flyers. Velocities greater than 10 km/s have been achieved with ≈ 1 g flyers using multi-stage guns at some national laboratories [33.83]. The second of these issues, the accurate measurement of projectile velocities, has already been dealt with in the section on high-strain-rate pressure-shear plate impact. The last two issues, the measurement of the stress state and the measurement of the particle velocity at these very short time scales are discussed in this section. There are several quantities that are of interest in a shock wave plate impact experiment. These are the normal stress, the transverse stress, the shock speed, the particle velocity in the direction of wave propagation (not all other particle velocity components are zero), the pressure and the temperature. Of the normal stress, the transverse stress and the pressure, only two of these have to be measured independently. Note that if the shock impedance of the material is known, then measurements of the particle velocity can be used to infer at least one of the stresses. Thus one needs in general to measure two stresses, the shock speed, one particle velocity and the temperature. All of these need to be measured with nanosecond resolution over time frames of several microseconds. Table 33.1 (adapted from [33.1]) lists some typical ways in which these quantities can be measured. Excellent summaries of experimental techniques for the measurement of these quantities are provided in the recent review by Field et al. [33.1] and in a chapter by Meyers [33.68]. The installation of stress gauges within specimens (stress gauges must of necessity be placed within specimens rather than on free surfaces) is a difficult task to perform consistently, and individual laboratories have developed specialized ways of performing these installations. Some of the difficulties associated with the use Table 33.1 Measurement approaches (after [33.1]) Variable of interest

Measurement approach

Normal stress Transverse stress Particle velocity Shock speed Temperature

Stress gauges Stress gauges (lateral gauges) Interferometry (e.g. VISAR) Velocity history or stress history Pyrometry [33.81], spectroscopy [33.82]

High Rates and Impact Experiments

of stress gauges are described by Gupta [33.84]. The use of lateral stress gauges to measure transverse stresses appears to be particularly difficult, and remains uncommon. However, excellent measurements of strength have been obtained using this technique [33.85]. The stress gauges that are used are either of the piezoresistive or piezoelectric types. Typical piezoresistive gages are manganin [33.86] or ytterbium [33.87] stress gauges. Manganin gauges are perhaps the most commonly used, being cheap and commercially available for use up to 40 GPa. Typical piezoelectric stress gauges include quartz [33.88] and PVDF [33.89], although PZT and lithium niobate gauges are also used. PVDF gages are typically used for stress levels up to 1 GPa, while quartz gages may be used up to 5 GPa. The particle velocity on the rear surface of the target is typically measured using laser interferometry. The rear surface may be a free surface, or may be interro-

33.4 Dynamic Failure Experiments

949

gated through a transparent window material. The most common approach to performing such interferometry is using the velocity interferometer system for any reflector (VISAR) developed by Barker et al. [33.90–92]. Versions of the system that use fiber optic probes, and versions that can provide data from multiple points (the so-called Line VISAR) have been developed; the latter can be used to look at heterogeneous wave structures [33.93]. Most shockwave experiments are designed to provide a one-dimensional strain state, so that single-point measurements should be representative of the events. However, some of the phenomena that develop during shock wave propagation lead to heterogeneous structures. These include the development of failure waves and spallation. In such circumstances, it is a distinct advantage to be able to obtain full field information using high-speed photography (this is most useful for transparent materials) [33.94].

33.3 Taylor Impact Experiments sired velocity, and the ability to measure the deformed shape of the rod. However, the experiment has the disadvantage that it essentially represents an inverse problem: one must try to determine material properties on the basis of the macroscopic measurement of an inhomogeneous deformation. Several improvements have been made to this experiment, including the use of high-speed photography to measure the process of deformation rather than simply the final deformation, and techniques for measuring the internal hardness variation within sections of the deformed bar. An excellent review of the Taylor impact experiment is provided by Field et al. [33.1].

33.4 Dynamic Failure Experiments One must distinguish between the kinds of experiments needed to understand the deformation of a material and the kinds of experiments needed to understand a failure process within the material or structure. The former are most easily studied with experimental techniques that develop homogeneous stress fields, homogeneous strain fields and well-controlled loading. In such experiments, as in the high-strain-rate experiments described earlier in this chapter, as the deformation evolves, the stress fields and deformation fields remain nearly homoge-

neous, and a small number of measurements (typically single point or area averaged measurements) are sufficient to obtain information about the deformation and stress fields of interest. However, once a failure process begins in the material, the deformation field rapidly becomes localized, and the experimental techniques used to extract information using single-point or areaaveraged measurements no longer provide adequate information. Understanding the failure process typically requires a different suite of experimental measurements,

Part D 33.4

The most commonly used direct impact experiment is the Taylor impact experiment. The experiment involves the impact (at normal incidence) of a rod onto a plate (assumed to be nearly rigid). Substantial dynamic deformations are developed at the site of the impact, leading to a local expansion of the rod and the development of a final shape that depends on the impact velocity and on the properties of the material of the rod. An alternative version of the Taylor impact experiment involves a symmetric impact of two rods. The Taylor impact experiment has the advantage that it is extremely simple to perform. All that is needed is a facility for launching the rod at the de-

950

Part D

Applications

Part D 33.4

and often requires a completely different experimental design. A simple example of this can be seen in the case of fracture: the uniaxial tension experiment is very well developed for the measurement of material behavior under uniaxial tension, but the utility of this experiment to determine the ability of the material to resist the propagation of cracks is minimal. Instead, specially designed fracture experiments are required. This is one reason why, although material scientists have a great affection for the simple tension experiment, they find it difficult to extract useful information about the toughness of a material by examining the elongation of the specimen in tension. A similar situation arises when understanding the dynamic failure process: specialized experiments are required to measure the dynamic failure process, and normal high-strain rate experiments must be interpreted with care when failure processes are occurring. The experimentalist in the dynamics of deformable solids does have one advantage over the experimentalist in the traditional mechanics domain. Most failure processes are inherently dynamic, and he/she may already have access to instruments with sufficient time resolution to resolve the failure process. However, there is also a potential complication. The failure process, itself dynamic, may be initiated by the arrival of a dynamic load. Thus, both slowly varying and rapidly varying loads may result in the development of dynamic failures. There are three primary modes of failure in most engineering materials: 1. void nucleation, growth and coalescence, 2. crack nucleation, growth and coalescence, and 3. shear band development. The first and third of these are typically associated with ductile failure, while the second is associated with brittle failure. However, all three modes may be observed simultaneously during a ductile failure event. Most experimental techniques in the dynamic failure literature are designed to examine the growth of a pre-existing void or crack, although the nucleation process may be of particular importance during dynamic failure [33.95]. The late stages of any failure process typically involve coalescence processes and can be the most difficult to resolve in an experiment. We begin by discussing experimental techniques associated with the measurement of void growth and shear band development under dynamic loading. We do not discuss experimental techniques for dynamic fracture in any detail, since this topic is heavily discussed in the literature [33.96]. It is important to remember that

because these complex failure processes may feature interactions between the failure process, the dynamic loading, nearby free surfaces, and high strain rate material properties, a theoretical understanding of the failure process can be a great asset in designing appropriate experiments. Early experiments in this area were primarily concerned with defining the phenomenology of these failure processes, helping to determine which mechanisms should be included in the modeling. The development of improved analyses and much greater computational power now affords us the opportunity to perform experiments with the appropriate measurements to be able to extract parameters that characterize the failure process. Indeed, an early decision that must be made in experimental design is whether the objective is to determine the micromechanisms associated with the dynamic failure process or to provide characteristic parameters that define the sensitivity of the material to this particular failure process.

33.4.1 Void Growth and Spallation Experiments At this time, analytical and computational papers on the dynamic growth of voids outnumber experimental papers by more than 10 to 1, attesting to the difficulty of performing controlled dynamic experiments on void growth. Most experiments that consider the dynamic growth of voids are plate impact experiments, although a few experiments have considered dynamic tension (for example using the tension Kolsky bar) with varying degrees of triaxiality [33.97] produced by inserting controlled notches. Dynamic tension experiments with controlled triaxiality are essentially dynamic versions of similar experiments that have been performed to examine the quasi-static process of void growth and coalescence [33.98]. Understanding these experiments requires a coupled computational and experimental approach (sometimes called a hybrid approach), since the stress state is deliberately nonuniform. The utility of the experiments is greatly improved if they are coupled with high-speed photography of the developing deformation, as well as the use of digital image correlation [33.99–105]. Comparison of computational and experimental results is particularly useful in examining such tension experiments [33.106]; we note, however, that such experiments alone are not sufficient for determining the mechanics associated with the failure processes that occur inside the specimen. Rather, such experiments help bracket the parameters and thus constrain the assumed failure model. Quantitative meas-

High Rates and Impact Experiments

Rarefaction fan

u0 = 0

x

Spall plane

Fig. 33.21 Schematic and wave propagation diagrams for

the spallation experiment. Note the location of the spall plane with respect to the two rarefaction fans

urements of the internal deformations associated with the dynamic growth of voids would be ideal, but such measurements are not available in solids except with very expensive stereoscopic x-ray equipment. The first steps in this direction have been taken by the PCS group at the Cavendish [33.107]. Plate impact experiments that examine void growth are of two types. In the first type, pioneered by Clifton and his group at Brown [33.109–112], a pre-fatigued crack is generated within a target plate that is subjected to impact, such that a reflected tensile wave loads the crack. The existence of the prior crack generates a well-defined region within which the void growth can commence. This approach has been used with both brittle materials [33.109, 110] and ductile materials [33.111], and the loading has included both tensile Critical stresses (GPa) 7

Maraging steel

Non-thermal critical stress Adiabatic critical stress Range of spall strengths

6 4340 steel

5 α-Ti

4

Tantalum

3 2

Ti–6Al –4V Cold drawn CU

Annealed Cu 7075-T6 Fe Al

Mild steel

1

6061-T6

2024-T4

0 0

0.05

0.1

0.15

0.2

0.25 0.3 0.35 Strain hardening exponent n

Fig. 33.22 Spall strength of a num-

ber of metals, as listed by Wu et al. [33.108]

951

Part D 33.4

and shear waves [33.112]. The technique has the advantage that the location of the prior crack is known, and the subsequent failure process can be studied using both the measured diffracted waves and the fracture surface morphology [33.113]. The second type of plate impact experiment of interest here is the spallation experiment [33.114, 115]. The basic approach is identical to that in the shockwave experiments described in Fig. 33.19, with the difference that the phenomenology of interest is developed when the release waves interact (Fig. 33.21). The flyer is normally thinner than the target, and may be made of a different material. The wave propagation within this arrangement is shown in Fig. 33.21. When the compressive shock waves reach the free surfaces of the flyer and the target plates, they are reflected as release waves. However, release waves unload along the isentrope rather than along the Rayleigh line (Fig. 33.18), so that a rarefaction fan is developed in each case. Very high tensile stresses are developed very rapidly at the location where the two rarefaction fans intersect, and the corresponding plane is called the spall plane. Voids nucleate, grow and coalesce along the spall plane, resulting in the separation of a piece of the target plate. This is called a spall failure or spallation (note the identical experiment can be performed for relatively brittle materials such as hardened steels). The development of the spall failure results in specific signatures in the particle velocity at the rear surface of the target plate which can be interpreted in terms of

t

Flyer Target

33.4 Dynamic Failure Experiments

952

Part D

Applications

the spall strength of the material [33.116]. An excellent review of spallation experiments, and a discussion of the approaches to interpreting the results is presented by Antoun et al. [33.117]. The spall strengths of a large number of materials have been determined (Fig. 33.22), and a wide range of values is quoted for some cases. One of the key difficulties in this experiment is that the rate of hydrostatic tensile loading (or the tensile strain-rate) is difficult to control accurately, since this quantity depends both on the process of void growth and on the local hydrostatic tension (which cannot be measured directly). However, Wu et al. [33.108] have shown that a lower approximation to the spall strengths of most metals can be obtained using a critical pressure model.

33.4.2 Shear Band Experiments

Part D 33.4

The initiation and development of adiabatic shear bands in materials have been reviewed by Bai and Dodd [33.118] and more recently by Wright [33.119]. Adiabatic shear bands represent thermo-viscoplastic instabilities, and are often observed in ductile metals subjected to high rates of loading, because there is insufficient time to conduct away heat during the event. Such shear bands can be the dominant mode of failure in an impact event, and are particularly common in impacts involving ductile metals subjected to overall compression and also in the perforation and punching of sheets and plates. Adiabatic shear bands are also observed in applications such as high-speed machining, where they limit the speed of the manufacturing process. The classical experimental technique used to study adiabatic shear bands is the dynamic torsion of thinwalled tubes in a torsional Kolsky bar experiment, as developed by Marchand and Duffy [33.48]. In this approach, an initially homogeneous torsional deformation of a thin-walled tube evolves into a strongly localized deformation with the development of an adiabatic shear band somewhere along the length of the tube during continued loading. Simultaneous recording of the stress history and the end displacement history together with time-resolved high-speed photography of the continuing deformation provides a direct measure of the time of occurrence of the adiabatic shear band. This technique has been used to study adiabatic shear band development in a variety of metals, including steel [33.120], Ti-6Al-4V [33.121] and tungsten heavy alloys [33.122]. Experiments of this type have guided much of the modeling that has been performed in the adiabatic shear

band literature, since they can be considered to be simple shear experiments (at least until the onset of the instability). However, the technique has been shown to be extremely sensitive to defects in the dimensions of the specimen [33.123] and defects in the material, as with all dynamic failure processes. To counter this, a predefined defect can be introduced into a thin-walled tube that is then subjected to dynamic torsion (note such experiments are essentially designed to study the failure process). Duffy and his coworkers introduced dimensional defects by hand using local polishing. Deltort [33.124] and Chichili and Ramesh [33.125] introduced standardized notches as defects in known locations using machining. Chichili and Ramesh [33.126] also developed a recovery technique, which allowed the recovery of specimens within which an adiabatic shear band had been grown under a single known torsional pulse, with a superimposed hydrostatic compressive load. A similar capability has been developed by Bai and co-workers [33.127]. Such approaches are critical for the development of an understanding of the deformation mechanisms [33.125] associated with adiabatic shear bands under welldefined dynamic loads. A number of other experimental techniques have been developed, in which the emphasis is on the development of adiabatic shear bands in a repeatable manner so that the deformation micromechanisms can be studied. These include hat specimens deformed in a compression Kolsky bar [33.128], dynamic perforation or punching experiments [33.129], explosive loading [33.130] and shear compression susceptibility experiments [33.131]. A very interesting approach is to consider the asymmetric impact of prenotched plates [33.132]. In all of these cases, the simultaneous use of high-speed photography is a great asset. Further, given that the temperature is a critical component of the thermoviscoplastic problem, direct dynamic measurements of the temperature of the shear band tip are of great interest [33.133]. Unfortunately, a complete characterization of the deformation state and temperature state at the shear band tip continues to be difficult, and is an area of continuing research.

33.4.3 Expanding Ring Experiments The tensile instability that is analogous to shear band development is the onset and growth of necking. However, the dynamic measurement of the necking instability is very difficult. The difficulties that arise are identical to the difficulties described in the high rate

High Rates and Impact Experiments

33.4.4 Dynamic Fracture Experiments We do not discuss experimental techniques for dynamic fracture in any detail, since this topic is heavily discussed in the literature [33.144–148] and there is a very recent book on the subject [33.96]. Unlike quasi

static fracture testing, dynamic fracture testing does not have an accepted set of testing standards. Thus there are a wide variety of specimen geometries, constraints, and loading conditions in use [33.96]. Most of these have in common the generation of a fatigued pre-crack and an attempt to measure the K-dominated crack-tip stress, strain or deformation fields, typically using optical diagnostics and/or high-speed photography. The specimens are often impulsively loaded using a Kolsky bar type configuration, or by direct impact in either 3-point bend, 1-point bend or asymmetric impact configurations (the Kalthoff experiment [33.149]). Useful results can also be obtained with crack opening measurements and properly located strain gauges [33.150], but it can be extremely difficult to identify the onset of crack propagation from the signals. The more commonly used optical diagnostics include photoelasticity [33.151], caustics [33.152], dynamic moiré [33.153], and coherent gradient sensing [33.154].

33.4.5 Charpy Impact Testing Relatively rapid measures of the dynamic toughness of a material can be obtained using the ASTM standard Charpy impact test (ASTM E23-06). The technique essentially uses a notched bar for a specimen and a pendulum machine as an impacting device. Commercial pendulum machines are available to perform such experiments. There is also a NIST recommended guide for maintaining Charpy impact machines, NIST Special Publication 960-4. Versions of these devices have been developed for subscale testing, as well as for the testing of polymers. A recent reference book on pendulum impact machines has also been published by ASTM [33.155].

33.5 Further Reading • • •

T. Antoun, L. Seaman, D.R. Currar, G.I. Kanel, S.V. Razorenov, A.V. Utkin: Spall Fracture (Springer, New York 2004) Y.L. Bai, B. Dodd: Adiabatic Shear Localization: Occurrence, Theories and Applications (Pergamon, Oxford 1992) D.S. Drumheller: Introduction to Wave Propagation in Nonlinear Fluids and Solids (Cambridge Univ. Press, Cambridge 1998)

953

• • •

J.E. Field: Review of experimental techniques for high rate deformation and shock studies, Int. J. Imp. Eng. 30, 725–775 (2004) G.T. Gray III: Classic Split-Hopkinson pressure bar testing. In: ASM Handbook, Vol. 8, ed. by H. Kuhn, D. Medlin, (ASM Int., Materials Park Ohio 2000) pp. 462–476 K.A. Hartley, J. Duffy, R.H. Hawley: The torsional Kolsky (Split-Hopkinson) bar. In: ASM Handbook,

Part D 33.5

tension experiment earlier in this chapter: the end conditions required to develop the tension typically modify the failure mode. The most effective way to generate tension and study the necking instability is to use an expanding ring experiment. The dynamic expansion of the ring produces an initially axisymmetric tension mode, and the onset of the necking instability represents the breakage of symmetry within the experiment. There are three fundamental approaches to developing dynamic expansion of rings. The first is to use an explosive expansion, the second to use an electromagnetically driven expansion, and the third to use the axial motion of a wedge to develop a radial expansion of an enclosing ring. In practice, the first and third approaches are very difficult to control. Thus the primary experimental method available for expanding rings is that of electromagnetic launch [33.134–139]. This approach is often used to examine the development of tensile fragmentation in metallic systems, and has recently been coupled with high-speed photography to provide unprecedented detail of the dynamic necking process [33.140]. Although these experiments are extremely difficult to perform, the analysis of such experiments is very attractive to the theoretician and has the benefit of providing a great deal of insight (recent analyse are presented by Zhou et al. [33.141] for the brittle case and by Guduru and Freund [33.142] for the ductile case). Related experiments involve explosive loading to generate expanding cylinders [33.143].

33.5 Further Reading

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• •

Vol. 8, ed. by H. Kuhn, D. Medlin, (ASM Int., Materials Park Ohio 1985) pp. 218–228 R.W. Klopp, R.J. Clifton: Pressure-shear plate impact testing. In: ASM Handbook, Vol. 8, ed. by H. Kuhn, D. Medlin, (ASM Int., Materials Park Ohio 1985) pp. 230–239 M.A. Meyers: Dynamic Behavior of Materials (Wiley Interscience, New York 1994)

• • •

T. Nicholas, A.M. Rajendran: Material characterization at high strain-rates. In: High Velocity Impact Dynamics, ed. by J.A. Zukas 1990, (Wiley, New York 1990) pp. 127–296 K. Ravi-Chandar: Dynamic Fracture (Elsevier, Amsterdam 2004) T.W. Wright: The Mathematical Theory of Adiabatic Shear Bands (Cambridge Univ. Press, Cambridge 2002)

References 33.1

33.2

33.3

33.4

Part D 33

33.5

33.6

33.7

33.8

33.9

33.10

33.11

33.12

J.E. Field, W.G. Proud, S.M. Walley, H.T. Goldrein: Review of experimental techniques for high rate deformation and shock studies. In: New Experimental Methods in Material Dynamics and Impact, ed. by W.K. Nowacki, J.R. Klepaczko (Institute of Fundamental Technological Research, Warsaw 2001) p. 109 C.S. Coffey, V.F. DeVost: Drop weight impact machines: a review of recent progress, JANNAF Propulsion Systems Hazards Subcommittee Meeting CPIA Publ., 1(446), 527–531 (1986) J.D. Winkel, D.F. Adams: Instrumented drop weight impact testing of cross-ply and fabric composites, Composites 16(4), 268–278 (1985) N. Banthia, S. Mindess, A. Bentur, M. Pigeon: Impact testing of concrete using a drop-weight impact machine, Exp. Mech. 29(1), 63–69 (1989) P.R. Sreenivasan, S.K. Ray, S.L. Mannan, P. Rodriguez: Determination of K(Id) at or below NDTT using instrumented drop-weight testing, Int. J. Fract. 55(3), 273–283 (1992) C.S. Coffey, V.F. Devost: Impact testing of explosives and propellants, Propellants Explosives Pyrotechnics 20(3), 105–115 (1995) H.M. Hsiao, I.M. Daniel, R.D. Cordes: Dynamic compressive behavior of thick composite materials, Exp. Mech. 38(3), 172–180 (1998) T. Nicholas, A.M. Rajendran: Material characterization at high strain-rates. In: High Velocity Impact Dynamics, ed. by J.A. Zukas (Wiley, New York 1990) pp. 127–296 S.M. Walley, J.E. Field: Strain rate sensitivity of polymers in compression from low to high strain rates, DYMAT J. 1, 211–228 (1994) G. Ravichandran, G. Subhash: A micromechanical model for high strain rate behavior of ceramics, Int. J. Sol. Struct. 32, 2627–2646 (1995) D. Jia, K.T. Ramesh, E. Ma: Effects of nanocrystalline and ultrafine grain sizes on constitutive behavior and shear bands in iron, Acta Mater. 51(12), 3495– 3509 (2003) H. Kolsky: An investigation of the mechanical properties of materials at very high rates of loading, Proceedings of the Physical Society, Vol. 62B (London, 1949) p. 676

33.13

33.14

33.15

33.16

33.17

33.18

33.19

33.20

33.21

33.22 33.23

33.24 33.25

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961

Delamination 34. Delamination Mechanics

Kenneth M. Liechti

This chapter deals with the related phenomena of interfacial fracture, adhesion, and contact. These topics provide some excellent examples of applications of experimental techniques and are important in the areas of structural adhesive bonding, microelectronics packaging, and micro-electromechanical systems (MEMS) devices. The theoretical background for each phenomenon is first summarized so as to set the stage for descriptions of the experiments that are used to extract the associated physical properties.

34.1 Theoretical Background ........................ 34.1.1 Interface Cracks.......................... 34.1.2 Crack Growth Criteria .................. 34.1.3 Delamination in Sandwiched Layers ................. 34.1.4 Crack Nucleation from Bimaterial Corners .............. 34.1.5 Nonlinear Effects........................

962 962 963 964 966 968

968 969 969 970 970 971 977 977 978 978 979 980 980

may be ignored. Thus, if the adherends are the same, the crack will appear to be cohesive, or one in a homogeneous material. If the adherend materials differ and a crack is growing in the adhesive layer, but it is being ignored, then the crack appears to be an adhesive one, growing along the interface between the two different adherends. When the adhesive layer is accounted for, then cohesive and adhesive cracking are again possible, albeit from a slightly different perspective. Cohesive delaminations can be dealt with using the concepts of fracture mechanics that have been described in Chap. 5 for crack growth in monolithic materials. Many of the concepts developed there can be extended to adhesive or interfacial fracture. The fact that the crack lies on the interface between two materials does affect the crack tip asymptotic fields and, as a result, the def-

Part D 34

Laminated materials or structures are quite rich in their array of potential fracture mechanisms. There can be cohesive cracks that grow entirely within a layer. Another possibility is cracks that grow along an interface to create adhesive fracture. Cracks may branch into substrates or oscillate within a layer. All these can be dealt with by fracture mechanics. Cracks are not the only sources of high stresses or singularities in laminated systems. There are generally a multitude of so-called bimaterial corners that also excite singular stresses. They are admittedly of a slightly different nature from those generated by cohesive or adhesive cracks, but nonetheless can often be handled in a similar manner. Delamination in bonded systems can be viewed at several scales. In many cases, the adhesive layer itself

34.2 Delamination Phenomena ..................... 34.2.1 Dynamic Interface Fracture and Coherent Gradient Sensing.... 34.2.2 Dynamic Interface Fracture and Photoelasticity .................... 34.2.3 Dynamic Matrix Cracking and Coherent Gradient Sensing.... 34.2.4 Quasistatic Interface Fracture and Photoelasticity .................... 34.2.5 Quasistatic Interface Fracture and Crack Opening Interferometry 34.2.6 Quasistatic Interface Fracture and Moiré Interferometry ............ 34.2.7 Crack Nucleation from Bimaterial Corners and Moiré Interferometry. 34.2.8 Bimaterial Corner Singularities and Photoelasticity .................... 34.2.9 Crack Growth in Bonded Joints and Speckle............................... 34.2.10 Thin-Film Delamination and Out-of-Plane Displacements . 34.3 Conclusions .......................................... References ..................................................

962

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Applications

inition of fracture parameters such as stress intensity factors. The potential for crack face contact adds one additional length scale, the contact zone, to scales such as the plastic zone, the K -dominant zone, etc. However, once fracture parameters are defined, concepts of fast fracture, fatigue fracture, time-dependent fracture, and environmentally assisted fracture can be carried over from fracture in monolithic materials in a relatively straightforward manner. The main emphasis of this chapter will be on interfacial fracture mechanics concepts with some extension to crack branching and crack nucleation from bimaterial corners. Most of the current fracture mechanics practice in testing adhesives and designing of adhesively bonded joints or laminated materials and structures is limited to linear elastic fracture mechanics concepts. The development of the background material presented here will therefore be similarly constrained, except for the last part, which deals with nonlinear effects. Historically, fracture mechanics developed from energy balance concepts and examinations of stresses around crack tips. The adhesive fracture and composite materials communities have tended to favor the former, but both have

useful features and will be carried forward in the discussions that follow. In adhesive delamination, cracks run along the interface between two different materials due to interactions between the stress field in the adhesive layer and spatial variations in fracture properties. The cracks are not generally free to evolve as mode I cracks, as was the case for cohesive cracks, and mixed-mode fracture concepts (combinations of tension and shear) have to be considered. Mode II or III shear components are induced, even in what appear to be nominally mode I loadings, due to differences in moduli about the interface. The theoretical underpinnings for these issues are presented in the first part of this chapter. There have been a number of experimentally based examinations of interface fracture and delaminations in layered materials. Those reviewed in the second part of this chapter are generally aimed at uncovering crack nucleation and growth mechanisms over a wide range of crack speeds. As a result, they are optical techniques offering the advantage of full-field measurements. The third section of the chapter offers some concluding remarks.

34.1 Theoretical Background In this section we will deal with crack growth along interfaces and in sandwiched layers, and crack nucleation from bimaterial corners. We will examine the nature of the stress states and fracture parameters.

Part D 34.1

34.1.1 Interface Cracks The dominant stresses near the tip of an interface crack with material 1 above material 2 (Fig. 34.1) are given by σαβ =

Re[Kr iε ] (2πr)

σI 1/2 αβ

(θ, ε) +

Im[Kr iε ] (2πr)1/2

(3 − νi )/(1 + νi ) for plane stress, and μi and νi are the shear moduli and Poisson’s ratios, respectively, of the upper and lower materials. The quantity K = K 1 + iK 2 is the complex stress intensity factor. Its real and imaginary parts are similar to the mode I and II stress intensity I (θ, ε) factors for monolithic materials. The functions σαβ II and σαβ (θ, ε) are given [34.2] in polar coordinates. The transformation to Cartesian coordinates is routine but tedious, so the results are not given here.

II σαβ (θ, ε) .

σ22

(34.1)

The bimaterial constant   1 1+β ε= , 2π 1 + β where μ1 (κ2 − 1) − μ2 (κ1 − 1) β= , μ1 (κ2 + 1) + μ2 (κ1 + 1) is one of the Dundurs [34.1] parameters for elastic bimaterials. The quantity κi is 3 − 4νi for plane strain and

x2

σ12 σ11

Material 1

Crack faces

r

σ11

σ12 σ22

µ1, υ1

θ x1

Material 2

Fig. 34.1 Coordinate system for an interface crack

µ2, υ2

Delamination Mechanics

The stresses are normalized so that the stresses ahead of the crack tip are given by σ22 + iσ12 =

(K 1 + iK 2 )

r iε ,

30

where r iε = cos (ε ln r) + i sin (ε ln r). This is an oscillating singularity, which leads to interpenetration of the crack faces as can be seen by examining the crack flank displacements δi = u i (r, π) − u i (r, −π), which are given by

20

δ1 + iδ2 =

(2πr)

8 K 1 + iK 2 (1 + 2iε) cosh(πε) E∗  r 1/2 × r iε , 2π

963

ω* 40

(34.2)

1/2

34.1 Theoretical Background

β=0 β = α/4

10 0 –10

(34.3)

∗ The function in Fig. 34.3. The quan ∗ ω ∗ is given tity α = E 1 − E 2 / E 1∗ + E 2∗ is one of the Dundurs [34.1] parameters for bimaterial systems. An examination of (34.6) reveals that, even though the loading is symmetric, the mode 2 component is nonzero, due

M µ1, υ1 2h

–30 – 40 –2

–1

0

1

2 α

Fig. 34.3 The variation of ω∗ with α (after [34.3])

to the difference in material properties across the interface. This so-called mode mix is defined as the ratio of the stress intensity factors through  iε  −1 Im (Kl ) , (34.7) ψ = tan Re (Kl iε ) where l is a suitable reference length. If l is chosen so that it lies within the zone of K dominance, then an equivalent expression for the mode-mix is   σ12 . (34.8) ψ = tan−1 σ22 r=l The choice of l is arbitrary, but is often based on some material length scale such as the plastic zone size. If ψ1 is the mode mix associated with a length l1 , then the mode mix ψ2 associated with a different length scale l2 is ψ2 = ψ1 + ε ln (l2 /l1 ) .

(34.9)

This transformation is very useful when comparing toughness data. The toughness of many interfaces is a function of the mode mix and it is important to note what length scale is used in the measure of mode mix. Two sets of toughness data reported on different length scales can be brought into registration using (34.9).

µ2, υ2 M

Fig. 34.2 Bimaterial double cantilever beam specimen under uniform bending

34.1.2 Crack Growth Criteria Just as for cohesive cracks, we distinguish between dynamic, quasistatic, and subcritical adhesive cracks.

Part D 34.1

where   1 1 1 1 = + (34.4) E∗ 2 E 1∗ E 2∗   and E i∗ = E i / 1 − νi2 for plane strain and E i∗ = E i for plane stress. The energy release rate for crack advance along the interface is    1 − β2  2 (34.5) G= K 1 + K 22 . ∗ E A number of stress intensity factor solutions have been developed over the years. Several solutions are given in the review article by Hutchinson and Suo [34.3]. One example is the stress intensity factors for a bimaterial double cantilever beam subjected to uniform bending (Fig. 34.2). In this case √ K 1 + iK 2 = 2 3Mh −3/2−iε  −1/2 ∗ × 1 − β2 ei (α,β) . (34.6)

–20

964

Part D

Applications

However, whereas cohesive cracks tend to follow a path, where K II = 0, adhesive cracks must by definition follow the interface. This usually means that a mixed-mode fracture criterion that involves tensile and shear components must be developed. For quasistatic growth, the most common approach has been to plot the critical value of the energy release rate as a function of the fracture mode mix ψ. We can think of this as a two-parameter criterion that involves energy and stress intensity parameters. One example for a glass–epoxy interface is shown in Fig. 34.4, where there are two sets of data, one for specimens 6 mm thick [34.4] and the other for a specimen thickness of 2 mm [34.5]. The toughness rises sharply for positive and negative shear, but not in the same way. There is an asymmetry to the shear-induced toughening, which in this case was caused by differences in the amount of plastic deformation that are induced by positive and negative shear [34.6]. Although a by-product of the explanation just given is that the toughness envelope for a bimaterial interface can now be predicted via cohesive zone modeling once the intrinsic (minimum) toughness and inelastic deformation characteristics are known, many designers of laminated materials or structures will make use of measured envelopes. Predictions of quasistatic delamination at bimaterial interfaces will involve a determination of energy release rate and mode mix in the cracked component followed by a comparison with the

Part D 34.1

Fracture toughness (J/m2) 60 Liechti and Chai (1992) Liang and Liechti (1995)

50

l = 1 μm

40 Closing shear

30

Opening shear

toughness envelope at the appropriate value of mode mix. Durability analyses of subcritical interfacial crack growth under fatigue or static loadings are usually conducted in the same way as was described earlier for cohesive cracking. The fracture parameter that is generally used for correlating with crack growth rates is the energy release rate. This generally seems to be sufficient for accounting for mode-mix effects [34.7, 8].

34.1.3 Delamination in Sandwiched Layers The analyses described in the previous sections can be applied to adhesive joints with, respectively, similar and dissimilar adherends, but ignoring the adhesive layer itself. However, it is possible to move down one scale level and account for various types of crack growth (Fig. 34.5) within the adhesive layer by making use of the results of Fleck et al. [34.9]. These are restricted to situations where the adherends are the same. Nonetheless they are rather powerful and simple, because they can be used to extend existing analyses at the macroscopic level to the microscopic one without extensive numerical computations. We first consider the case where a straight cohesive crack is growing at some distance c from the lower adherend in a layer of thickness h. At the scale level of the adherends, we can suppose that we have the stress intensity factors K I∞ and K II∞ for a cracked homogeneous joint. Then the macroscopic or global energy release rate is  1  ∞2 2 K I + K II∞ . (34.10) G∞ = E¯ 1 At the same time, the energy release rate associated with the crack in the adhesive layer itself is  1  2 K 1 + K 22 . (34.11) G= E¯ 2 From the path independence of the J integral [34.10], which is also the energy release rate when linear elastic

20

10

0 – 90

– 60

– 30

0

30

Straight cohesive

Alternating

Wavy cohesive

60 90 Mode mix (deg)

Fig. 34.4 Mixed-mode toughness envelop for a glass–

epoxy interface (after [34.4, 5])

Adhesive

Fig. 34.5 Crack paths in sandwiched layers

Delamination Mechanics

fracture mechanics holds, we have that G = G ∞ . This, together with the fact that stresses are linearly related, allows the local stress intensity factors to be related to the global ones through   1 − α 1/2 (K 1 + iK 2 ) = 1+α   ∞ × K I + iK II∞ eiϕ(c/h,α,β) , (34.12) where ϕ = ψ − ψ ∞ can be thought of as the shift in phase angle between the global and local stress intensity factors. Fleck et al. [34.9] have shown that     c 1 h −1 +2 − ϕ˜ (α, β) , (34.13) ϕ = ε ln c h 2 where ϕ˜ is given in Fig. 34.6. If a joint is loaded under globally mode I conditions and the crack runs along the mid thickness of the layer, then (34.12) and (34.13) indicate that   1 − α 1/2 ∞ KI and K 2 = 0 . (34.14) K1 = 1+α Thus, if a polymeric adhesive layer is being used to join two stiffer metallic, composite, or ceramic adherends α > 0 and K 1 < K I∞ . This means that the crack in the adhesive layer is shielded from the global loading. For an interface crack we have    1 − α 1/2  ∞ K I + K II∞ h −iε eiω , K 1 + iK 2 = 1 − β2 (34.15)

where the ω function is plotted in Fig. 34.7.

–2

–4

–6

–8

–10

–12 –2

–1

ω (deg) 5 β=0 β = α/4

0 KI KII

–5

Re(Khiε ) 1

–10

Im (Khiε )

2 1

–15 –1

– 0.5

0

0.5

1 α

Fig. 34.7 The dependence of ω on α and β (after [34.3])

One might think that an initially straight cohesive crack (Fig. 34.5) would turn (kink) upon the slightest application of K II∞ . However, it turns out that the elastic mismatch that is contained in (34.15) allows a straight crack to continue as such for approximately K II∞ ≤ 0.1K I∞ . To see this, we make the common assumption that cracks in homogenous materials grow in such a way that the local mode II component K II = 0. This condition, when substituted into (34.15), yields a relationship between the location of the crack (c/h) and the global mode mix ψ ∞ : ϕ (c/h, α, β) = −ψ ∞ .

β = α/4 β=0

0

1

Fig. 34.6 Dependence of ϕ˜ on α and β (after [34.9])

2 α

965

The solution is plotted in Fig. 34.8 for several values of α and β = α/4. There can be no straight paths for zero mismatch (α = β = 0). The 10% level of K II∞ that was referred to earlier can lead to straight cracks when the mismatch is relatively large. The next question that arises is whether or not an initially straight cohesive crack can remain so in the presence of slight perturbations. This was first addressed for cracks in monolithic materials by Cottrell and Rice [34.11]. They postulated that the T -stress controls the directional stability of cracks. Chai [34.12] made the extension to adhesively bonded joints and identified an oscillating crack path, where the crack periodically touched the top and bottom adherends (composite or aluminum) under nominally mode I loading. The period was quite consistent at 3–4 times the

Part D 34.1

φ 0

34.1 Theoretical Background

966

Part D

Applications

bond thickness. Fleck, Hutchinson, and Suo [34.9, 13] and Akinsanya and Fleck [34.13, 14] conducted an analytical investigation of the problem. Crack paths were categorized into those that settled on the centerline no matter what the original elevation, those that oscillated gently about the centerline, those that approached the interface gradually, and those that approached it at a large angle. Which pattern dominates in any particular situation was driven by the signs of two parameters: the local T -stress and ∂K II /∂c. For nominally mode I loading (K II∞ = 0), the local T -stress is given by T=

  1−α ∞ 1 − α 1/2 K Ic cI √ , (34.16) T + σR + 1+α 1+α h

which shows that it is controlled by the global T -stress T ∞ , thermal and intrinsic residual stresses, and the value of the global mode I stress intensity. The latter is related to the local toughness K Ic via (34.11). The coefficient cI (and its mode II partner for cases where K II∞ = 0) was tabulated for several values of c/h, α, and β. The global T -stress T ∞ had previously been determined by Larsson and Carlsson [34.15] for several common bonded joints. Chen and Dillard [34.16] controlled the residual stress levels in the adhesive layer by prestretching double cantilever beam specimens prior to fracture testing. This gave them sufficient control over the local T -

stress that a number of the crack patterns that had been predicted were indeed observed. Interestingly, the toughness of the joints was quite similar, irrespective of crack pattern. A subsequent paper [34.17] dealt with the effects globally mixed-mode loading and load rates. It was found that crack paths were stabilized at the interface for all T -stress levels when the mode II component was greater than 3%. Higher crack propagation rates led to more cohesive cracking and waviness. The toughness of specimens decreased with increasing mode II component. While seemingly at odds with previous results on mode-mix effects, this latter result can be explained by the different crack paths that were taken as the mode-mix changed.

34.1.4 Crack Nucleation from Bimaterial Corners Laminated structures abound with bimaterial corners. These can be sources of crack nucleation due to the stress concentrations that can be associated with them. The simplest situation arises when one of the materials is comparatively rigid. In that case we have a plate which is clamped along one boundary and free on the other (Fig. 34.9). This is one of several combinations of boundary conditions at a corner that Williams [34.18] considered. With a polar coordinate system originating at the corner, it can be shown that the stresses have the form σ ∝ r λ−1 .

Part D 34.1

c/h 1 c

(34.17)

The value of λ depends upon the corner angle θ1 of the plate and the Poisson’s ratio of the material (Fig. 34.10 with ν = 0.3). For θ1 > 60◦ , λ < 1, which leads to singular stresses that tend to infinity as the corner is approached. At the same time it can be seen that, for θ1 < 60◦ , the stresses are nonsingular and the stress con-

h

0.8 α = 0.8 α = 0.4

0.6

0.4 E1, υ1 0.2 β = α/4

0 –0.15

θ1

r θ

– 0.1

– 0.05

0

0.05

0.1

0.15 KII∞/KI∞

Fig. 34.8 The dependence of crack location on mode mix

and material mismatch (after [34.3])

Fig. 34.9 The clamped–free corner of a plate

Delamination Mechanics

34.1 Theoretical Background

967

Minimum Re (λ) 2.5 υ = 0.3 E1, υ1

2 θ1

1.5

r θ

1 θ2

0.5

0

E2, υ2

0

50

100

150

200

250 300 350 400 Wedge angle θ1 (deg)

Fig. 34.10 The singularity associated with a clamped– fixed corner (after [34.18])

σαβ =

N  i=1

σ0 (MPa)

P (N ) 1200 1000

K ai r λi −1 σ¯ αβi (θ)

+ K a0 σ¯ αβ0 (θ) (α, β = r, θ) .

global boundary conditions. When there is only one singular term, the interpretation of (34.17) is analogous to the crack problem, where λ = 0.5. It turns out that 90◦ corners give rise to just one singular term. Quite a lot of work has been done with them by Reedy and coworkers both on the stress analysis front and in determining the corner toughness K ac for various combinations of adhesives and adherends [34.21–25]. Other corner angles and material combinations that give rise to single eigenvalues have been studied by Qian and

Data (average) CZM

120

P

P

100

800

(34.18)

This indicates that there can be N singularities with strength (λi − 1). The angular variation in the stresses is given by the σ¯ αβi (θ) terms and depends on the elastic constants of the materials, the corner angles (θ1 , θ2 ) in each material, and the boundary conditions on the sides that are not joined. The singularities are determined on the basis of asymptotic analyses that satisfy the local boundary conditions near the corner. The singularities can be real, as was the case for the crack in a monolithic material, or complex [34.20], as in the case of the interface crack. The stress intensity factors K ai have to be determined from the overall or

80 α

600

1

60

2

400

40

200

20

0 –22.5

0

22.5

45

67.5

90

112.5

135

0 157.5 α (deg)

Fig. 34.12 Measured and predicted loads for interfacial crack nucleation from a bimaterial corner (after [34.19])

Part D 34.1

centration does not exist. Thus, where possible, corner angles should be kept below 60◦ . Sometimes this can happen naturally when adhesive spews from the joint during processing. Notice that, when θ1 = π, the crack configuration is recovered and λ = 0.5. When both materials are compliant, singular stresses can also occur. The situation is more complicated because now there may be multiple singularities and they depend on the elastic properties of each material in addition to the corner angles in each material (Fig. 34.11). There are several ways to present the state of stress near a bimaterial corner. One common approach is

Fig. 34.11 A general bimaterial corner

968

Part D

Applications

Part D 34.2

Akinsanya [34.26] and Dunn et al. [34.27]. As a result, it has been possible to predict when cracks in one configuration will nucleate, based on experiments to determine K ac on another. Needless to say, the corner angle and material combination have to be the same, only the global conditions differ. If one has a choice of corner angles in a particular design then a consequence of this approach is that the corner toughness K ac must be found for each corner angle [34.28]. Mohammed and Liechti [34.19], recently remedied this situation for cracks that nucleate along the interface by making use of an energy approach instead. A cohesive zone model (CZM) was used to represent the behavior of the interface. The cohesive zone model parameters, which can be thought of as representing the toughness and strength of the interface, were determined from experiments on and analysis of an interface crack (α = 0) between the materials of interest. Since the interface was still the same for all corner angles and the crack nucleated along the interface, the same cohesive zone parameters were used with other corners and found to predict (Fig. 34.12) the nucleation load and near corner displacements very well. Thus in this approach, only one corner was needed for calibration purposes. Another advantage is that it can be used for corners that give rise to multiple singularities. Such an energy-based approach could presumably be used even if the crack did not grow along the interface, although a suitable calibration specimen would have to be employed. Corner cracks may also initiate under fatigue loading. In fact this may be the most common form of nucleation. Nonetheless, this problem seems to have received relatively little attention in the open literature. Lefebvre and Dillard [34.29, 30] considered an epoxy wedge on an aluminum beam under cyclic loading. They chose corner angles (55◦ , 70◦ , and 90◦ ) that resulted in one singularity. A stress intensity factor based fatigue initiation envelope was then developed.

34.1.5 Nonlinear Effects All of the analyses of cracks and corners that have been described here were based on the linearly elastic responses of the materials. It was recognized that this would lead to some physically unreasonable results very close to the crack front or corner where yielding would more likely occur. Nonetheless, as long as this yielding zone was small in scale, the elastic analysis was sufficient for characterizing the stress state. Nonlinear effects enter when the yielding or plastic zone becomes a dominant feature. Other nonlinear effects can arise due to large nonlinear elastic deformations, for example, in rubber/metal joints, large rotations in peeling joints, and nonlinear viscoelastic effects. For cohesive cracks in layers that exhibit strain hardening behavior, the Hutchinson, Rice, Rosengreen (HRR) [34.31, 32] type of singular behavior described in Chap. 5 can be applied if the HRR zone is smaller than the layer thickness. The extension of this approach to cracks between strain hardening materials was made by Shih and coworkers [34.33–35] and Zywicz and Parks [34.36]. One interesting consequence of the plastic deformation, even at low strains, was that that the pathological behavior of oscillating singularities and crack face interpenetration that was seen under linear elastic behavior and small strains disappeared. For nonlinear elastic materials and considering finite deformations, Knowles and Sternberg [34.37] also showed that the oscillating singularities disappeared. These observations were consolidated by Rice [34.38], who introduced the concept of small-scale contact for cracks at interfaces as the basis for ignoring the pathological behavior in appropriate cases. Larger-scale frictionless contact, particularly under shear loadings, can be handled by the types of analyses conducted by Comninou [34.39] and Dundurs [34.40] for elastic materials and Aravas and Sharma [34.41] for elastoplastic materials. Frictional effects have also been accounted for by Comninou and Dundurs [34.42] and Deng [34.43].

34.2 Delamination Phenomena A rich array of optical techniques has been applied to the measurement of delaminations in laminated materials or structures. By providing wholefield displacement or stress, these techniques have provided improved understanding of delamination mechanics in a variety of situations. Some tech-

niques such as photoelasticity, moiré interferometry, and coherent gradient sensing have been described in the first part of this Handbook. However, other techniques such as crack opening interferometry and projected grating techniques will be described here as needed. This section is organized in

Delamination Mechanics

34.2 Delamination Phenomena

969

terms of crack speed, starting with dynamic fracture.

34.2.1 Dynamic Interface Fracture and Coherent Gradient Sensing Tippur and Rosakis [34.44] applied coherent gradient sensing (CGS) to interfacial crack propagation between polymethylmethacrylate (PMMA) and aluminum. A drop load was applied in asymmetric three-point bending. Nonzero mode mix manifests as tilt in the fringe lobe emanating from the crack tip (Fig. 34.13). Shear dominance was reduced as waves reflected from the specimen boundaries. More remarkably, the crack speed was a large fraction of the Rayleigh wave speed in PMMA. This finding stimulated more analytical work and further experiments by Lambros and Rosakis [34.46]. Working in transmission with a PMMA–steel bimaterial and a blunt notch, they were able to produce crack speeds that approached the longitudinal wave speed in PMMA. This intersonic crack speed was accompanied by large-scale contact. Although there was evidence of fringe compression, suggesting a shock wave, CGS is unable to visualize discontinuity in the shear stress. This led to the use of dynamic photoelasticity.

34.2.2 Dynamic Interface Fracture and Photoelasticity

t = –28 μs

t = –14 μs

t = 0 μs

t = 14 μs

t = 28 μs

t = 0 ... Crack initiation

PMMA

Al

Fig. 34.13 CGS interferograms of dynamic crack propagation between PMMA and Al (after [34.44])

fracture, Coker et al. [34.51] used photoelasticity to examine dynamic frictional effects between similar materials. All these laboratory results shared similarities with observations of intersonic rupture speeds in shallow earthquakes. a)

b)

10 mm

Fig. 34.14a,b Comparison of measured and predicted isochromatics during intersonic crack growth along a weak plane (after [34.39, 45])

Part D 34.2

Singh and Shukla [34.47] used dynamic photoelasticity to examine interfacial crack growth between aluminum 6061 and Homalite 100 polyester. A dynamic off-axis compressive load was applied to the aluminum part, resulting in a shear stress on the stationary crack. They noted, for the first time, shock waves as crack speeds exceeded the shear speed in Homalite. A largescale contact zone in the direction of propagation was also observed. The result was confirmed analytically by Liu et al. [34.48]. Much earlier, Broberg had predicted [34.49] that intersonic growth would be possible in homogeneous materials, if a forbidden regime between the Raleigh wave speed cR and the shear wave speed cS could be crossed. Rosakis et al. [34.45] achieved this by using a homogeneous specimen with an initial crack on a weak plane in order to ensure coplanar growth under shear loading. As the crack propagated, shock waves were observed (Fig. 34.14) emanating at about 45◦ from the weak plane, indicating √ that the delamination was propagating steadily at 2cS , as had been predicted by Freund [34.50]. In some work which has a connection to dynamic shear

t = – 42 μs

970

Part D

Applications

a)

b) North Shear wave 1 + GR rupture

Shear wave 2

β1

North + GR rupture

β2 East

Subrayleigh Supershear West

Fig. 34.15a,b Bilateral rupture along a laboratory fault line (af-

For smaller tilt angles and lower compressive load levels, the speed in the negative direction was lower than the Rayleigh speed in the slower material and dependent on the tilt angle and load level. On the other hand, for larger tilt angles and load levels, the speed of the rupture tip √ in the negative direction was super shear at roughly 2c2S , the shear wave speed in the slower material. These observations were related to the 1999 Imitz earthquake in Turkey, which exhibited positive direction rupture√at the Rayleigh speed and negative direction rupture at 2 times the shear wave speed in crustal rock.

ter [34.52])

More recently, Rosakis et al. [34.52] have conducted experiments with bimaterial (Homalite 100 and polycarbonate) plates in order to simulate slip and typical differences elastic properties between rocks. The fault line was placed at various scarf angles to a fixed compressive load in order to simulate a range of tectonic conditions. The friction between the plates represented the resistance to slip provided by the fault core. An exploding wire centrally placed on the fault line was used to trigger slip. Dynamic photoelasticity (Fig. 34.15) was again the diagnostic tool. Prior to this experiment, analyses had suggested that slip would only occur in one (the so-called positive) direction along the fault. However, the experiments established that, for all combinations of fault line angle and compressive load, slip occurred in a bilateral manner. The speed of the positive tip was always sub shear at the general Rayleigh speed of the material combination.

34.2.3 Dynamic Matrix Cracking and Coherent Gradient Sensing Rosakis and coworkers examined dynamic matrix cracking or interfiber splitting in unidirectional composites in mode I [34.54] and II [34.53]. Reflection CGS was used as the diagnostic method (Fig. 34.16). In mode I, crack speeds were limited by the Rayleigh wave speed. Intersonic cracking was again observed in shear, with crack speeds approaching the longitudinal wave speed in the fiber direction. Once again the presence of shock waves could be inferred by an increase in fringe density in the third and fourth frames.

34.2.4 Quasistatic Interface Fracture and Photoelasticity Photoelasticity has been a popular tool in examinations of quasistatic interfacial fracture. A number of

Part D 34.2 –5.6 μs

1.4 μs

Fig. 34.16 Reflection CGS applied

4.2 μs

7 μs

to dynamic fiber splitting in unidirectional fiber-reinforced polymer composite (after [34.53])

Delamination Mechanics

studies have been concerned with using the technique to extract stress intensity factors. These range from classical applications of photo elasticity in early times [34.55, 56] to automated [34.57, 58] and digital [34.59] photoelasticity more recently. Load stepping or temporal phase unwrapping [34.60] have been used to reduce the effects of systematic noise. Phase-stepping photoelasticity has been used to determine shear stress distributions at fiber–matrix interfaces and fiber breaks [34.61]. In experiments with sapphire fibers embedded in epoxy, the interfacial shear stresses obtained from the technique were compared with those calculated from the axial stresses measured by fluorescence spectroscopy [34.62]. Quantitative relations between interfacial strength and chemistry were made by Zhao et al. [34.63] who used the phase-stepping technique to examine stress distributions in fiber fragmentation experiments where the degree of cross-linking was varied. Photoelasticity was also used [34.65] to distinguish between delamination, the propagation of an active sliding zone, and full sliding in fiber pull-out experiments with optical glass fiber coated with a layer of acrylate a)

34.2 Delamination Phenomena

or gold–palladium alloy and embedded in an epoxy matrix. The coatings caused varying degrees of interface cracking and stick–slip frictional behavior. Competing failure mechanisms in a brittle–ductile–brittle laminate were also examined via photoelasticity [34.64]. Two Homalite 100 strips were bonded to a thin aluminum layer with three different adhesives and the single edgenotched specimens were loaded in three-point bending. The type of adhesive that was used altered the sequence of failure from a starter crack (perpendicular to the interface) in one of the Homalite 100 strips. Specimens with one adhesive failed entirely by interface cracks (Fig. 34.17a) that branched from the starter crack. A second adhesive produced some delamination, which was followed by crack initiation (Fig. 34.17b) in the upper Homalite 100 strip. Specimens with the third adhesive failed either by complete delamination or the combination of delamination and crack initiation in the upper Homalite 100 layer. A third example of the use of photoelasticity for identifying failure sequences and interfacial defects comes from the use of an infrared grey-field polariscope (IR-GFP) [34.66], which was used to distinguish between small particles and bubbles at bonded silicon interfaces. The defects produced different residual stress signatures in such a way that defects smaller than the 39.1 μm wavelength of the IR could be resolved. The applications of photoelasticity in dynamic interfacial cracking presented earlier and for elucidating slow cracking mechanisms and interfacial defects, just now presented, coupled with its relative simplicity, bring out the principal advantages of the technique.

Fig. 34.17a,b Photoelastic fringe patterns reveal (a) delamination and (b) reinitiation in the upper Homalite 100 strip (after [34.64])

Crack opening interferometry (COI) is a technique for measuring the gap or normal crack opening displacements (NCOD) between crack faces in transparent materials. Fizeau fringes are formed which are contours of NCOD. The principal advantages of the technique are that NCOD can be obtained to high resolution very close to the crack front, crack closure can be followed, and variations in crack front geometry during propagation and other three-dimensional effects can be examined. The main principle behind the technique is that, when light is reflected from two opposing surfaces, the reflected beams may interfere with one another and the resulting interference fringes can be used to determine the distance between the two surfaces. The technique

Part D 34.2

34.2.5 Quasistatic Interface Fracture and Crack Opening Interferometry b)

971

972

Part D

Applications

has been applied to study the topology of surfaces and the properties of thin films and metals [34.67]. It has also been used in a wide variety of fracture studies. In any case we have the combination of two beams represented by light vector magnitudes E 1 and E 2 of essentially equal amplitude, a, but having phase angles φ1 and φ2 at some particular position. Thus, if E 1 = a cos(φ1 − ωt)

S

P

n1

(34.19)

A φ E

n2

and

D

C F

Surface A

h γγ

E 2 = a cos(φ2 − ωt) ,

(34.20)

Then their combination E is

1 E = a cos (φ1 + φ2 ) − ωt , 2 where



a = a 2 [1 + cos(φ2 − φ1 )] .

(34.21)

(34.22)

(34.23)

we have a = 2a cos (πδ/λ) .

(34.24)

The intensity I of the combined beams is proportional to the square of the amplitude a. ¯ Thus I ∼ (a)2 = 4a2 cos2 (πδ/λ) ,

(34.25)

Part D 34.2

and it can be seen that the intensity of the combined beams has minima and maxima depending on the value of the phase difference δ. That is I = Imin = 0 for

δ = (2n + 1) λ/2 , n = 0, 1, 2, 3, . . .

(34.26)

and I = Imax

for δ = nλ ,

B

Fig. 34.18 Ray diagram for the formation of frings of

Since the phase angle difference (φ2 − φ1 ) is related to the linear phase difference δ through φ1 − φ2 = 2πδ/λ ,

Surface B n1

n = 1, 2, 3, . . . (34.27)

Equations (34.26) and (34.27) represent the interference effects (fringes) that are used in classical interferometry. Dark and light fringes are formed wherever conditions (34.26) and (34.27) are satisfied, respectively. We now consider how the linear phase difference arises between beams reflected from two opposing surfaces. The situation is shown schematically in Fig. 34.18 where the opposing surfaces A and B are inclined at an angle φ to one another and are separated by a medium having a refractive index n 2 such that n 2 < n 1 and

equal separation

n 2 < n 3 . We consider the combination of two rays SDP and SABCP that are incident on surfaces A and B, respectively at locations B and D whose separation BD is h. Since the optical path length of a beam passing through distances N L i in N media having refractive inn i L i , the optical path difference δ0 dices n i is i=1 between the rays SABCP and SDP is δ0 = [n 1 (SA + CP) + n 2 (AB + BC)] − n 1 (SD + DP) .

(34.28)

If the optical arrangement is such that the distance AC is small compared to SD and DP, then n 1 SD ≈ n 1 SA + n 2 AE , n 1 DP ≈ n 1 CP + n 2 FC , and (34.28) can be simplified to δ0 = n 2 (EB + BF) = 2n 2 h cos γ ,

(34.29)

where γ is the angle of incidence on surface B. Because the reflection at B involved a rare/dense interface, the ray SABCP incurs a further retardation of λ/2. The total linear phase difference between the beams arriving at P is therefore δ = 2n 2 cos γ + λ/2 .

(34.30)

From (34.26) and (34.30) we see that destructive interference occurs for nλ (34.31) , n = 0, 1, 2, 3, . . . . h= 2n 2 cos γ In addition, from (34.27) and (34.30), it can be seen that constructive interference occurs for (2n − 1)λ (34.32) h= , n = 1, 2, 3, . . . . 4n 2 cos γ

Delamination Mechanics

When the rays SDP and SABCP pass through a transparent body and are incident on crack faces (Fig. 34.19a) as surfaces A and B, the separation, h in (34.32) corresponds to the normal crack opening displacement Δu 2 . Thus dark fringes can be viewed (Fig. 34.19b) over the plan view of the crack wherever Δu 2 = u 2 |θ=π − u 2 |θ=−π =

nλ , 2n 2 cos γ

n = 0, 1, 2, 3, . . . .

(34.33)

This arrangement is known as crack opening interferometry (COI). Figure 34.19 brings out one of the principal advantages of the technique, namely that three-dimensional effects can be examined by measuring Δu 2 at various x3 locations through the thickness of the cracked component. If the crack is filled with air and a beam splitter is used to provide coincident illumination and viewing, then Δu 2 =

nλ 2

n = 0, 1, 2, 3, . . . .

(34.34)

For λ = 546 nm, the resolution in NCOD is therefore 273 nm, which can be improved to 136 nm by locating bright and dark fringes. The resolution of the technique can be further improved by using a liquid to fill the crack and by digital image analytical techniques, as follows. The light intensity of the fringe pattern is essentially a cosine function, as proposed by Tolansky [34.68] and implemented in [34.69–71]. The following equation

D x2, u2 r, θ x1, u1 BD = u2D –u2B = Δu2

B

b) Location of dark fringes Cracked

Uncracked

x1, u1 n 5

4

3

2 1 0 x ,u 3 3

Component thickness

Fig. 34.19 Measurement of NCOD in a cracked transpar-

ent body

can be used to interpolate between fringes:  

1 I 4πΔu 2 = ±1 ∓ cos , Ipp 2 λ

973

(34.35)

where I is the light intensity at a particular location and Ipp is the intensity difference between the bright and dark fringes being interpolated. These intensities can be measured from intensity profiles taken from the framegrabber signal behind the crack front. Depending on the magnification and working distance of the microscope used to view the fringes, resolutions in NCOD from 10–30 nm have been obtained at distances of 300 nm from the crack front. Fowkes [34.72] has shown that the variation of the index of refraction in the crack tip region can lead to uncertainty in the location of the crack tip. He estimated this error based only on the dilatation in the crack tip region. Recently, Kysar [34.73] explored the change of the index of refraction in greater depth. In his formulation, the change of birefringence due to the stress fields near the crack tip includes the optical path differences of parallel and perpendicular polarized light, the stress intensity factor, and the size of the K -dominant region. For the sandwich specimens used in [34.70] and [34.71], using Kysar’s [34.73] analysis gave the maximum uncertainty introduced by birefringence in the glass as 16 nm (for G = 12.5 J/m2 and a K -dominant region size of 100 μm). This value is negligible compared to the spatial resolution of 300 nm. Liang [34.74] showed that, since the slope of both fracture surfaces are not exactly normal to the incident light, there will be a relative uncertainty in the interpolation of NCOD. The uncertainty is equal to tan2 δ, where tan δ is the slope Δu 2,1 . In the current experiments, tan δ < 0.1, making the uncertainty in NCOD due to the slope of the surfaces less than 1%. For interpolation of NCOD values between fringes, this uncertainty was small, because the NCOD difference between fringes was only 136 nm. The experimental results showed an uncertainty of 10 nm in the NCOD due to either uneven reflections from the surfaces or nonuniform background reflections. Large deviations due to electronic noise in the camera signal or bright reflections from the other interface could be ignored, but small deviations could not be distinguished from the experimental uncertainty. The 10 nm uncertainty dominated in the interpolation of NCOD. Some examples of the utility of NCOD measurements in extracting stress intensity factors, traction–separation law parameters, and examining three-dimensional effects follow.

Part D 34.2

a) Central profile of a crack

34.2 Delamination Phenomena

974

Part D

Applications

Stress Intensity Factors Since the crack opening displacements of a cracked component are related to the stress intensity factor, it was a natural development to use the previously described displacement measurements to determine stress intensity factors. The extraction of mode I stress intensity factors from NCOD was originally proposed by Sommer [34.75], further developed by Crosley et al. [34.76], and has been reviewed by Packman [34.77] and Sommer [34.78]. If mixed-mode conditions are in effect then the NCOD are, by themselves, insufficient to determine the mode I and II stress intensity factors and additional information must be supplied [34.79]. We will first examine the pure mode I situation in a homogeneous material and consider the associated u 2 displacements near the crack front (Fig. 34.19a). We have  r 1/2 2(1 + ν) u2 = K1 E  2π  θ θ × sin (34.36) 2 − 2α − cos2 , 2 2

where α = ν for plane strain and α = ν/(1 + ν) for plane stress. The NCOD are then given by Δu 2 = u 2 |θ=π − u 2 |θ=−π  r 1/2 8(1 − ν2 ) = K1 E 2π

(plane strain) (34.37)

and

Part D 34.2

Δu 2 =

 r 1/2 8 K1 E 2π

(plane stress) ,

(34.38)

where r is now the distance behind the crack front. If the NCOD have been measured using crack opening interferometry, and considering, only for convenience, plane strain and dark fringes, then (34.33) and (34.37) may be used to extract K I . By noting the r location of a particular fringe n, we have  r 1/2 nλ 8(1 − ν2 ) , = K1 2n 2 cos γ E 2π n = 0, 1, 2, 3, . . . . (34.39)

Δu 2 =

Thus, for any fringe nλ E 16n 2 cos γ (1 − ν2 ) n = 0, 1, 2, 3, . . . .

K1 =



2π r

1/2 , (34.40)

The cost that crack opening interferometry pays for providing information on the three-dimensionality of any configuration is that only one displacement component, the NCOD, can be measured. Thus crack opening interferometry cannot, by itself, be used to extract mixed-mode stress intensity factors. In this situation, a second displacement component must be measured or a hybrid experimental/numerical stress analytical technique must be used. In the latter case, finite element solutions for the NCOD of interface cracks (which are inherently mixed mode) have been matched with measured NCOD in the K -dominant region and then used to extract mode I and II interfacial stress intensity factors [34.79, 80]. Traction–Separation Laws Some recent examples of data for epoxy–sapphire interfaces are shown in Fig. 34.20. The data is compared with finite element analyses that accounted for the viscoplastic nature of the epoxy and the constitutive behavior of the interfaces in terms of traction–separation laws. The first case (Fig. 34.20a) was a bare sapphire/epoxy interface: the sapphire was not treated with any self-assembled monolayer prior to bonding with epoxy. The K -dominant region can be identified from the regime with a slope of 0.5. Closer to the crack front, the crack faces were being pulled together so the NCOD had a steeper slope within a cohesive zone size of 0.35 μm. The origin of the tractions on the crack faces, which have also been noted in glass–epoxy [34.70] and quartz–epoxy [34.81] interface cracks, have yet to be identified, but humidity and electrostatic effects have been eliminated [34.70]. The traction–separation law that provided such a good fit (Fig. 34.20a) with the NCOD data was independent of mode mix and had a maximum traction higher than the yield strength of the epoxy. As a result, all increases in toughness with increasing shear were due to dissipation in the bulk epoxy [34.71]. When the sapphire was coated with a self-assembled monolayer (SAM) consisting of 10% bromoundecyltrichlorosilane (BrUTS) and 90% dodecyltrichlorosilane (DTS) prior to bonding the toughness of the interface increased and the cohesive zone size increased to 5.6 μm, leaving a much smaller K -dominant zone. The SAM covalently bonds to the sapphire and the Br end group makes an ionic bond with the epoxy. The traction separation law was a function of mode mix and, because the maximum traction was lower than the yield strength of the epoxy, no plastic zone was observed. In this case all the increase in toughness with mode mix

Delamination Mechanics

was accounted for in the traction–separation law which represented the pull-out of epoxy chains that had made ionic bonds with the Br end groups in the SAM [34.71].

Δu 2 = Ar λ .

a) Log NCOD (μm) – 0.4 – 0.6

Sapphire/epoxy Gss = 1.6 J/m2 Ψ = 20°

– 0.8 –1 – 1.2 CZS (μm)

– 1.4

Data 0.35

– 1.6 –0.5

0

0.5

1

1.5 Log r (μm)

b) Log NCOD (μm) 0 – 0.2 – 0.4

Coated sapphire 10% BrUTS Gss = 6.4 J/m2 Ψ = 2.6°

– 0.6 – 0.8 CZS (μm)

–1

Data 4.9 5.6 7 10.5

– 1.2 – 1.4 – 1.6 –0.5

0

0.5

1

1.5 2 Log r (μm)

Fig. 34.20a,b NCOD versus distance from the crack front (a) bare sapphire and (b) sapphire coated with a 10% BrUTS / 90% DTS self-assembled monolayer prior to epoxy bonding (after [34.71])

(34.41)

The value of λ decreased towards the specimen edge (Fig. 34.23) for all values of fracture mode mix that were considered. This leads to a stronger strain singularity near the edges. For bond-normal loading ψ = 16◦ and a negative shear-dominant mode-mix ψ = −54◦ the values of the exponent in the central portion of the specimen were in good agreement with plane-strain finite element analyses. The predicted plane-strain exponent

975

Fig. 34.21 Tunneling crack in a glass–polyurethane sandwich under mode I loading (after [34.82])

Part D 34.2

Three-Dimensional Effects Crack opening interferometry has been used to examine three-dimensional effects in interfacial cracks. One striking example came in the early work of Liechti and Knauss [34.82] in which cracks in a glass/polyurethane sandwich specimen tunneled (Fig. 34.21) along the interface without the crack front ever intersecting the free surfaces of the specimen. The width of the crack was about a third of the specimen width. This form of crack growth was due to the incompressibility of the polyurethane and the constraint provided by the glass, which gave rise to high stresses in the center of the layer. The tunneling was less pronounced when shear loading was applied to the specimen. However shear provided its own interesting features where crack closure occurred at the mouth of an existing crack and the crack then propagated as a bubble along the interface, opening at the leading front and closing at the trailing one. Three-dimensional effects in glass–epoxy bimaterial strip specimens have also been examined in a preliminary fashion using crack opening interferometry [34.83]. The fronts of the cracks were all convex in the direction of crack growth and did not change with mode mix as drastically as those in the sandwiched polyurethane sandwich specimens. However, the glass adherends were slightly beveled. When the bevels were removed, the crack front became concave in the direction of crack propagation (Fig. 34.22) as suggested by an analysis of Barsoum [34.84]. The beveling introduced an additional mode III component near the edges and apparently enough additional toughening was therefore encountered that the crack front was pinned back near the specimen edges. An elastic singularity analysis across the thickness of a cracked bimaterial strip specimen revealed a through-thickness variation in the power-law component λ in the expression for the NCOD

34.2 Delamination Phenomena

976

Part D

Applications

Part D 34.2

of 0.4 was only attained near the edges for a mode mix of ψ = 86.5◦ , although other results from the specimen center [34.4] had been much closer to the predicted plane-strain value. These results also suggest that interfacial crack front geometries may change with fracture mode mix, which may complicate fracture toughness calculations. Such complications arose in some work with strip blister specimens [34.5]. For short cracks, the crack front was smooth and straight (Fig. 34.24a). However, crack initiation from long cracks was complicated by the formation of undulating crack fronts, with the lagging portions acting as sources of shear bands (Fig. 34.24b). The undulating crack front and accompanying shear banding were presumably responsible for the much larger toughness values that were associated with long cracks. The phenomenon did not occur in strip blister specimens in which the epoxy layer was sandwiched between glass and aluminum. However NCOD profiles and crack extension characteristics were strongly affected by the additional constraint applied by the aluminum. Another example of undulating crack fronts has been observed in glass–epoxy, quartz–epoxy, and sapphire–epoxy sandwich specimens. In this case (Fig. 34.25), left-to-right propagation of the crack was accomplished via lateral tunneling of many small cracks. A tunneling event would initiate at a void that opened up at a random location along the crack front. The width of the tunneling crack was approximately similar to the diameter of the nucleating void. Several laterally propagating cracks can be seen in Fig. 34.25 along with the serrations that they produce in the crack front. This lateral tunneling mechanism leaves behind

very characteristic ridges on the epoxy fracture surface, which can be seen in the interferogram (Fig. 34.25) and more clearly in atomic force microscope scans of the fracture surface [34.70, 71]. The ridges were spaced about 2 μm apart and had a residual height of 20 nm under nominally mode I loading. More complex ridge patterns were left behind after mode II dominant loading. Exponent λ 0.7

0.6

Magnification

Run

X64

#1

X64

#2

X64

#3

X32

#4

0.5 ψ = 16°

0.4 – 0.5

– 0.25

0

0.25

0.5 x3/b

Exponent λ 0.7

Magnification

uos = 15.24 μm υo = 2.54 μm

uos = –12.7 μm υo = 2.29 μm

X64 X32 ψ = –54°

0.6

0.5 ψ = 86.5°

0.4 – 0.5

– 0.25

0

0.25

0.5 x3/b

Fig. 34.22 Crack growth (R → L) at a glass–epoxy inter-

Fig. 34.23 Through-thickness variation of NCOD expo-

face. The crack front is concave in the direction of crack growth (after [34.83])

nents for various fraction mode mixes in a glass–epoxy bimaterial strip specimen (after [34.83])

Delamination Mechanics

34.2 Delamination Phenomena

977

a)

Crack tip ux field

Crack tip uy field

Fig. 34.26 Near-crack-tip fringe patterns from a moiré interferometry diagnosis of an aluminium epoxy specimen under normal loading (after [34.85])

b)

34.2.6 Quasistatic Interface Fracture and Moiré Interferometry

Fig. 34.24 (a) Short and (b) long cracks in a glass–epoxy strip blister specimen. Note the shear bands in (b) (after [34.5])

34.2.7 Crack Nucleation from Bimaterial Corners and Moiré Interferometry Scale 0

50

100

150

200

250 μm

Fig. 34.25 Serrated front of a glass–epoxy interface crack. The serration are due small cracks tunneling along the main crack front (after [34.70])

Mohammed and Liechti [34.19] considered crack nucleation from epoxy–aluminum corners. Four corner angles α were considered (Fig. 34.27) and moiré interferometry was used to extract traction–separation parameters for the epoxy–aluminum interface using the

Part D 34.2

Moiré interferometry has been used to analyze interface cracks since the early 1990s. At that time the analyses were used in conjunction with finite element analyses in order to extract fracture parameters and determine the interfacial toughness of welds [34.86] and adhesively bonded joints [34.87,88]. Wang et al. [34.89] developed a six-axis mixed-mode fracture tester for determining the toughness of interfaces in microelectronics components. Moiré interferometry was used to determine the crack length in lap shear specimens. Nishioka et al. [34.85] used near-tip displacement fields measured by phase-shifting moiré interferometry (Fig. 34.26) to extract stress intensity factors for cracks at aluminum– epoxy interfaces in specimens where the mode mix was controlled by loading angle. Separated mode I and II energy release rates were also obtained and compared with values obtained from finite element analyses. The first report of moiré interferometry being used to measure real-time crack tip displacements in a bimaterial specimen under fatigue loading was by Zhong et al. [34.90]. Thermal loading led to mode I dominant stress states near the crack tip of a silicon–epoxy–FR04 specimen.

978

Part D

Applications

Fig. 34.27 Specimen geometry for crack nucleation from bimaterial corners with various angles α (all dimensions in mm)

Thickness = 12.7

P

95.3

P

76.2

α Aluminum

Grating

12.7

Strain gage 38.1

177.8

specimen with α = 0 (interface crack) and then validate the choice of parameters with specimens having angles α of 45◦ , 90◦ , and 135◦ . The specimen was loaded in a four-point bending rig. A stepper motor provided 0.1 mm/step and the load cell and the linearly varying deferential transformer (LVDT) that were used had sensitivities of 0.1 N and 0.1 mm, respectively. A fourbeam achromatic moiré interferometer [34.91] was used to measure displacements near the corner. A He–Ne laser source with a wavelength of 633 nm was used to generate a coherent beam. The four-beam arrangement made it possible to measure both in-plane displacement components. A 2048 × 2048 digital camera yielded a spatial resolution of 12.4 mm/pixel. Digital image analysis was carried out following each experiment in order to obtain the u 1 and u 2 displacement fields to a resolution of 0.417 mm (Fig. 34.28). b)

c)

d)

Part D 34.2

a)

38.1

A three-dimensional image analysis technique [34.92] was used to examine crack nucleation and growth from a bimaterial corner in glass–aluminum specimens under thermal cool down. Slow thermal cracks grew according to a G II = 0 criterion.

34.2.8 Bimaterial Corner Singularities and Photoelasticity Perhaps the classic paper in this area is the one by Parks [34.93], who applied photoelasticity to the stress analysis of corner singularities. The results were compared with theoretical analyses and limitations of the technique were explored. Digital photoelasticity was applied to the extraction of generalized stress intensity factors for a series of bimaterial wedges [34.94]. A multiparameter technique that made use of higherorder terms was used for the extraction. Results compared well with finite element analyses that made use of special higher-order elements at the corner. Thermal stresses were the focus of the photoelastic stress analysis conducted by Wang and coworkers [34.95, 96]. Differences in stresses for heating and cooling suggested that viscoelastic effects were important. Comparisons of measured stresses with several analyses were also made. Particular emphasis [34.97] was placed on reducing the edge singularity by introducing convex corners in polycarbonate–aluminum butt joint specimens. The photoelastic measurements confirmed that the singularity had been eliminated and that the ultimate strength of the joint was increased by 81% compared to a classical butt joint with 90◦ corners.

34.2.9 Crack Growth in Bonded Joints and Speckle

Fig. 34.28a–d Moiré interference fringes for an interface crack between epoxy and aluminum. (a) and (b) are the u 1 -fringes in the unloaded and loaded states, (c) and (d) are the corresponding u 2 fringes (after [34.19])

Speckle interferometry has been applied to a number of delamination problems. In 1992, Chiang and Lu [34.98] used a laser speckle technique to examine the displacement fields around a central crack in a bimaterial plate under remote shear loading. Local shear opening displacements agreed well with

Delamination Mechanics

predictions. However the normal crack opening displacements were much larger than predicted and no crack face contact was observed. This is in contrast to crack opening interferometry measurements, where crack closure was observed [34.99] and cracks traveled as bubbles under shear, opening at the leading crack front and closing at the trailing one. Wang and Chiang [34.100] made an innovative use of speckle in a scanning electron microscope (SEM) to examine the properties of the interphase region in a metal matrix SCS-6/Ti-6-4 composite. An effective Young’s modulus of the interphase was determined. Machida [34.101] made use of speckle photography with Fourier transformation processing to measure crack opening displacements in steel–epoxy specimens under mixed-mode loading. The extracted stress intensity factors agreed well with those obtained from a finite element analysis. Laser speckle interferometry was used [34.102] to measure strains in T-stiffeners bonded to tape laminate skin. Braided and tape layup stiffeners were compared. The former gave rise to more uniform strains, which resulted in a higher load-carrying capacity and more-benign failure mechanisms. The measured strain distributions were well predicted by the binary model of braided composites. Electronic speckle pattern interferometry was used [34.103] to determine strains in wood joints. The measurements confirmed that the phenol–resorcinol– formaldehyde and polyurethane adhesives that were used in the study infiltrated the wood cells and stiffened them, with the first adhesive providing a greater stiffening effect.

Thin film delamination in blister and superlayer specimens often involves out-of-plane deformations. For relatively large deflections, these can be measured by techniques such as shadow moiré and projected gratings. Such full-field techniques are useful for validating analytical or numerical models of delamination or they can be used to extract traction separation law parameters in conjunction with analysis. An example of the use of projected grating comes from the work of Shirani and Liechti [34.104]. The technique was applied to pressurized circular blister specimens of PMDA ODA on aluminum to measure the blister shape and volume. The competing requirements of sufficient stand-off distance and fringe visibility could not be reconciled in classical shadow

979

5 mm

Fig. 34.29 Projected grating image of a pressurized circular blister (after [34.104])

moiré. Parallel lines were projected on the surface of interest and the lateral deflection Δ of the projected lines is related to the out-of-plane displacement of the surface. Viewing from above provides a precise way of determining the shape of the crack front. A light source with grids of 20, 10 or 5 lines/mm was used to project a shadow of the lines on the specimen. A video system recorded the change in line spacing while the test was in progress Fig. 34.29. Prior to each test, a two-dimensional calibration of length/pixel was necessary for each magnification level. The following procedure was adopted for determining the deflection w of the blister at any particular location. The image analysis system was used to digitize recorded frames and then scan the video image to obtain profiles of intensity of light versus pixel position. The location and order of the dark or bright lines

1

2

1

Fig. 34.30 Shadow moiré fringe pattern of a delamination in a peninsula blister specimen (after [34.105])

Part D 34.2

34.2.10 Thin-Film Delamination and Out-of-Plane Displacements

34.2 Delamination Phenomena

980

Part D

Applications

were determined by identifying maxima and minima in the intensity profiles. The relative positions of the same bright or dark line in the undeformed and deformed configurations allowed the shift Δ to be determined. The deflection is related to the shift and the angle of illumination α through u3 =

Δ . tan α

(34.42)

The lines were projected at an angle of 30◦ , giving rise to a resolution of 50 μm with a grating of 20 lines/mm and counting bright and dark fringes, which could be located to within 5 mm. The overall accuracy of this technique was found to be 3% by using a calibrated 45◦ wedge [34.106].

Shadow moiré was used in analyses [34.105] of pressurized peninsula blister specimens. The smaller plan form dimensions of this specimen led to smaller deflections, which could be well accommodated by the technique. The blister shapes were measured using shadow moiré in which a master grating of 10 lines per mm was used, providing a half-fringe resolution of 50.8 μm. The fringe patterns (Fig. 34.30) were used to track the crack front position and displacement profiles as a function of time. They were also used in bulge tests that were conducted after total delamination along the peninsula. In this case, they were used to validate approximate membrane analyses of the specimens and extract the Young’s modulus of the delaminating film and the residual stress.

34.3 Conclusions

Part D 34

It should be clear from this review of delamination mechanics that optical techniques have played a large role in developing our understanding of the nucleation and growth of delaminations. The experiments, particularly in the dynamic regime, have shed light on the behavior of faults in earthquake-prone regions. At the same time, other investigations have led to improved understanding of failure mechanisms in very small components such as microelectronic devices. In addition, the techniques described have been applied to a wide variety of materials. Thus it is hoped that this chapter will provide some guidance for students and researchers who are trying to understand some new aspect of delamination mechanics. In terms of challenges for the future, one area in which optical techniques offer a lot of promise is in the determination of traction–separation laws for interfaces. This is due to the fact that such techniques allow for displacements near the crack front to be measured. Currently, many traction–separation laws are determined iteratively. However, this is time consuming and can raise questions regarding uniqueness, so the challenge is to determine traction–separation laws directly. Some

progress is being made in this direction, but a lot more can be done. Another area where optical techniques may be invaluable is in tracking the diffusion of solvents near interfaces and determining the effect that moisture has on traction–separation laws. Perhaps the biggest challenge in measurements of adhesion is likely to come from miniaturization as microelectronics devices shrink in size and micro(MEMS) and nano-electromechanical systems (NEMS) devices become more common. These may all require spatially programmable adhesion. New tools will be needed to assess the strength and durability of adhesion at increasingly smaller scales. Some progress has been made in this area with the introduction [34.107, 108] of the interfacial force microscope (IFM) and related analyses of the force profiles that it measures for the extracting of the traction–separation laws of functionalized surfaces [34.109]. The mechanics of cellular and biological adhesion is a developing area where new probes will be needed. Although the contact zones for this class of problems lie in the mesoscale, the ability to span a large range of forces in a variety of environments presents some interesting challenges.

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34.2

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Part D 34

985

Structural Tes 35. Structural Testing Applications

Ashok Kumar Ghosh

A structure is an engineered system that must fulfill multiple requirements. Testing of a structural system has a definite role to play in the overall process of development, be it in the context of developing new products or assessing existing products. The design-bytesting methodology (Fig. 35.1) seems to work well in this development process.

35.1 Past, Present, and Future of Structural Testing ............................. 987 35.2 Management Approach to Structural Testing ............................. 35.2.1 Phase 1 – Planning and Control ..... 35.2.2 Phase 2 – Test Preparation ............ 35.2.3 Phase 3 – Execution and Documentation .....................

990 990 993 996

35.3 Case Studies ......................................... 997 35.3.1 Models for High-Rate and High-Temperature Steel Behavior in the NIST World Trade Center Investigation ..................... 997 35.3.2 Testing of Concrete Highway Bridges – A World Bank Project ...... 1002 35.3.3 A New Design for a Lightweight Automotive Airbag ....................... 1009 35.4 Future Trends....................................... 1012 References .................................................. 1013 of concrete highway bridges. The final case study details the development of a lightweight automobile airbag from inception through innovation. This case study also illustrates the close ties between structural testing and numerical simulation. The chapter closes with examples of a few future structural systems, highlighting the complexity involved in their testing.

Testing expands the knowledge base regarding different parameters that influence the design and development. As one learns more about a particular process, one tends to cut efforts on physical testing to reduce time and cost, resulting in greater emphasis on predictive modeling. Three primary reasons attribute to such a shift in emphasis.

Part D 35

This chapter addresses various aspects of testing of a structural system. The importance of the management approach to planning and performing structural tests (ST) is emphasized. When resources are limited, this approach becomes critical to the successful implementation of a testing program. The chapter starts with illustrations of some of the past structures that were built using concepts developed through testing. Most often, these structures were built even before the principles of engineering mechanics were understood. At present, due to the unprecedented expansion of computing power, numerical and experimental techniques are interchangeably used in simulating complex natural phenomena. Despite encouraging results from simulation and predictive modeling, structural testing is still a very valuable tool in the industrial development of product and process, and its success depends on judicious choice of testing method, instrumentation, data acquisition, and allocation of resources. A generic description of the current test equipment and types of measurements is included in this chapter. After careful selection, three case studies are included. The complexity involved with the modeling of structural steel retrieved from the collapse site of the World Trade Center (WTC) under high-rate and high-temperature conditions is highlighted in the first case study. The second case study highlights the importance of the planning phase in providing the basis for manageable and high-quality testing

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First, the unprecedented expansion of computing power in the last decade has enabled us to develop highperformance scientific computing hardware. Between the 1990s and the present, this computing speed has improved 20 times from 2 tera-operations per second (OPS) to 40 tera OPS. All these computational capabilities are ideal for supporting highly complex and interdisciplinary application-oriented developments. Second, with these improved supercomputing capabilities, it is often possible to model a complex physical phenomenon. For most of the conventional problems, there is extraordinary agreement between the simulation and the real situation, thus making the requirement for experiments less important. The initiative Engineering Sciences for Modeling and Simulations-Based Life-Cycle Engineering is a collaborative research program by the National Science Foundation (NSF) and the Sandia National Laboratories (SNL) that focuses on advancing the fundamental knowledge needed to support advanced computer simulations. SNL is moving towards engineering processes in which decisions are based heavily on computational simulations, thus capitalizing on the available high-performance computing platforms. In the future, dominant use of analysis will be a part of the design process. Third, engineering design and development through experimentation will be used as a fact-finding tool that allows one to understand the circumstances surrounding a given problem and the main variables controlling it. Characteristically, complex systems are nonlinear, coupled, and exhibit multiple subsystems having mul-

Identification of the need

Definition of the problem

Design

Physical prototyping

tiple lengths and time scales. Reliable predictions of such systems need conscious coordination and integration of experiment, theory, and computer simulation. The challenge is to increase predictability by creating a robust design with reduced sensitivity to any influencing variable. A characteristic of future development will be simulation-driven product development (Fig. 35.2), where repeated tests are eliminated to cut the time and cost. Numerical and experimental techniques are interchangeably used in simulating complex natural phenomena. In the past, scientists such as Leonardo da Vinci, Galileo, and many others adopted experimental techniques to obtain solutions to complex problems whereas continuum mechanics methods would lead to partial differential equations. Many argue that, in the future, numerical techniques will take over the discipline of experimental techniques to solve so-called not-socomplex problems. This seems like the end of the road for a fading discipline. In the past, considerable efforts were made towards the development of knowledge based on actual testing of structural systems. Most of these test results were used to validate predictive models. Validating a predictive model based on experimental test data is not an efficient use of resources. Instead, simulation-driven predictive models will be developed with minimal involvement of experimentation. Structural testing would be imperative for problems that are qualified as either an unknown or important situation. Future structural systems will be active, intelligent, and adaptive, similar to live biological structures. Live

Testing

Analysis and optimization

Stop

Production

Fig. 35.1 Design-by-testing methodology process flowchart

Part D 35

Virtual prototyping Identification of the need and definition of the problem

Simulation-driven product development

Design

Parametric design and optimization

Fig. 35.2 Simulation-driven product development process flowchart

Physical prototyping and testing

Production

Structural Testing Applications

in this context means that these structures will have characteristics like build-in sensors, being adaptive to the prevailing environment, and self-diagnostics and self-repairing/healing. Understanding biological complexity requires sophisticated approaches to integrating scientists from a range of disciplines. These include biology, physics, chemistry, geology, hydrology, social sciences, statistics, mathematics, computer science, and engineering (including mechanics and materials). These collaborations cannot be constrained by institutional, departmental or disciplinary boundaries. The NSF is in the process of initiating a model-based simulation (MBS), which integrates physical test equipment with system simulation software in a virtual test environment aimed at dramatically reducing product development time and cost [35.1]. MBS will involve combining numerical methods such as finite element and finite difference methods, together with statistical methods and reliability, heuristics, stochastic processes, etc., all combined using supercomputer systems to enable simulations, visualizations, and virtual testing. Expected results could be less physical testing or, at best, strategically planned physical testing in the conduct of research and development (R&D). The manufacturing of the prototype Boeing 777 aircraft, for example, was based on computer-aided design and simulation. The training of engineers and technicians in testing and laboratory activities has decreased significantly since the 1960s due to reduced funding. Funding available for conducting tests, acquiring and maintaining equipment, and developing new techniques continues to decrease, and yet more detailed information is expected from the limited testing performed. Thus, it is absolutely imperative to plan the test program so that it is possible to understand the system from the minimum number of tests performed. This chapter deals with different aspects of modern ST, where the material used is inert and the loadings

35.1 Past, Present, and Future of Structural Testing

applied are conventional. This chapter will start with a discussion of the past, present, and future of structural testing in Sect. 35.1 with some historical illustrations on structural testing; salient features of structural testing that are characteristic of the past will be touched upon. Section 35.2 covers the management approach to structural testing, describing guidelines for planning and performing tests. A generic description of the test equipment and types of measurements is included. Section 35.3 consists of three case studies. Case study 1 describes modeling for high-strainrate and high-temperature steel behavior in the National Institute of Standard and Technology (NIST) World Trade Center (WTC) investigation. A major part of the investigation was the metallurgical analysis of structural steel recovered from the two WTC towers. The case study highlights the challenges faced in terms of cataloging the recovered steel to characterize the failure modes, quantifying the temperature excursions that the recovered steel experienced, and characterizing and modeling its mechanical properties. The second case study highlights how management plays a crucial role in the planning of a major testing program funded by the World Bank. The planning phase provides the basis for a manageable and high-quality testing process. The project involves testing of concrete highway bridges. Case study 3 details the development of a lightweight automobile airbag from inception through innovation to engineering development. The study demonstrates the engineering progression of the design and the role that testing and analysis played in the development. New test and analysis methods were developed as required during the engineering phase as the design was eventually completed and qualified. This case study also illustrates the close ties between structural testing and numerical simulation.

and vibration, acoustics, thermal, radiation, etc. The building is also required to be aesthetically pleasing. Through testing, the response of the structure under applied loads (force, pressure, temperature, shock, vibration, and other loading conditions) is determined. It is also common for a structure to be required to fulfill new, different or extended functions after it has been designed and built. Most structural testing is considered

Part D 35.1

35.1 Past, Present, and Future of Structural Testing A structure can be a large system (building, bridge, ship, or aircraft) or it can be something as small as a mechanical robotic actuator or an electronic component made to perform multiple functions; for example, a building structure has the primary function of withstanding loads (self-weight/dead, superimposed/live, wind, seismic, etc.), while providing the users with safety and comfort from the effects of external loads, from shock

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Part D 35.1

to be an art performed by experienced engineers who are familiar with the structure, who will then evaluate and interpret the test results based on their knowledge and experience as much as through prescribed procedures and formulae. In this chapter, an effort is made to present this knowledge of structural testing in a systematic manner. The history of documented structural testing is as old as the time of Leonardo da Vinci’s tensile strength test. The test involved loading iron wires of different lengths. Leonardo da Vinci used a setup similar to the one shown in Fig. 35.3. He used wires of similar diameter and different lengths to suspend the basket (b) from an unyielding support (a). The basket was filled slowly with sand, fed from a hopper (c) with an arrangement so that, when the wire breaks, a spring closes the hopper opening. Da Vinci experimentally recorded the failure loads for different length wires. He observed that longer wires were weaker than shorter wires. Early investigators of the time had no explanation for this observation based on the concept of the classical mechanics of materials. What was true for steel as observed by Leonardo da Vinci in the 15-th century is true today for any engineering material. The compressive strength of concrete cylinder (length/diameter = 2), a heterogeneous material, shown in Fig. 35.4, clearly demonstrates the same phenomenon [35.2]. Galileo Galilei demonstrated how structural shape and geometry can influence load-carrying capacity. He arrived at the correct conclusion that the bending strength of a rectangular cantilever beam is directly proportional to its width but proportional to the square of its height. After Galileo came the English physicist Robert Hooke, who observed that the force with which a spring attempts to regain its natural position is proportional to the distance by which it has been displaced. Around the same time, a group of mathematicians and physicists contributed to concepts of force, mass, and formulations of the principles of effect and counter effect. Towards the end of the 17-th century, the first theoretical investigations were carried out on the static behavior of vaults, which is among the most important elements of contemporary engineering structures. The next noteworthy work was that by Leonard Euler, who published an exhaustive treatise on plane elastic curves (bending lines). Knowledge of the properties of building materials has gradually developed into an important branch of engineering science. In the later part of the 18-th century, numerous strength tests, mainly compression of stone and mortar and bending tests with iron-reinforced

Fig. 35.3 Test setup used by Leonardo da Vinci (see text)

a

c

b

Strength (% of std 15×30 cm cylinder) 120 115 110 105 100 95 90 85 80

0

15

30

45

60

75

90

105

Diameter of cylinder (cm)

Fig. 35.4 Concrete cylinder under compression

stone beams, were carried out by a large number of scientists. Thanks to the ceaseless research work of untold, able engineers with theoretical backgrounds, the methods of structural analysis have since been steadily perfected and extended to new tasks. The knowledge base generated through these efforts has contributed to the development of numerous predictive models. Noone contributed as much as Navier did in the later part of 18-th century and early part of the 19-th century. In his project for the suspension bridge over the river Seine, Navier [35.3] carefully selected the material and designed the shape of the anchors based on actual calculations of mechanics Fig. 35.5. In the past, structures were often developed before the principles of engineering mechanics were understood. The era of theoretical mechanics followed the era of applied mechanics, and personalities such as Poisson, Cauchy, Lamé, Saint-Venant, and many others contributed to the growth of the knowledge base on the theory of structures.

Structural Testing Applications

35.1 Past, Present, and Future of Structural Testing

989

Fig. 35.5 Navier’s concept for an-

choring the cable (see text) Anchor

Test specimen Pivot

Scale

Fig. 35.6 Capp’s

single-lever extensometer

ories. Among these problems is the concentration of stresses, which occurs in complex structural elements in the immediate vicinity of abrupt changes of crosssection, where the concept of stress optics can be ideally applied. As far as measurement devices are concerned, for a long time experimentalists from all over the world developed various methods independently without any standardization. Capp’s single-lever extensometer [35.4] was one of the early measurement devices of surface strains (Fig. 35.6). For more information on the theory and application of various kinds of extensometers see Chap. 13. The next generation of extensometers were mounted with mirrors replacing the moving pointer of the rackand-pinion system, resulting in greater accuracies. They were difficult to calibrate, sensitive to vibrations, bulky, and had restrictions on where they could be mounted on the structure or specimen. Other optical methods are the moiré method (for surface displacement), photoelasticity (for complicated assemblies and parts), brittle coating (for surface crack pattern), holography (for in-plane displacements), speckle pattern (for online monitoring of surface deformation), and digital image correlation. Earlier descriptions of mechanical measurements of displacement showed the use of mechanical gages. The first electrically based displacement measurement system was the linear variable differential transformer (LVDT). It was not until the invention of electric resistance strain gages and their application in various devices that experimental mechanics began to have a great impact on the design of structures. Most technologies associated with structural testing have been revised or improved in the past few decades; several new technologies have also been introduced. Portable equipment has been developed, permitting field investigation of a wide range of potential problems with capabilities to perform complicated tests

Part D 35.1

The Industrial Revolution in England transformed the whole world with coal, steam engines, and railways. Great Britain was the first country to produce iron and steel in large quantities. With improved quality (high strength, elasticity and uniformity) came improved economy, as engineers could determine the dimensions of the structural elements very accurately. With the invention of the hydraulic binding agent called cement by Joseph Aspdin came the age of cement and reinforced concrete. During the 19-th century, the dimensions of structures were generally determined so that the admissible stress, calculated on the basis of Hooke’s law, was limited to a certain fraction of the breaking stress (the safety factor). It was found, however, that the safety factor thus defined did not represent a reliable measure of the actual safety of the structure, especially for indeterminate structures and structures made with concrete and masonry. With the statically indeterminate structures, however, the local application of a stress beyond the elastic limit will generally lead to a compensation of forces, inasmuch as overstressed parts are relieved by the action of less heavily stressed members. It is obviously only with the aid of extensive tests that these highly complicated conditions can be clarified. There are other problems connected with the strength of materials, which are impossible or difficult to solve on the basis of the classical methods of the mathematical the-

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under adverse conditions. Most of the data-mining methods/techniques are discussed in Parts B & C of this handbook. Despite encouraging results from simulation and predictive modeling, industry is still far from

reaching the point where it can discard experimentation. Testing remains an important activity, and its success depends on judicious choice of testing method, instrumentation, data acquisition, and allocation of resources.

35.2 Management Approach to Structural Testing Test purposes and evaluation criteria are the necessary guides in planning and performing tests as well as the basis for interpreting the test results. For any type of structural testing the activities can be grouped into three phases

• • •

Phase 1, planning and control Phase 2, test preparation Phase 3, execution and documentation

An important part of testing is the effort spent during the planning and preparation. As a rule of thumb, the distribution of efforts is 20% for planning, 40% preparation, and 40% for the actual execution and documentation of the test program. This means that, during the creation of the functional specification, a master test plan is established. The master test plan describes who performs which type of test and when these tests are performed. On the basis of the agreed test plan, more detailed test plans are made. It is worth noting that there may be legal or regulatory requirements that must be considered while planning for such tests.

35.2.1 Phase 1 – Planning and Control

Part D 35.2

The planning and control phase starts during the specification of the functional requirements. The planning phase provides the basis for a manageable and highquality testing process. After the test assignment has been approved, the test team is to be assembled carefully and the test strategy is defined. Manpower selection is crucial for successful outcomes from the testing program. Most testing requires specialist knowledge and skills and the accompanying education to derive high-quality results. A test team typically consists of personnel belonging to a large variety of disciplines. Adequate participation of testing specialists is essential, both in the area of test management and in the area of testing techniques. Test strategy is basically a communication process with all parties involved, trying to define the parts and allotting the necessary time

frame. The aim is to have the most feasible coverage of the testing organization and to define the test infrastructure. The objective of the next part of this phase is to manage the progress of testing with regard to the time and resources used. In accordance with the test plan, the testing process and the quality of the test object are documented and reported. The most important deliverable of a test program is a quality test report which also describes the accompanying risks. From the start of the testing process, testers develop a view of quality. It is important that quality indicators are established during all phases of the testing process. Periodically, and when asked, management receives a quality report on the status on the testing process to decide if the testing should be continued or discarded. Major issues that influence the decision-making process during this phase are described in the following subsections. Safety Requirement Potential hazards associated with conducting structural tests can be introduced anywhere from a minor source such as electrical shock to a major source like the failure of a hoisting crane, the snapping of a prestressed tendon during tensioning, or the inhalation of toxic fumes. Insights into safety-related issues can be very useful during the planning stage. There are many agencies and regulations such as the Occupational Safety and Health Administration (OSHA) [35.5]), the National Environmental Policy Act (NEPA) [35.6], and the Resource Conservation and Recovery Act (RCRA) [35.7]) that enforce safety laws in the workplace. Accuracy and Resolution Some types of tests (e.g., modal analysis) often require many cycles of testing to develop statistically sufficient insights into structural behavior. Test quality is defined through two parameters, obtaining sufficient information to describe structural response, and verifying that the information obtained is valid. A well-executed structural test provides objective evidence of structural behavior.

Structural Testing Applications

Specifications (Requirements of Code) Codes and specifications are developed to bring standardization to experimentalist all over the world. Structural testing has been profoundly affected by the development of codes and specifications. The American Society for Testing and Materials (ASTM [35.8]) has developed test procedures for various kinds of materials. There are organizations who develop test procedures and specific requirements for structural testing. The American Railway Engineering Association (AREA [35.9]), the American Association of State Highway Transportation Officials (AASHTO [35.10]), and the American Institute of Steel Construction (AISC [35.11]) are a few. The way a technique is applied depends on the way the specifications are structured. Based on the testing strategy and the development documentation, appropriate test specification techniques are selected and tailored during the preparation phase. Chapter 17 of the International Building Code 2003 (IBC [35.12]) deals with structural tests and special inspections. Where proposed construction cannot be designed by approved engineering analysis, or where the proposed construction does not comply with the applicable material design standard, the system of construction or the structural unit and the connections shall be subjected to the tests prescribed in section 1714 [35.12]. Classification of Structural Testing There are different ways in which structural testing can be classified: full scale and models, calibrations, acceptance or quality assurance types, proof tests, material characterization, etc. These tests are aimed to determine stiffness or load deformation, energy absorption, strength or failure load, vibration mode or other structural characteristics.

Load Test. The purpose of conducting load tests is usu-

ally to prove that a structure or its components can withstand the anticipated loads. For example, a highway bridge is designed to withstand a particular traffic load.

991

Through load tests on the actual structure, this value can be determined. Section 35.3.2 deals with the load test of a bridge structure. Destructive/Failure Test. Often tests are carried out

to understand the failure characteristics of a structure. Failure tests also indicate the reserve strength and deformation capabilities in a structure. Destructive testing is performed to gain knowledge of the failure modes. At present, the knowledge base of the structural response to a blast load is not well established. To develop design criteria, destructive tests are performed to understand how a structural system will respond to blast loads. Once this design basis is established, computer simulation will replace future destructive testing to save time and cost. Partial Load Test. Often, samples are taken from the test structure in order to understand the characteristics of the material used for the structure. This test is performed when it is necessary to predict the behavior of the structure, but the design and construction drawings are not available or have changed. More on this type of testing is presented in Sect. 35.3.2. Nondestructive Test. Nondestructive tests are done to

understand the elastic response of a structure and can be classified into diagnostic and proof tests. In a diagnostic load test, the selected load is placed at designated locations on the structure and the response of an individual member of the structure is measured and analyzed. During a proof load test, a structure is carefully and incrementally loaded until the structure approaches its elastic limit. At this point, the load is removed and the maximum applied load (the proof load) and its position on the structure are recorded. Scaled/Model Test. Full-scale tests are time and re-

source intensive. Often, scaled tests are done to predict the behavior of a structure by extrapolating observations based on scaled/model tests. There can be serious concerns with any scaled test, and the output should be properly analyzed to arrive at a proper prediction. Sometimes these concerns can be addressed by conducting a series of tests with different scale factors to determine the relation of the observed behavior with size; for example, modal analysis (mode shapes and frequencies) can predict how a structure will store deformation energy under a given loading condition. Modal analysis judgment and experience give insight

Part D 35.2

Full-Scale Test. Characteristics of a structure are influenced by many parameters such as material (nonhomogenous) properties, construction and fabrication techniques, load environment, boundary conditions. The influence of these parameters is not linear, thus making prediction of scaled models difficult. When possible, full-scale structural tests are recommended for accurate prediction of the characteristics of the system.

35.2 Management Approach to Structural Testing

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Identified: • Scope of the test program is clearly defined and understood. • Resources are recognized, including information on the structure to be tested (new or old). • Specific requirements including the time frame are determined. • Deliverables are defined.

Visualize the structure in service. Define: • loading environment, • boundary conditions, • simulate a test program fulfilling all the testing requirements.

Is it possible to account for all possible loading?

No

Is there a code of specification in place?

Yes

Study code specifications on • test specimen requirement, • loading requirement, • data acquisition systems needed. Evaluate potential hazards and determine the controlling measures.

Yes

No

Carry out load sensitivity analysis and determine possible loading that can be accounted for in the test program. Develop a predictive model to superimpose various load cases.

• Prepare the test structure. • Simulate the boundary conditions. • Arrange the instrumentation and data acquisition system. • Check the connections carefully. • Check the safety requirements. • Prepare for the testing and documentation.

Fig. 35.7 Flowchart of the test preparation during phase 2

Part D 35.2

into the dynamic response of a structure or component and avoid instability. To perform a modal test, the test item must be excited (naturally or artificially) in the same manner as the structure would vibrate in different modes. The response to the excitation can be measured by the conventional transducers, which were discussed in Parts B & C of this Handbook. A typical modal test includes an array of accelerometers placed on a structure and connected to amplifiers and a data recording system. An excitation can also be fed to the recording system to collect simultaneous data. Thus a modal analysis is often used to troubleshoot a vibrating system. Material Test. Structural testing and materials testing must both be considered to construct an accurate predictive model. For material testing, specimens should be extracted from similar structures to be analyzed. Chapters from part B & C of this Handbook contains detailed descriptions of various conventional tests that are performed to characterize materials used to build the structural system.

Forensic Tests. Forensic tests are performed to un-

derstand the real cause of any failure of a structure. Failure is a quick phenomenon to which a number of factors contribute. Analysis of such a phenomenon is very challenging. Forensic tests that determined the sequences of various phenomena that contributed to the eventual collapse of the Twin Towers are highlighted in Sect. 35.3.1. The output of the planning and control phase is a statement or description of the work to be performed based on a feasibility study. Feasible means that a test is possible on the basis of availability of resources, such as manpower, funding, equipment, management tools, and time. This phase will also clearly define

• • • •

the tests to be performed the available resources the specifications (administrative, legal, codal and safety) to be met the output and information on the quality and accuracy of the results

Structural Testing Applications

35.2.2 Phase 2 – Test Preparation After the planning and control phase, preparation for actual tests will start. In this phase a systematic study is carried out to understand the test program and deliverables to ensure preparation for a specific test will be thorough and adequate. The structure to be tested is to be designed and developed as if it was a new structure. In the case of an existing structure, the test structure will be identified. Figure 35.7 shows the flowchart of the test preparation during this phase. Loading on Structure The loading conditions used in the laboratory and field tests should simulate the service or use conditions. Often, equivalent static loads are used to adequately represent and simplify some complex dynamic environment. In addition to the magnitude of loading, the point of application and the directions are factors that influence any testing program. When the structural system behaves linearly, it is possible to apply loading sequentially. Again, for a linear-elastic system, it is possible to determine the forcing functions mathematically from the measured responses. Dead Load Dead loads do not typically change with time, or they change so slowly with time that they produce only static

35.2 Management Approach to Structural Testing

responses. Dead loads include the weight of the structure and loads arising from any permanently attached elements. Dead loads can be accurately estimated knowing the dimensions and density of the elements. Superimposed Load Loads that are temporary in nature come under the superimposed load category, like the weight of people in a building. Loads due to wind, earthquake, vibration of moving vehicles, etc. are treated as live or superimposed loads. Often transient loads that arise from impacts, gusts, and other rapidly varying loads are accounted for by additional multiplying factors. Based on safety and serviceability requirements, multiplying factors for the building structure are recommended in the International Building Code (IBC) 2003 [35.12]. When these values are not available, laboratory tests can be conducted to determine them. Thermal Load Simulations of thermal loading are rather complex, especially when full-scale tests are conducted. Case study 1 illustrates the challenges encountered while modeling the behavior of structural steel under high temperatures due to the combustion of the aircraft fuel inside the Twin Towers before their collapse. The towers were subjected to dead loads, imposed loads, thermal loads, and impact loads all at the same time.

Impact hammer

Spectrum analyzer

Heavy support

Accelerometer

Vibration meter

Fig. 35.8 Schematic diagram of vibration test setup

Part D 35.2

Line drive supply

Target beam

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Applications

Chapter 7 of the IBC 2003 deals with construction requirements for fire-resistance design for materials used in structures. ASTM E119 contains the necessary test procedures for fire testing. Dynamic/Vibration Load Dynamic testing is a widely accepted test method to characterize a structure and its components subjected to time-varying loads. The structure is excited at varied frequencies to determine the proof load, damping characteristics, dynamic modulus, mode shape, etc. Figure 35.8 shows a schematic of a typical experimental setup used to obtain dynamic characteristics of laminated composite beams under an impact load. Vibration of the beam causes the support to vibrate, which in turn influences the vibration of the beam. Thus, for proper interpretation of the test observation, the coupling effect between the support motion and the structure should be carefully accounted for. Blast and Explosive Load After the 9/11 terrorist attacks on the Twin Towers, the importance of investigating structures under blast loads has intensified. Blast loads create waves in the air which impact on exposed structures. Loading systems for blast waves and explosive detonations usually require special facilities. Even for very small quantities of explosives, rugged facilities are needed to contain, or at least focus or direct, the released energies. The overpressure/duration relationship for a high explosive or nuclear explosion in air is shown in Fig. 35.9, where p is overpressure (or air blast pressure) and t is time.

Wind Load The design and development of a structure that will be exposed to wind loads for most of its service life will usually be tested in a wind tunnel to understand the response of the structure. Aerodynamics plays an important role in the design of an aircraft, automobile, or a high-rise building. The Tacoma Narrows suspension bridge collapsed due to loss of rigidity because of a 42 mph (18.78 m/s) wind, even though the structure was designed to withstand winds of up to 120 mph (53.64 m/s).

p/p0 1

Part D 35.2

0.8 p/p0 = (1–t/t0) e –t/t0 0.6 0.4 0.2 Positive phase

0 –0.2

Negative phase

0

0.2

0.4

0.6

0.8

In the figure, the decay of pressure after the first instantaneous rise is expressed exponentially. The value of the peak instantaneous overpressure p0 will depend on the distance of the point of measurement from the center of the explosion. In a TNT (trinitrotoluene) explosion, p0 might be 200 psi (1.38 MPa) or 300 psi (2.07 MPa) at the point of burst of the explosion, but would rapidly diminish with distance. In a nuclear explosion equivalent to 1000 tonnes of TNT, the peak overpressure would be 2000 psi (13.78 MPa) at 30 m from the center of the explosion. When an explosive charge is detonated in contact with or very close to a structure in air, the loading can no longer be considered uniform over the area of component faces. The accurate measurement of dynamic loads on structures is a demanding subject. The physical properties of blast waves as they strike the structure are most commonly recorded in terms of pressure. When an explosion occurs within an enclosed room, the impulse that loads the structure may be broken into two distinct parts, the shock front and a quasistatic gas pressure load. This initial blast load is due to the primary shock front and secondary reflected shock fronts. Air shock loadings typically have very short time durations, but may have peak pressures of several thousand pounds per square inch (1 psi = 689.47 Pa) or more. At later times, the structure is loaded primarily by a low-intensity longduration gas impulse that is due to the expansion of the explosive byproduct gases within the structure. Recording the pressure and acceleration data is a challenge by itself. High-speed photography (typically 2000–3000 frames per second or higher) is also used to measure structural response.

1

1.2

1.4

1.6

1.8 t/t0

Fig. 35.9 Blast load–duration relationship

Probabilistic Loads It is well known that structures and material characteristics are seldom known deterministically. This phenomenon was observed by Leonardo da Vinci in the 15-th century, and is still true today. Degrees of

Structural Testing Applications

uncertainty in determining material characteristics call for probabilistic approaches to determine the strength of material (as illustrated in case study 2). Loading Systems Loading systems are used to apply loads during structural testing. The objective is to replicate service load as nearly as possible. It is very difficult to create exact loading situations in the laboratory, and thus approximations must be made. Simulation Consider the simplest situation of applying a uniformly distributed load (UDL) on a plate structure. Figure 35.10 shows at least three ways one can achieve a UDL on a plate structure. The UDL remains uniform as long as the plate does not deform excessively. As soon as the plate deforms, the curvature effect will make the load nonuniform. While trying to simulate UDL, careful consideration is needed to make sure that the UDL is accurately represented throughout the loading history. One way to account for this difference between true UDL and simulated UDL is by adjusting for curvature effects in the response. Thus, it is important to check the method used to apply the loads. This check includes the locations, directions, and manner of applications (point, line, distributed, or inertial). It should also include loading rates and other time-dependent descriptions of the loading, such as magnitudes as the test progresses.

35.2 Management Approach to Structural Testing

995

Rows of hydraulic jacks Pneumatically

Using sand bags

Fig. 35.10 Application of uniformly distributed loads can be done different ways, all of which are different when the beam starts to bend excessively

Loading Equipment Testing machines are mechanical systems with various controls, including manual, mechanical/hydraulic, and electrical control with feedback systems. Two basic types of controls have the ability to impose either loads or displacements. An excellent description of test control systems is provided in [35.4]. Three main types of force testing machines have been developed and are in use – the dead weight, screw-driven, and hydraulically actuated. Primarily, there are two ways to control loading conditions (interaction between diagnostic and feedback).

1. An open system: performing tests in which the operator retains all the control on loading/unloading and the loading rate 2. Closed-loop systems: performing tests under an environment where the machine controls loading/

Readout equipment and data/control processor

Load signal Feedback selector

Load cell

Part D 35.2

Stroke signal Specimen

Closed loop Servo valve Control signal LVDT connector

Fig. 35.11 Components with connections for a closed-loop control system

Valve drive or controller

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unloading and the loading rate. However, the operator retains a manual override option. A simple block diagram of a closed-loop actuator with a servo– valve combination system is shown in Fig. 35.11. Calibration Measurement processes depend on the calibrations of the instruments used. Inconsistent calibrations are common and should be determined in any testing program and the proper corrections should be applied in the response data. By performing periodic checks, one can determine inconsistency in instruments behavior. Often, the instruments used for measurements could be the source of undesirable responses in a testing program. Calibration activities are essential in structural testing because they give credibility to the data generated. Test Control There are many ways to apply control and feedback systems in structural testing. It is important to select the proper test equipment to match the requirements.

35.2.3 Phase 3 – Execution and Documentation

Part D 35.2

In order to satisfy the agreed delivery schedule, the execution phase should start as soon as the preparation for the test program is complete. Sometimes, it may be necessary to conduct exploratory tests to determine the test domain. There are other times when part of the data is used to support a later part of the testing process. The difference between the actual test results and the expected results can indicate a problem area. This can be due to some error with the infrastructure, an invalid test case, or poor judgment. During the entire test execution stage, one should allow for quick and reliable reporting. Possible questions to be addressed are what percentage of the program is complete, how do these tests compare with the prediction, what are the trends, should testing be continued, etc. The infrastructure for testing includes all facilities and resources needed for structural testing. The selection of the correct infrastructure is strongly dependent on the type of resources and hardware platform available. Obtaining Data There are wide ranges of transducers, gages, sensors, and other measuring equipment available, as described in Parts B & C of this Handbook. The purpose of a transducer is to convert the physical changes occurring in

a structure due to applied loading into an electrical change, usually by altering voltage or current. There are transducers that provide a point response of the structure and there are others that represent full-field or whole-field results. Judicious selection of the testing machine, the right kind of sensors to collect data, and proper processing tools will yield the expected results. Often it is expected to combine both point and full-field techniques to better understand the response of the structure. Point Techniques. Measurement devices such as strain gages, capacitance gages, load cells, accelerometers, and thermocouples are used to collect structural response in terms of accelerations, displacements, forces, velocities, temperatures, and strains. Response data obtained through the point technique are then fed into an analytical model to analyze the response of the whole structure. For more information on these techniques, refer to Chap. 12 for strain gages. Full-Field Techniques. This technique examines a por-

tion of the structure or at times, the whole structure. Most of the optical methods provide full-field structural response. The use of such methods (moiré, holography, speckle, photoelasticity, brittle coating, optical-fiber strain gages, thermoelastic stress analysis, or x-ray analysis) prevent the use of another method in the same test. The theories behind most of the full-field techniques are provided in Chaps. Chapter 29 (optical methods), Chapter 20 (digital image correlation), Chapter 21 (geometric moiré), Chapter 22 (moiré interferometry), Chapter 24 (holography), Chapter 23 (speckle methods), Chapter 25 (photoelasticity), Chapter 14 (optical fibers), Chapter 26 (thermoelastic stress analysis), and Chapter 28 (x-ray stress analysis). Instrumentation and Data Acquisition and Processing The term instrumentation encompasses all the devices used to sense, connect, condition, display, and record the data. Selection of the instrumentation system is dictated by the requirements of the loading environment. For more information, refer to [35.4, Chap. 10]. Data Transmission. To avoid hazards in many test

situations, sensitive instruments and personnel can be located remotely. Thus the cables that connect transducers with the signal conditioner and recorder should be such that their signal attenuation is minimized. This

Structural Testing Applications

attenuation is more pronounced in the case of highfrequency signals. Signal Conditioning. Once a mechanical response is

sensed, the output of the sensor needs to be amplified for use in the recorders. This is known as signal conditioning, and is done after the response is converted into an electrical signal. Computer Data Acquisition. This topic is covered ex-

clusively in ([35.4, Chap. 10]). Completion Phase The completion phase is launched after the execution phase is completed. These activities are generally less structured or often forgotten, and when under high pressure, concessions are made. There should be some provisions on how to carry out additional tests (if so warranted) after evaluating test results based on the main test program. Experimental Methods – Evaluation Criteria. Test

evaluation criteria consist of methods according to which test results can be interpreted and understood. Evaluation criteria are usually specified through methods and techniques of engineering mechanics and

35.3 Case Studies

997

material science. The data obtained from the associated measurement devices used in testing will make correlations and comparisons of theory and test data more direct and easier to understand. Rarely do test results and theoretical predictions agree completely, even more so in the case of a complicated system. Much of the time needed for a test setup is used to ensure that the loading conditions are realistic, while attempting to comply with the test purposes and evaluation criteria. On rare occasions, when large discrepancies between theoretical predictions and test results occur, the experiment is systematically debugged in the following order: 1. Questions will arise about the validity of the calibration of the instruments, i. e., the relationships between known inputs and observed outputs and the uncertainties associated with the two measurement processes are quantified and defined; 2. Questions will arise on the signal conditioning and data acquisition; 3. Questions will arise about the capabilities of the test personnel. It is better for everyone to get involved and discover the problem as early as possible in the laboratory or under controlled conditions.

35.3 Case Studies disaster of September 11, 2001. That investigation addressed many aspects of the catastrophe, from occupant egress to the factors that affected how long the Twin Towers stood after being hit by the airplanes [35.13,14]. A major part of the investigation was the metallurgical analysis of structural steel recovered from the two WTC towers. The analysis included tasks to catalog the recovered steel, characterize the failure modes of the columns, quantify the temperature excursions that the recovered steel experienced, and characterize and model its mechanical properties. The last task consisted of two major subtasks:

35.3.1 Models for High-Rate and High-Temperature Steel Behavior in the NIST World Trade Center Investigation

1. Assess the quality of the steel recovered from the collapse site to ascertain whether it had the yield strength called for in the design drawings, and determine whether its properties were consistent with those expected of construction steels from the WTC construction era. 2. Provide material models for room-temperature stress–strain behavior, high-strain-rate stress sen-

Description In September 2002, the NIST became the lead agency in the investigation of the World Trade Center (WTC)

Part D 35.3

After careful selection, three case studies have been included in this chapter on structural testing. The first study is based on modeling aspects of high-rate and high-temperature tests on steel retrieved from the collapse of the World Trade Center disaster. This study only highlights the issues that were not addressed in earlier chapters. This investigation falls under forensic study. The next case study deals with the challenges faced while conducting nondestructive and load tests on multiple concrete highway bridges. The third case study illustrates how structural tests can innovate a product design.

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sitivity, and high-temperature plasticity and creep behavior to the groups that conducted the finite element modeling of the collapse sequence [35.15]. This case study summarizes the findings and methodologies of the second subtask. Challenges and Strategy Challenges. The team faced five challenges in complet-

ing the subtask of the metallurgical analysis:

Part D 35.3

1. Very little of the structural steel from the fire and impact floors remained when NIST assumed control of the investigation. Although the salvage teams recovered many important perimeter columns from near the area of impact in WTC 1, most of the structural steel was recycled very soon after the excavation of the site. 2. Identifying and cataloging the recovered steels was extremely time consuming. After the recovered columns arrived at NIST, team members attempted to identify their locations in the building using the original structural plans and various remaining identifying marks. 3. The WTC towers were constructed from a much wider variety of steels than is common in high-rise building construction. Unlike many tall buildings, which might use only three or four grades of steel supplied to ASTM specifications, the WTC employed 12 yield strength levels, many of which were proprietary grades from foreign steel mills. Four different fabricators, who purchased steel from both domestic and foreign mills, worked on the fire and impact floors alone. The wide variety of steels made it impossible to completely characterize each one. 4. Locating suitable areas to harvest test coupons was frequently difficult. The collapse and subsequent recovery efforts damaged many of the columns by plastically deforming them. Truss elements proved extremely vexing in this regard. Not only were their original locations in the building unidentifiable, but they were compressed into tight balls for removal from the collapse site during the recovery. This handling rarely left any material suitable for mechanical testing. 5. The accelerated time schedule necessary to deliver evaluated properties to the modeling groups frequently required the team to evaluate properties using literature values, even when recovered steel existed in the NIST inventory.

Strategy. To evaluate the quality of the steel and to es-

tablish baseline mechanical properties, the group tested at least one tensile specimen from each yield strength level. Additional specimens from relevant steels from the fire and impact zones were also characterized. To characterize the high-rate stress sensitivity, the focus was on perimeter and core column steels from near the impact zone. These columns were most relevant for determining the severity of damage to the buildings caused by the impact of the aircraft. To characterize the hightemperature deformation behavior, the focus was on steels from core columns and the floor trusses. Creep characterization focused on the floor truss steels only. In parallel to the experimental characterization, the team developed methods to estimate properties of other steels not characterized, using literature data and models as well as test results on WTC steels. Test Methods Room-Temperature Tensile Tests. Evaluating the qual-

ity and establishing the baseline tensile behavior of the steels from the fire and impact zones required several hundred room-temperature tensile tests that encompassed all the relevant strength levels and forms, such as plates, rolled shapes, and truss components. The test protocol generally followed ASTM E 8 (and in some cases ASTM A 370) for comparison to mill-test report data. All tests retained the extensometer on the specimen past the point where the specimen began to neck to capture the full stress–strain behavior. Results of these tests were used to assess the quality of the steel, as well as to form the basis of the stress–strain models for the high-rate and high-temperature behavior. High-Strain-Rate Tests. Relevant steels from the im-

pact zones were tested at rates up to 500 s−1 to establish the strain rate sensitivities of their strengths. The test suite comprised eight perimeter column steels with 50 ksi ≤ Fy ≤ 100 ksi (345 MPa ≤ Fy ≤ 689 MPa) and five core column steels with 36 ksi ≤ Fy ≤ 42 ksi (248 MPa ≤ Fy ≤ 290 MPa). The core column group included examples of both plates from built-up box columns and wide-flange shapes. Specimens from both shapes were machined from sections flame-cut from the original columns. Specimen gage sections were located at the one-half or one-quarter depth positions (as required by E 8 and A 370) and well away from any flame-cut edge. Tension tests in the range 50 s−1 ≤ ε˙ ≤ 500 s−1 employed a servohydraulic test machine with a special slack adapter grip that allowed the actuator to reach full speed before loading the specimen [35.16].

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imens with d = 12.7 mm. Strains were measured on a 25.4 mm gage length. Test temperatures were 400 ◦ C, 500 ◦ C, 600 ◦ C, and 650 ◦ C, which spans the temperature region where the strength changes most rapidly. To maximize the number of different steels tested and define the temperature dependence of the strength, generally only a single test was made at each temperature. A second group of short-time creep tests characterized the time-dependent deformation behavior of steels from the floor trusses. These tests employed smaller specimens, with a 32 mm-long uniform cross-section that is 6.0 mm by 3.17 mm. Characterizing the creep behavior required about 20 tests per steel, at the same temperatures used in the high-temperature tensile tests.

The gage section of the typical test specimen was 32 mm long; the cross section was 6.35 mm × 3.17 mm. This specimen was geometrically similar to the larger, standard tensile specimen used in the quasistatic tests, which allowed pooling of the data for evaluating the effect of strain rate on ductility. In the high-strainrate tests, the specimen stress was measured by using a strain gage bonded to the nondeforming grip end of the specimen, while the specimen strain was measured from a high-elongation gage bonded to the gage section. The 1% offset yield strengths YS01 for all tests were evaluated using the European Structural Integrity Society (ESIS) procedure, which is described graphically in the inset to Fig. 35.12 [35.17]. Choosing the 1% offset yield strength YS01 instead of the more common 0.2% offset YS002 removes most of the spurious effects from load cell ringing, which can be misinterpreted as increased strength.

Deformation Models High Strain Rate. High-strain-rate tensile tests, as dis-

played in Fig. 35.12, show the strain-rate sensitivity of the steels. Although several models exist for describing the change in strength with strain rate, a particularly simple one is  m ε˙ , (35.1) σ = σ0 K ε˙ 0

High-Temperature Tests. Two types of tensile tests

characterized the high-temperature behavior of representative specimens of steels from perimeter columns, core columns, floor trusses, and truss seats. A first group of standard, high-temperature tensile tests, which followed ASTM E 21, complemented the roomtemperature tensile tests and formed the basis for the high-temperature plasticity models. In general, the high-temperature tensile tests employed round specEngineering stress (MPa)

35.3 Case Studies

where the strength σ can represent the 1% offset yield strength YS01 , for example. Figure 35.13 plots the calculated strain rate sensitivity m of the 1% offset yield strength YS01 for the perimeter and core column steels.

Engineering stress (ksi) 120

800

100 600 80 (5)

(4)

(1) dε/dt = 417 1/s (2) dε/dt = 260 1/s (3) dε/dt = 100 1/s (4) dε/dt = 65 1/s (5) dε/dt = 6.1×10–5 1/s

200

800

(2)

(3)

True stress (ksi) 120

1% offset

(1)

80

400

40

ESIS procedure for YS01

0

0

0.1

0

40

60

3rd order polynominal fit for 0.01 < ε< 0.05

200 0

60

100

YS01

600

Perimeter column flange WTC1 column 130 floors 90–93

0.01 0.02 0.03 0.04 0.05

20

20

0 0.06

True strain

0.2

Part D 35.3

True stress (MPa)

400

(1)

0 0.3 0.4 Engineering strain

Fig. 35.12 Example stress–strain curves for high-rate tensile tests. The inset describes the procedure for estimating the 1% offset yield strength YS01

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The plot also includes the strain-rate sensitivities calculated from the reported yield strengths for some other construction and low-alloy steels from the WTC construction era [35.18–24]. The strain-rate sensitivities of the yield strength of the WTC steels are similar. The accelerated delivery schedule for the strain-rate Strain rate sensitivity (m) 20 0.07

40

60

Fy (ksi) 80

100

120

1% offset yield strength YS01 α (dε/dt)m

0.06

Perimeter columns Core columns Literature

0.05 0.04 0.03

High Temperature. NIST supplied three models for cal-

culating the high-temperature behavior of the relevant steels from the fire zone: 1. normalized yield and tensile strength as a function of temperature 2. tensile stress–strain (plastic) behavior as a function of temperature 3. creep strain as a function of temperature, stress, and time

0.02 0.01 0

sensitivity data required that the team supply a single strain-rate sensitivity for all steels, which was evaluated using a subset of the steels in Fig. 35.13. Because that strain-rate sensitivity was based primarily on data from the higher-strength perimeter column steels, it slightly overpredicts the strength increase of the lower-strength core column steels. Significantly, none of the perimeter column steels suffered from brittle failure at the high deformation rates expected from the aircraft impact. At rates of up to 500 s−1 , the total elongation to failure El t was never less than 20% for any of the steels characterized.

200

400

600

800

Fy (MPa)

Fig. 35.13 Strain-rate sensitivity m of the 1% offset yield

strength YS01 for WTC steels and structural steels of similar strength and composition f = YS002/YS002 (T =20 °C) f = A2 +(1–A2)exp(–0.5[(T/s1)m1 +(T/s2)m2] A2 = 0.074 m1 = 8.07 m2 = 1 s1 = 635 °C s2 = 539 °C

1

0.8

0.6

Figure 35.14 compares the measured 0.2% offset yield strength YS002 of the recovered steels to a suite of literature data for structural steel [35.25–31]. The solid line represents the model for the temperature dependence of the yield strength, which was developed from the literature data in Fig. 35.14 before significant testing was complete. The data for the WTC steels generally lie slightly below the bulk of the literature data for T > 500 ◦ C, probably because the WTC tests employed a slower testing rate. In addition to the generic yield and tensile strength behavior model of Fig. 35.14, NIST developed a second set of models to predict the stress σ as a function of strain ε and temperature T for all steels in the fire zone

Part D 35.3

σ = RTS K (T )εn(T ) . Perimeter columns Core columns Truss seats Truss components Literature data Model Eurocode ε = 2 % yield strength = fmax Eurocode proportional limit fp

0.4

0.2

0

0

200

400

600

800 T (°C)

Fig. 35.14 High-temperature yield strength YS002 for steels recovered from the WTC, structural steels from the literature and the model for the normalized yield strength f

(35.2)

The temperature-dependent terms in this stress–strain model account for the decreasing work hardening with increasing temperature. The form of the functions K (T ) and n(T ) is the same as the master curve for hightemperature normalized yield strength in Fig. 35.14. The accelerated delivery schedule required NIST to develop models for all relevant steels based on test data from only two steels. The behavior of A 36, Fy = 36 ksi (248 MPa) steels was estimated from literature stress– strain curves [35.25]. The behavior for all other steels, including the wide-flange core column in Fig. 35.15, is

Structural Testing Applications

based on the stress–strain response of nominally ASTM A 242, Fy = 50 ksi (345 MPa) micro-alloyed steel recovered from the WTC floor trusses. To calculate the stress–strain behavior of an uncharacterized steel, the stress is scaled by the ratio, RTS = TSref /TS, of the room-temperature tensile strength of the reference steel on which the model was developed to the tensile strength of the uncharacterized steel. Scaling by the room-temperature tensile strength TS produced greater fidelity than scaling by the ratio of room-temperature yield strengths, as previous studies have done [35.32]. Figure 35.15 compares the stress–strain curves at different temperatures for specimens from the flange of a wide-flange core column to the prediction of the global model of (35.2). Note that the parameters used to predict the stress–strain curves were developed using data from the ASTM A 242, Fy = 50 ksi (345 MPa), micro-alloyed truss steel, and not the plotted wide-flange core column steel. The quality of the prediction confirms the choice to scale the results by the ratio of the room-temperature tensile strengths, RTS , in (35.2). The third set of models NIST supplied included two models to represent the creep behavior of all the different steels exposed to the fires. The first, for the low-strength A 36 steels, came directly from previous NIST research, and was based on literature data for the creep of low-strength structural steel [35.32, 33]. The second model, used for all other higher-strength steels, was based on the creep data from the ASTM A 242, Fy = 50 ksi (345 MPa) micro-alloyed steel recovered from the WTC floor trusses. Both models broke the temperature T , stress σ, and time t dependence of the creep strain εc into separate functions εc = A(T ) (RTS σ)C(T ) t B(T ) .

(35.3)

Engineering stress (ksi)

700 k0 = +183 MPa k4 = +845 MPa k1 = +6.576 k2 = +6.597 tk1 = +512 °C 20 °C tk2 = +512 °C 400 °C

500

403 °C

400

200

80

n0 = +0.034 n4 = +0.151 n1 = +10 n2 = +10 tn1 = +520 °C tn2 = +500 °C

500 °C

300

60

40

600 °C 600 °C 650 °C Experiment: dε/dt = 1×10–3 1/s Model

100

0

100

σ = RTS K(T)ε K(T ) = (k4 – k0) exp{–1/2[(T/tk1) k1+(T/tk2) k2 ]}+k0 n(T ) = (n4 – n0) exp{–1/2[(T/tn1) n1+(T/tn2) n2 ]}+n0 n(T )

600

0

0.05

1001

20

12WF92 Fy = 42 ksi core column WTC1 column 605 floors 98–101

0.1

0.15

0 0.25

0.2

Engineering strain

Fig. 35.15 High-temperature stress–strain curves and the prediction

of the global model (35.2) for high-strength steel Engineering creep strain (ε0) 0 0.5

1

(1) 297 MPa (2) 278 MPa (3) 252 MPa (4) 200 MPa

0.06 X(1)

t (h) 2

1.5

Truss angle ASTM A242 500 °C

Experiment Global model εc = A(T) (RTS σ)C(T )t B(T ) A(T ) = exp(a0 +a1T+ a2 T 2 ) X(2) B(T ) = b0 +b1(T/1°C)b2 C(T ) = c0 +c1T

0.04

X(3)

a0 = –55.45 a1 = +9.476 ×10–3 °C a2 = –3.52×10–5 °C2 b0 = +0.398 b1 = +3.553×10–11 b2 = +3.698 c0 = +3.233 c1 = +1.167×10–2 °C –1 RTS = +1

0.02

(4)

0

0

2000

4000

6000 t (s)

Fig. 35.16 Creep curves and the prediction of the global model

(35.3) for creep of floor truss steels that conformed to ASTM A 242

method deviated the most from the literature experimental data. Figure 35.16 compares the experimentally determined creep curves for the Fy = 50 ksi (345 MPa) micro-alloyed floor-truss angle steel to the prediction of the global model in (35.3).

Part D 35.3

The exact forms of B(T ) and A(T ) differ slightly between the two models. Like the high-temperature stress–strain models, the difference in creep response of the various steels was captured by scaling the stress by the same tensile strength ratio RTS . An extensive comparison of the results of this method on a suite of literature data sets for creep of structural steel established the superiority of this method over two other possible scaling ratios, no scaling and room-temperature yield-strength scaling [35.34, 35]. No scaling of the applied stress (i. e., setting RTS = 1) is equivalent to assuming that all steels have identical creep behavior. Scaling the stress by the room-temperature yield strength ratio proved to be particularly unsuitable. Creep curves from this

Engineering stress (MPa)

35.3 Case Studies

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This shows the excellent fidelity of the prediction of the global strain–time model for that steel. However, because NIST did not characterize the creep response of any other WTC steels, it is not possible to assess the fidelity of the prediction on the WTC steels. Summary NIST conducted more than 500 mechanical property tests to assess the quality of the steels and develop material models in support of the World Trade Center collapse investigation. These data supported models for room-temperature stress–strain behavior, strain-rate sensitivity, and high-temperature deformation. The case study highlights the challenges faced in terms of cataloging the recovered steel to characterize the failure modes, quantifying the temperature excursions that the recovered steel experienced, and characterizing and modeling its mechanical properties.

35.3.2 Testing of Concrete Highway Bridges – A World Bank Project Introduction The National Highways Development Project (NHDP) launched in 1999 was India’s largest ever highways project, covering a length of nearly 24 000 km (15 000 mile). The project cost (estimated to be US $25 billion) was funded by the World Bank, the Asian Development Bank, and the Indian Government [35.36]. The objective of this project was to develop a world-class highway network with uninterrupted traffic flow. A large part of this network involves improving the existing system, which was constructed during the 1950s. Most of the bridges in these highways had signs of distress due to lack of regular maintenance, and in most cases, records on their design and

construction details were nonexistent. The traffic on these structures had increased dramatically since their original construction. This situation called for tests to estimate the load-carrying capacity, particularly to determine if they had developed material deterioration and loss of section strength over the years of usage. Due to budget and time constraints, only a few bridges could be completely tested and assessed. This case study describes how the management approach, explained in Sect. 35.2, was implemented to test a large number of bridges within the constraints of time and cost. Challenges Involved The investigating team came across three major challenges in completing this project.

1. Test structures were located on National Highways that connect different states. The traffic volume on these highways is very high. Even for testing purposes, highway closure for more than 15 min was not permitted. Thus a testing method that allowed intermittent loading, allowing normal flow of traffic between two successive tests, was adopted. Intermittent loading is possible only by using articulated trucks. A market survey demonstrated that no such test truck with a telescopic working platform existed in the country (Fig. 35.17). Importing such a truck was not feasible from the perspective of cost and time. 2. Most bridges had visible distress marks and had no design/construction documentation available. As a result, a number of additional tests (mostly nondestructive and partial destructive) had to be performed to determine the characteristics of the concrete. It is a well-established fact that no single

Telescopic working platform

Part D 35.3

Test truck

Bridge deck slab

Railing

6m Transverse girders Longitudinal girder

13 m

Fig. 35.17 Test truck with telescopic working platform

Structural Testing Applications

test would be sufficient to assess the condition of any of these structures. 3. The greatest challenge was developing a theoretical model to predict the load–deflection behavior of a distressed structure. This model was needed during load testing of the highways.

• •

35.3 Case Studies

1003

characteristics of the concrete used in various elements of the structures – this was done by conducting nondestructive and partial-destructive tests; A theoretical model to predict the response of the structure.

Phase 3 – Load Testing. Load testing of the actual

Planning and Control Phase In order to meet the objectives of the test program, it was necessary to test these structures in three phases. Phase 1 – Visual Assessment. Visual assessment was

carried out in order to detect all symptoms of damage and defects as per specifications given in a special publication (SP) of the Indian Road Congress (IRC:SP35) [35.36]. A carefully selected assessment team, represented by experts from each of the contributing agencies (i. e., funding agencies, users, consulting firms, and the state highway department) were involved during this phase. On the basis of visual assessment, all of the bridge structures were categorized into three groups

• • •

Group 1 – bridges with little or no distress at all, which were not recommended for testing Group 2 – bridges with minor distress, which were recommended for testing to determine the severity of the distress Group 3 – bridges having considerable distress, for which tests were not needed as their replacement can be clearly justified

bridge structure was performed to ascertain the live load-carrying capacity. The procedure for rating the live load capacity requires knowledge of both the actual physical condition of the bridge and the actually applied loads. This phase involves the following tasks:

•

• •

Task 1 – Based on the traffic survey report and following the specification given in IRC:SP37 [35.37], the test load was determined. Load test was done at four different stages of loading as given in the procedure. The chosen loading arrangement should be such that these values can be achieved quickly. Sand bags, which can be added or removed easily, were used to achieve the required loading. Task 2 – Fig. 35.18 shows the distribution of sand bags to achieve the code-specified axle-wise load distribution. Task 3 – Before conducting the actual load test, a theoretical model was developed to predict the de-

Sand bags to load

Bridges that fell into group 2 were identified and grouped, and only representatives from each group were recommended for testing. The objective of this test program was to evaluate safe load-carrying capacity and establish a common procedure for posting a structurally deficient structure.

Part D 35.3

• •

Phase 2 – Generation of a Scientific Database. Since

there were no design/construction related data available, this phase was adopted to generate following information:

• •

physical/geometrical information of various elements; a map of crack (structural or nonstructural) patterns existing in the structure;

Fig. 35.18 Sand bags used to achieve code-specified load with the proper distribution

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•

flection of the structure at various stages of loading, assuming – dead load of the transverse girders as concentrated loads – twisting of the transverse girders due to eccentrically-applied live loads (in the form of the influence factor of the girder) – railing dead load, as a uniformly distributed load (UDL) – Deck slab and girders constructed monolithically (i. e., longitudinal girders worked like T-beams) – Prestressed concrete longitudinal girders, originally designed as balanced sections with the soffit stress due to self-load as −5 kg/cm2 , and the profile of the tendon was assumed to be parabolic. Task 4 – Actual load test.

After the field tests, a rating analysis was performed and the safety of the bridge structures was determined and reported, after proper documentation.

•

•

Task 2 – The working platform was designed, fabricated, and tested in the laboratory before being transported into the field. The platform was made using rectangular steel tubes in a telescopic arrangement to be easily transported in pieces and assembled at site quickly. Figure 35.20 shows the simple but elegant arrangement of the suspension system for the working platform. Using a proper counterweight, the requirement for a cable at the site was avoided. Task 3 – The instruments intended to be used for nondestructive, partially destructive, and load tests were properly calibrated in the laboratory.

Execution and Documentation The test program constitutes three different kinds of tests:

• • •

Test 1 – Nondestructive tests Test 2 – Partially destructive tests Test 3 – Load tests

Brief descriptions of each of these tests are given below. Preparation Preparations for the testing program included the following tasks:

•

Task 1 – On the basis of the traffic survey, the heaviest vehicle plying on the bridges was determined to be 25 tonnes. The code specifies to use the next heavier vehicle, i. e., an articulated truck trailer of 35.2 tonnes. Figure 35.19 shows the geometry of the test vehicle, along with the axle-wise load distribution to meet specifications of the code. The spatial distribution of the sand bags to achieve different stages of loading was ascertained through proper calibration.

Test 1 – Nondestructive Test. Four subtests were per-

formed under nondestructive tests 1. subtest 1 – to determine compressive strength of the concrete by impact hammer 2. subtest 2 – to determine the quality of concrete using ultrasonics 3. subtest 3 – to determine the reinforcement cover by cover meter 4. subtest 4 – to measure the corrosion of reinforcing steel using a half-cell meter Subtest 1 – Impact Hammer Test (Also Known as Schmidt Hammer Test or Rebound Hammer). This

Part D 35.3

test indicates the hardness of the concrete surface and indirectly reflects the strength of the concrete. ASTM C 805 provides the necessary specification for this test. Characteristics of this test method are 1.18 m

3.2 m

4.05 m

16 %

30 %

5.6 t

10.6 t

1.36m 1.35m

27 % 27 %

9.5 t

9.5 t

Fig. 35.19 Geometric of the test vehicle with axle-wise

load distribution

• • • •

a large number of readings can be taken in short duration, a large number of locations on the structure can be selected, a statistical approach with a high degree of confidence can be applied, the actual structure is tested, therefore the test result reflects the totality of the final product,

Structural Testing Applications

35.3 Case Studies

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Fig. 35.20 A single arrangement on how the working platform was analyzed, designed, fabricated in the laboratory before being transported to the site for field test

String

String pull Reaction from side rail Pin connection Counter weight

Wheel

Rectangular hollow tube in tube (telescopic) Deck reaction

•

the strength of concrete in a structure can be determined with an accuracy of ±15%. When little information is available about the concrete quality, the error can be up to ±25%.

ASTM C 805 specifies that ten rebound numbers be recorded for each location. The minimum number of locations where tests should be performed is ten per element. Thus for any element, there are 100 test readings. These are analyzed statistically as: f ck (most probable characteristics strength, MPCS, of concrete) = f m − 1.64 S, where f m is the mean value of the strength data and S is the standard deviation (SD) of the data. Figure 35.21 shows the frequency distribution of concrete in the deck slab of the bridge with identification number 38/1.

• • •

surface condition and moisture content can influence pulse velocity by up to 20% existence of cracks in the material has a strong influence the presence of steel reinforcement can increase pulse velocity by about 40% age, aggregate type, and size of the concrete all influence pulse velocity

Subtest 3 – Cover Test. Major causes for the deterioration of reinforced cement concrete (RCC) structures have been the corrosion of the embedded reinforcement. The main cause for the occurrence of corrosion is identified as the absence of proper cover, which provides the protective shield. When reinforcement details are Frequency 40 Samples = 10 Data size = 100 Max. value = 40 MPa Min. value = 26 MPa Mean = 33.96 MPa Sd = 3.1 MPa Mpcs = 28.85 MPa

35 30 25

Part D 35.3

Subtest 2 – Ultrasonic Test. In an ultrasonic test, compressive strength estimates are derived from timedomain signals. The density of a material affects both pulse velocity and strength. Thus there is a unique correlation between the longitudinal (compressive) wave velocity and material strength. ASTM C 597 outlines this test procedure. The operational principle of modern testing equipment includes the measurement of longitudinal, transverse, and surface waves. The receiving transducer detects the onset of the longitudinal waves and determines which is the fastest. From the velocity of this pulse, estimation can be made on strength. Table 35.1 gives the classification of concrete on the basis of pulse velocity. The zero setting of the equipment is done during the calibration of the device. Characteristics of this test method are

•

20 15 10 5 0

0

10

20

30

40 50 Strength (MPa)

Fig. 35.21 Frequency distribution of concrete strength

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Table 35.1 Classification of the quality of concrete on the basis of pulse velocity Longitudinal pulse velocity km/s 103 ft/s

Quality of concrete

> 4.5 3.5 – 4.5 3.0 – 3.5 2.0 – 3.0 < 2.0

Excellent Good Fair Poor Very poor

> 15 12 – 15 10 – 12 7 – 10 250 mV and < 350 mV 350 mV > 350 mV

Insignificant Corrosion initiated Corrosion is certain High to severe

not known, the position of the reinforcement close to the surface can be determined by a cover meter. A cover meter has a probe and an indicator on the front panel. The position and direction of the reinforcement embedded in the concrete were determined by sweeping the probe symmetrically over the surface. The needle of the indicating instrument will deflect if the probe nears a reinforcement bar. This is measured by placing the probe in different positions until a maximum needle deflection is obtained.

of these grid lines are marked numerically as shown in Table 35.3. Assessment of the rate of corrosion on the basis of half-cell potential is given in Table 35.2. ASTM C 876 has the necessary specifications for this test. Test 2 – Partially Destructive Test. Of all the tests

available for the determination of compressive strength of concrete, a core test may be the most direct test of in situ concrete. In this test method, cylindrical cores are cut using a rotary diamond drill (ASTM C 42) from the candidate structure and then polished and tested under uniaxial compression. While identifying a location for core cutting, regions where embedded reinforcement steel is present should be carefully avoided. Cores can be used to determine not only the compressive strength of concrete but also some of the other characteristics of the structure, i. e., carbonation depth, cement content, chloride content, and sulfate content in the mortar (ASTM C 1084). The presence of excessive chloride and sulfate in the concrete can cause corrosion of the embedded reinforcement. The process of drilling, including the vibration and impact, can weaken the interface between the cement paste and the aggregates. Core testing has an accuracy of approximately ±12% if a single core is tested. If multiple cores are tested at the same location, √ the mean core strength will have an accuracy of ± (23/n)%, where n is the number of cores [35.38]. Table 35.3 shows typical core test observations.

Subtest 4 – Corrosion Test. When reinforcing steel is

Part D 35.3

embedded in concrete, it does not normally corrode. The inherent cement-alkaline environment acts as a protective passive layer on its surface. However, if the depth of cover of the concrete is insufficient, then the passive layer can break in the presence of excessive amount of chloride ions. The chloride can originate from sodium chloride (common salt) in marine locations or from de-icing applications, or from the use of a particular admixture, e.g., calcium chloride (accelerator), or from surrounding soil, from the contaminated unwanted aggregates themselves or even from the mix water or curing water. The breakdown of the passive layer will force the steel to rust and expand in volume. As a result, the concrete cracks. In order to assess the rate of corrosion of embedded steel in concrete qualitatively, half-cell potential measurement, an electrochemical nondestructive technique, was adopted for the detection of the state of re-bars within the concrete structure. The procedure for this test is to divide the surface under test into a grid system of suitable dimensions as per site conditions. Intersections

Test 3 – Load Test: Deflection Prediction. From the

available geometrical and material data, a theoretical model is developed. Theoretical deflection is due to both the self-weight of the structure and the superimposed load due to the truck trailer loaded with sand bags. Equation (35.4) gives the deflection due to a uniformly distributed dead load of intensity w.  y=

1 EI



  −wL x 3 − wx 4 − 0.0416wL 3 x . (35.4)

As per IRC SP 37 [35.39], the position of the trucktrailer should be such that it causes the absolute maximum bending moment in the girder. The bending moment under the load will be at the absolute maximum when a load and the resultant of the set of loads are equidistant from the center of span for the various possible combinations of loads. When more than one truck-trailer are used, as per the code spec-

Structural Testing Applications

ification, the clear distance between vehicles CD is 18.5 m. Figure 35.22 shows the longitudinal placement of two truck-trailers, one completely inside the span, i. e., DEFG and the other partially, i. e., ABC on the span.

0

0

10

20

5

30

40

Procedure for Load Testing. As per the requirement of

the code:

• Deflection due to self-weight of the bridge

15 10.6 t

9.5 t

20 25

•

Deflection due to superimposed load

10

A B C

•

9.5 t

•

5.6 t

D

E

F G

Truck 1

•

Fig. 35.22 The longitudinal placement of two truck-

•

Truck 2 30 Deflections (mm)

trailers as per code specification

For this position of live loading, deflection can be found by the method of superposition given by equations (35.5,35.6) as   1 y= 6E I   −P(L − a)x 3 + Pa(L − a)(2L − a)x ,

a

P

x y L

Fig. 35.23 Deflection of a simply supported beam due to

a concentrated load P at mid span

• •

observations are made to find the existence of any cracks. If found, they should be marked and their width should be measured; for ease of observation of the behavior of cracks and their new formations during the test, a lime whitewash was applied the day before testing at the critical section; the load test was done during the morning hours of the day when the variation in temperature was low. the test load was applied in stages of 0.5W, 0.75W, 0.9W, and W, where W was the gross-laden weight of the test vehicle (i. e., 35.2 tonnes); for each stage, the corresponding loaded vehicle was brought to the marked position and the observations of deflections was made instantaneously, and again after 5 min; after the placement of the load, the development of new crack and the widening of the existing cracks were observed; prior to the start of testing, the theoretical deflections at various stages of loading were calculated and plotted. If the in situ deflections exceeded these values by more than 10%, the test was discontinued; for testing with multiple test vehicles, the individual vehicles were gradually brought into position and the resulting deflections were continuously monitored; the test vehicle was taken off the bridge, and instantaneous deflection recovery and the deflection recovery after 5 min were noted.

Figure 35.22 shows that the deflection caused by selfweight (uniformly distributed over the span) was around ten times that for the superimposed load. Deflection was measured at mid span using a dial gage with a resolution of 0.01 mm. Rating Analysis. After the load test, a load versus deflection graph is drawn. The load corresponding to a deflection of 1/1500 of the span is determined after extrapolating the load–deflection curve to obtain the acceptance load. Most of the bridges demonstrated nearly 100% recovery of deflection with unloading of the test vehicle. Test observations are summarized in Table 35.3.

Part D 35.3

(35.5) for x < a ,   −Pa y= 6E I   2     (L − a2 )x (L − x)3 + − L 2 − a2 , L L (35.6) for x > a .

1007

Total deflection can be obtained after combining the live load and dead load deflections.

• Span (m) 50 60

35.3 Case Studies

1008

Part D

Applications

Table 35.3 Summary of tests conducted on the bridge identified as no. 38/1 over the River Subarnarekha on National Highway 60 Test

Location of test

Observations

Test 1,

Deck slab

Subtest 1

Pier

Hammer test

Longitudinal girder

(Strength in

Transverse diaphragm

MPa)

Pier cap

MPCS 28.85 MPa; MPCS 19.37 MPa; MPCS 35.81 MPa; MPCS 30.85 MPa; MPCS 24.88 MPa;

Test 1, Subtest 2 Ultrasonic pulse test Test 1,

Deck slab Web of girder Flange of girder Diaphragm Soffit of central girder from upstream (U/s) face Web of central girder from U/s Central diaphragm between central and U/s girder Soffit of deck slab between central and U/s girder Pier no. 2 from west side Pier cap no. 2 west face Abutment (2.8 m from top) Grid area = 0.09 square meter around the exposed bar as shown on the rightmost cell in Fig. 35.25. Distance from 1 to 8 = 600 mm Distance from 1 to 2 = 150 mm

Subtest 3 Cover (mm)

Test 1, Subtest 4

Corrosion intensity around an exposed bar

Test 2

Part D 35.3

Partially destructive test (core test)

Test 3 Load test

Structural Strength element (MPa) Top slab 30.1 Abutment 28 cap Kerb 26 Girder Weak Upper abutWeak ment Lower 24 abutment Pier 16 Recovery of deflection Rating of the bridge on the basis of load test

Inference ;

SD = 3.1 MPa

Good

;

SD = 9.12 MPa

Average

;

SD = 5.80 MPa

Good

;

SD = 4.14 MPa

Good

;

SD = 8.01 MPa

Average

30

Good Good Good Fair Adequate as per code

40

Adequate as per code

40

Adequate as per code

25

Adequate as per code

55 60 1 – 580 mV 2 – 630 mV

Adequate as per code Adequate as per code Rate of corrosion is very high

3 – 640 mV 4 – 690 mV 5 – 688 mV 6 – 750 mV 7 – 594 mV 8 – 650 mV 9 – 620 mV Carbonation depth (mm) Nil Nil

Cement (%)

Chloride (%)

Sulfate (%)

15.41 10.83

0.033 0.009

0.045 0.033

Nil 2 2

13.33 17.91 11.25

0.013 0.04 0.1

0.058 0.14 0.033

Nil

10.83

0.015

0.055

1 Full recovery

10.83

0.007 0.065 Loading is within the elastic limit

The bridge is safe for a class-A train of vehicle, 70R tracked, and wheeled vehicles

Structural Testing Applications

Bridge characteristics:

•

prestressed concrete girder bridge with 13 spans (1 × 21.3 + 11 × 48.8 + 1 × 12.5 = 570.6 m long) with a cross section as shown in Fig. 35.24. N is the number of transverse girder.

Largest span = 48.8 m; N = 7

•

Fig. 35.24 Cross section of the bridge identified as no. 38/1

visual distress includes spalling, reinforcement bar exposure, and cracks in the deck slab and on supporting structures

Summary of the test observations is given in Table 35.3.

35.3.3 A New Design for a Lightweight Automotive Airbag

1

2

3

4

5

6

7

8

9

Fig. 35.25 Grid

points around the exposed bar as marked numerically from 1 to 9

1009

It was anticipated that a lightweight fabric airbag would require a new design to pass the rigorous testing of an automotive industry component. The incumbent driver-side airbag, a two-piece circular design, is not very structurally efficient. To check this notion, an incumbent two-piece, circular airbag was fabricated using the lightweight fabric and subjected to a burst-pressure test. The airbag industry employs burst-pressure testing to assist in the design qualification of prototype airbags. During this test, the airbag is inflated with ambient air over several seconds until the airbag fails structurally. While each automotive manufacturer requires different qualification standards, the result of the lightweight-fabric, two-piece circular design burstpressure test conclusively demonstrated that lightweight fabrics would require a new design to produce a viable airbag. Challenges Involved Early lightweight fabric prototype designs converged on gore-shaped structures similar to solid-canopy parachutes and hot-air balloons, as depicted in Fig. 35.26. An initial design review concluded that the proposed prototypes might satisfy the engineering specifications, but certainly would not represent a costeffective, marketable product. Using cost as the exclusive constraint, a subsequent design session produced a radically different yet economically viable airbag. The simple design used a single square piece of fabric and the enclosure was completed by folding the corners toward the center of the square, as shown in Fig. 35.27. Seams were constructed along the diagonals where the free edges formed, and a hole was cut in the center to accept the attachment hardware. This pattern ensures with its very simple design that the airbag produced the strongest seams and no parent fabric was wasted. Planning and Control Phase To prove this design, a significant quasistatic pressurization and dynamic-deployment cold-air inflation test series was initiated. The experimental test lab consisted of a high-pressure source and serially plumbed solenoid valves that could supply a dynamic burst of air simulating an inflator deployment. High-speed video cameras (1000 fps) were the primary diagnostic tools that assisted the post-test evaluation of the early prototypes. Simple pressure-transducer measurements monitored the progress of the evolving prototypes. This enabled the airbag deformation pressure–time history to be quantified during these inflations.

Part D 35.3

Description In 1991, Sandia National Laboratories and a fabric manufacturing company initiated a project to develop a lightweight fabric automotive airbag. The objective of the project was to address performance issues inherent with incumbent, heavyweight automotive airbags. For example, heavyweight fabric airbags, constrained by imposed pack volume, must be small, necessitating high pressures in the airbag to decelerate the occupant during an accident. Conversely, lightweight fabric airbags could be considerably larger, resulting in lower pressures exerted on the occupant during an accident. Also, the lightweight fabric is soft and smooth, which eliminates the facial abrasion experienced with heavyweight fabric airbags. Finally, the momentum of airbag deployment imparts a large impulse to the occupant’s face, a phenomenon termed face-slap, which is mitigated by the smaller mass of lightweight fabrics.

35.3 Case Studies

1010

Part D

Applications

Quasistatic tensile tests of the fabric were performed to determine the ultimate capacity of the fabrics, along with the stiffness, or modulus of elasticity. To correctly determine the modulus, a uniaxial straining of the material is required. Wrap grips were used in these tests to wrap the fabric over a cylindrical fixture to avoid the hourglass deformation patterns common with other test fixtures. The crosshead of the test machine in these quasistatic tests moves at a maximum rate of 10 in/min (0.0042 m/s) while the load and deformation are recorded. The modulus was measured along the original fabric roll direction (the warp direction), across the fabric roll direction (the fill direction), and in-between or at 45◦ to each of these directions (the bias direction). This modulus data is used as the basis of the structural analysis model definitions. Prototype

Conceptual gore-shaped airbag design

Sandia National Labaratories

Fig. 35.26 Early prototype design appears to be gore

shaped

Structurally efficient airbags

Airbag Innovations Research

Part D 35.3

Attachment hardware

Fold lines

Seams

Fig. 35.27 A simple design using square fabric to reduce

wastage

seams were also evaluated using the tensile tests, with construction variables such as seam type, thread size, needle size, stitches per inch, type of stitch, and reinforcement materials. Simulation A structural analysis investigation of the new design was started to understand how load is distributed in the airbag. The chosen approach called for finite element analyses using the nonlinear code Abaqus [35.40]. This code allows for gross deformation of the airbag during inflation using membrane elements that incorporate the orthotropic behavior of the fabric. A simulation of the deployments from the cold-air inflation testing was chosen to allow direct comparisons with a known loading. High-speed video coverage of the airbag deployments was compared with the deformation from the analyses, showing that the curvatures of the airbags were dissimilar. These analyses used the loading measured as pressure–time histories and the quasistatic material properties described above. The testers conjectured that the inflation dynamics (which occurs in the time frame of 10–20 ms) were altering the effective stiffness of the fabric. Dynamic tests were then justified to determine the strain-rate sensitivity of the fabric. Postprocessing of the analysis strain showed a strain rate of 5–10 /s in the fabric during the pressurization (analysis is a very good tool for determining the strain rates of loading, and is generally not intuitive). Using this information, tensile tests were designed utilizing a high-speed MTS load frame. This test machine is capable of producing a velocity of 200 in/s (5.1 m/s) in the crosshead. A photograph of this test setup is shown in Fig. 35.28, where a seam test sample is shown in a temperature-conditioning fixture. The gage length of the sample was adjusted until the desired strain rate was achieved, and verified using high-speed video coverage of the test. This was accomplished by running the test machine at its maximum speed and adjusting the test sample length until the desired strain rate was measured using the video coverage. The correct specimen length was iteratively determined by repeating the test until the strain rate of interest was finally achieved. The desired strain rates were bracketed to ensure consistency of the measurements and the loading technique for the test specimen. Testing at the loading rate of the airbag deployment showed an apparent stiffening of the fabric matrix, and produced as much as a 30% increase in the elastic modulus of the fabric. The strain-to-failure was also

Structural Testing Applications

frequently reduced depending on the amount of crimp in the fabric weave (essentially an expression of how tightly the fabric is woven). There was also a large modulus discrepancy between the fabric construction directions, warp and fill. Warp fibers are the yarns in the direction of the fabric roll, and fill fibers are passed in between and normal to the warp yarns, to make up the fabric matrix. Typical tensile test results of nonoptimized fabrics are shown in Fig. 35.29, where the load–displacement response of the fabric is shown for both the warp and fill directions during high-speed tests. The complete loading history of the same fabric in the warp and fill direction is shown in the figure (machine startup dynamics occur early in the load trace, as noted) with the actual tensile loading being represented by the second half of the load trace. Note that, not only is the slope or modulus of the material not the same, but the strain-to-failure of the material is different by almost a factor of three. The strength of the fabric is similar in the warp and fill directions (the normal way fabric is designed or ordered). Most importantly, the elastic properties of this material cannot produce a balanced state of strain in the resulting airbag deployment with this mismatch in moduli (the two slopes of the load traces approximated by the lines in the same figure). Future fabric construction for this airbag was designed to balance out the moduli in the warp and fill directions.

35.3 Case Studies

1011

Load (lbs for 2 in. sample) 250

Fill modulus/warp modulus ~ 2

200

Machine startup dynamics

Fill Warp

150 100 50 0 0

0.01

0.02

0.03

0.04 0.05 0.06 Displacement of crosshead (in.)

Fig. 35.29 Load–displacement history of fabric in the wrap and fill directions (1 lbs = 4.4 N; 1 in = 2.54 cm)

the airbag’s high-speed deployments improved greatly. A still image during deployment is shown in Fig. 35.30. In this figure, the photo of the airbag is overlaid with the deformed mesh lines (shown in black) of the finite element membrane model. The curvature of the front and rear panels of the airbag now match very well. This proves graphically that the dynamic modulus infor-

Validation of the Model When this new understanding of the fabric properties was incorporated into the analyses, the curvature of

Part D 35.3

Fig. 35.30 Photograph and analysis overlay of inflated Fig. 35.28 High-speed MTS load frame

airbag

1012

Part D

Applications

Fig. 35.31 Warp stress distribution of the mounting area of

the airbag

mation, replacing the earlier quasistatic tests, produces the correct state of stress and deformation in the fabric structure. The airbag seams have warp- and fill-directed fabric on opposite sides of the seam due to the geometry and simplicity of the design. Any mismatch in the properties of the warp and fill will cause shear stress to develop at the seam interface. Conversely, shear stress control down the seam line is made easier by the balancing of modulus of the dissimilar directions. Any imbalance in the moduli across the seam must be balanced by producing shear across the seam, which leads to tearing or premature failure. The analyses showed these effects of airbag construction. The warp stress

contours around the mounting area of the airbag are shown in Fig. 35.31. Developing seams to move with the deforming panels of fabric was the next step in project optimization. It is clear that different levels of bunching occur during the loading process due to the different amounts of fabric adjacent to the stitching. This relates directly to the shear stress formation down the line of the seams. This seam behavior was also optimized using the highspeed test frame so that the radii of the stitched material increased the ability of the material to handle the dynamics of the inflation. This seam optimization is an example of an analysis that has identified an issue and its root cause, which was then addressed through testing. Once the state of stress in the airbag is understood, it is possible to determined how this stress can be balanced across the seams. Moduli balancing was accomplished by literally designing the fabric weave to act in a balanced fashion. Yarns in the fabric weave were removed to achieve a stronger fabric. This was a radical concept to achieve a dynamically balanced fabric response. This concept led to altering the yarn properties in conjunction with the weave design to dynamically and statically optimize the fabric action [35.41]. A case study that details the project from inception through innovation and engineering development has been presented. It was noted that the implementation of different constraints led to radically different design proposals during the innovation phase. The study also demonstrated the engineering progression of the design and the role that testing and analysis played in the development. New test and analysis methods were developed during the engineering phase as the design was eventually completed and qualified.

35.4 Future Trends Part D 35.4

In nature, shape and structure spring from the struggle for better performance, thus they are adaptive and multifunctional. Exciting progress is being made in developing engineered systems with ideas taken from the biological world. These systems are complex and call for an interdisciplinary effort encompassing a breadth of science and engineering disciplines, including biological sciences. Living organisms produce a striking array of structural concepts with a wide variety of biological functions. For example, consider the case of a daffodil stem, a biological beam with a high bendto-twist ratio of ∼ 13, while that of an isotropic circular

beam would be 1.0–1.5. This natural beam will twist and bend in response to wind loads to reduce drag by up to 30%. Thus, structural testing of a biological beam should incorporate loading under bending and twisting simultaneously. The particular combination of these loading cases will be function of the characteristics of biological structures or materials. Generating engineering concepts with ideas from the biological world will be the emphasis of future research efforts. The growth of the field (biomechanics, biomimetics, biomaterials, bioengineering, biological soft tissue, etc.) can be seen by its maiden presence in

Structural Testing Applications

Chaps. 7, 31, and 32 of this Handbook. More than half a dozen journals specific to biological materials have been started in the last decade. The number of publications in this area during last 5 years exceeds those published during the previous 50 years. Advances in medical imaging (e.g., 3-D Doppler echo and magnetic resonance imaging), medical image reconstruction techniques, and computer simulation methods are playing important roles in our health care system, both for the design of new medical devices and for the rehabilitation simulators that provide realtime feedback for health-care planning and training. For proper validation of these structural systems, (medical

References

1013

devices and simulators), state-of-the-art measurement, control, and analysis techniques will be needed. Nanoscale technology will influence the future in many ways. Multifunction nanotubes are wonder material with unusual combination of characteristics: strength (more than 460 MPa), creep resistance, electrical conductivity, damping, and many other desirable characteristics at high temperature (450 ◦ C) through to cryogenic environments. Naturally, a structure developed using this wonder material should be tested in coupled modes. Chapters 16 and 30 of this Handbook contain more information on carbon nanotubes.

References 35.1

35.2 35.3

35.4

35.5

35.6 35.7 35.8 35.9 35.10 35.11

35.13

35.14

35.15

35.16

35.17

35.18

35.19

35.20

35.21

35.22

35.23

Mechanical and Metallurgical Analysis of Structural Steel (NIST, Gaithersburg 2005), available at http://wtc.nist.gov W.E. Luecke, J.D. McColskey, C.N. McCowan, S.W. Banovic, R.J. Fields, T. Foecke, T.A. Siewert, F.W. Gayle: Federal Building and Fire Safety Investigation of the World Trade Center Disaster: Mechanical Properties of Structural Steels (NIST, Gaithersburg 2005), available at http://wtc.nist.gov D.M. Bruce, D.K. Matlock, J.G. Speer, A.K. De: Assessment of the strain-rate dependent tensile properties of automotive sheet steels, SAE Technical paper 2004-01-0507 (SAE, 2004) European Structural Integrity Society: Procedure for Dynamic Tensile Tests. In: ESIS Designation P7-00, ed. by K.H. Schwalbe (ESIS, Delft 2000) D. Chatfield, R. Rote: Strain Rate Effects on Properties of High Strength, Low Alloy Steels, Publication No. 740177 (Society of Automotive Engineers, Warrendale 1974) H. Couque, R.J. Asaro, J. Duffy, S.H. Lee: Correlations of microstructure with dynamic and quasi-static fracture in a plain carbon steel, Met. Trans. A 19(A), 2179–2206 (1988) R. Davies, C. Magee: The effect of strain-rate upon the tensile deformation of materials, J. Eng. Mater. Tech. 97(2), 151–155 (1975) A. Gilat, X. Wu: Plastic deformation of 1020 steel over a wide range of strain rates and temperatures, Int. J. Plast. 13(6-7), 611–632 (1997) J.M. Krafft, A.M. Sullivan: On effects of carbon and manganese content and of grain size on dynamic strength properties of mild steel, Trans. ASM 55, 101–118 (1962) M. Langseth, U.S. Lindholm, P.K. Larsen, B. Lian: Strain rate sensitivity of mild steel grade ST-52-3N, J. Eng. Mech. 117(4), 719–731 (1991)

Part D 35

35.12

R. Sack: Model Based Simulation (National Science Foundation, Arlington 1999), White paper, See also: [www.eng.nsf.gov/cms/] for update W.H. Price: Factors influencing concrete strength, Proc. J. Am. Concr. Inst. 47(6), 417–432 (1951) H. Straub: A History of Civil Engineering – An Outline from Ancient to Modern Times (MIT Press, Cambridge 1964) R. Reese, W.A. Kawahara: Handbook of Structural Testing (Society of Experimental Mechanics, Bethel 1993) H. Erwin: OSHA 1910 General Industry Standards Made Easy, 2nd edn. (Erwin Training Institute, Edmonton 1989) National Environmental Policy Act under US DOE at http://www.eh.doe.gov/nepa/ http://www.epa.gov/region5/defs/html/rcra.htm American Society for Testing and Materials: http://www.astm.org/ The American Railway Engineering and Maintenance Association: http://www.arema.org/ (2008) http://www.aashto.org/ and http://www.transportation.org/ American Institute of Steel Construction http:// www.aisc.org/ International Code Council: International Building Code (International Code Council, Country Club Hills 2003) National Institute of Standards and Technology: Federal Building and Fire Safety Investigation of the World Trade Center Disaster: Final Report of the National Construction Safety Team on the Collapses of the World Trade Center Towers (NIST, Gaithersburg 2005), available at http://wtc.nist.gov F.W. Gayle, R.J. Fields, W.E. Luecke, S.W. Banovic, T. Foecke, C.N. McCowan, T.A. Siewert, J.D. McColskey: Federal Building and Fire Safety Investigation of the World Trade Center Disaster:

1014

Part D

Applications

35.24

35.25

35.26

35.27

35.28

35.29

35.30

35.31

M.J. Manjoine: Influence of rate of strain and temperature on yield stresses of mild steel, Trans. ASM 66(A), 211–218 (1944) T.Z. Harmathy, W.W. Stanzak: Elevated-temperature tensile and creep properties of some structural and prestressing steels. In: Fire Test Performance, ASTM STP 464, ed. by ASTM (American Society for Testing and Materials, Philadelphia 1970), pp. 186– 208 R. Chijiiwa, Y. Yoshida: Development and Practical Application of Fire-Resistant Steel for Buildings, Nippon Steel Techn. Rep. 58, 47–55 (1993) S. Goda, T. Ito, H. Gondo, I. Kimura, J. Okamoto: Present status of weldable high strength steel, Yawata Techn. Rep. 248, 5086–5163 (1964), in Japanese J.M. Holt: Short-time elevated-temperature tensile properties of USS “T-1” and USS “T-1” Type A constructional alloy steels. In: United States Steel Report (Applied Research Laboratory, Monroeville 1963) J.M. Holt: Short-time elevated-temperature tensile properties of USS Cor-Ten and USS Tri-Ten high-strength low-alloy steels, USS Man-Ten (A 440) high-strength steel, and ASTM A 36 steel, US Steel Corp. Rep. 57, 19–901 (1964) G.F. Melloy, J.D. Dennison: Short-time elevatedtemperature properties of A7, A440, A441, V50, and V65 grades. Internal memorandum to J.W. Frame (Bethlehem Steel, Bethlehem 1963) United States Steel Corp.: Steels for Elevated Temperature Service (United States Steel Corporation, Pittsburgh 1972)

35.32

35.33

35.34

35.35

35.36 35.37

35.38

35.39

35.40 35.41

B.A. Fields, R.J. Fields: Elevated Temperature Deformation of Structural Steel. NISTIR 88-3899 (National Institute of Standards and Technology, Gaithersburg 1989) D. Knight, D.H. Skinner, M.G. Lay: Short Term Creep Data on Four Structural Steels. Report MRL 6/3 Melbourne Research Laboratories (The Broken Hill Proprietary Company, Clayton 1971) G. Williams-Leir: Creep of structural steel in fire: analytical expressions, Fire Mater. 7(2), 73–78 (1983) M. Fujimoto, F. Furumura, T. Abe, Y. Shinohara: Primary creep of structural steel (SS41) at high temperatures, J. Struct. Construct. Eng. 296, 145–157 (1980) National Highways Authority of India: http://www. nhai.org/ IRC: SP:35-1990, Guidelines for Inspection and Maintenance of Bridges, Indian Roads Congress (R. K. Puram, New Delhi 1991), http://www.irc.org.in/ S.S. Seehra: Specimen Testing vis-á-vis insitu Testing, Nondestructive Testing of Concrete Structures (Indo-US Workshop on NDT, Roorkee 1996) IRC: SP:37-1991, Guidelines for Evaluation of Load Carrying Capacity, Indian Roads Congress (R K Puram, New Delhi 1991), http://www.irc.org.in/ Abaqus Inc.: Abaqus Users Manual (Abaqus, Providence 2005) K. Gwinn, J. Holder: Fabric with Balanced Modulus and Unbalanced Construction, Patent 0930986B1 (2004)

Part D 35

1015

Electronic Pa 36. Electronic Packaging Applications

Jeffrey C. Suhling, Pradeep Lall

Electronic packaging is a field in rapid evolution due to strong and competing customer demands for increased functionality and performance, further miniaturization, heightened reliability, and lower costs. Such product drivers cause a myriad of reliability challenges for the engineer involved in the mechanical design of electronic systems, and several methods of experimental mechanics have become critical tools for the design and development of electronic products. In this chapter, we present an overview of important experimental mechanics applications to electronic packaging. Mechanics and reliability issues for modern electronic systems are reviewed, and the challenges facing the experimentalist in the packaging field are discussed. Finally, we review the state of the art in measurement technology, with the presentation of selected key applications of experimental solid mechanics to the electronic packaging field. These important applications are grouped by the goal of the measurement being made, including delamination detection, silicon stress characterization, evaluation of solder joint deformations and strains, warpage measurements, evaluation of behavior under transient loading, and material characterization. For each application, the important experimental techniques are discussed and sample results are provided.

36.1 Electronic Packaging ............................. 1017 36.1.1 Packaging of Electronic Systems ... 1017 36.1.2 Electronic Packaging Failure Modes ...................................... 1018

36.3 Detection of Delaminations ................... 1022 36.3.1 Acoustic Microscopy .................... 1022 36.4 Stress Measurements in Silicon Chips and Wafers .......................................... 1024 36.4.1 Piezoresistive Stress Sensors ........ 1024 36.4.2 Raman Spectroscopy ................... 1027 36.4.3 Infrared Photoelasticity............... 1029 36.4.4 Coherent Gradient Sensing .......... 1030 36.5 Solder Joint Deformations and Strains .... 1031 36.5.1 Moiré Interferometry .................. 1032 36.5.2 Digital Image Correlation............. 1034 36.6 Warpage and Flatness Measurements for Substrates, Components, and MEMS .. 1036 36.6.1 Holographic Interferometry and Twyman–Green Interferometry .... 1036 36.6.2 Shadow Moiré............................ 1037 36.6.3 Infrared Fizeau Interferometry ..... 1038 36.7 Transient Behavior of Electronics During Shock/Drop ................................ 1039 36.7.1 Reference Points with High-Speed Video ............... 1039 36.7.2 Strain Gages and Digital Image Correlation ...... 1040 36.8 Mechanical Characterization of Packaging Materials.......................... 1041 References .................................................. 1042

a typical product based on microelectronics is a set of silicon chips that contain devices, integrated circuits, micro-electromechanical systems (MEMS), and photonic components that perform desired functions. These

Part D 36

Modern electronic packaging [36.1–3] is the technology area dealing with the assembly and connection of electronic, electromechanical, and electro-optical components within an electronic system. The core of

36.2 Experimental Mechanics in the Field of Electronic Packaging ......................... 1019 36.2.1 Role of Experimental Mechanics and Challenges for the Experimentalist ............... 1019 36.2.2 Key Application Areas and Scope of Chapter ................................. 1020

1016

Part D

Applications

Part D 36

semiconductor die are connected and interfaced to the outside world by an elaborate series of interconnection layers that provide for input/output (I/O) of electrical signals and power, mechanical support, conduction of heat, and protection from the environment. The packaging field is highly interdisciplinary, involving several branches of engineering (electrical, mechanical, industrial, chemical, and materials), and a broad range of technology areas such as electrical circuit design and routing, electromagnetic interference (EMI), mechanics and structural analysis, material processing and characterization, thermal management and heat transfer, and manufacturing science. Another dominant characteristic of electronic packaging is that it is a field in rapid evolution. There is an ever-present push for miniaturization and weight reduction of electronic products. While the sizes of the basic electronic devices on the chip (e.g., transistors) are certainly decreasing, the majority of the size reduction at the product level is accomplished through more efficient packaging. To be competitive in the marketplace, these size reductions typically occur in parallel to increases in functionality (circuit density and I/O count), electrical performance (processing speed), and power density (heat generation). Finally, the most innovative and successful products obtain all of these improvements while concurrently increasing the reliability and lowering the cost. The competing goals of size reduction, increased performance, and lower cost have led to a continual stream of thermomechanical reliability challenges for the packaging engineer. Perhaps the most commonly faced problem is the internal battle fought by the various materials in the electronics assembly due to mismatches in the coefficients of thermal expansion (CTE). Temperature changes occur in electronic products during shipping, storage, and operation. These often lead to thermally induced stresses and strains in the various assembly materials, particularly solder joints and thin layers or films within the package. Other sources of poor reliability include stresses induced during assembly and processing, mechanical loadings caused by handling and human interaction, and transient phenomena such as shock/drop incidents. In the past, reliability issues in electronic products were often sorted out at the end of the design process in a problem-solving mode. However, the ever-shrinking product development cycle time has led to more appreciation of the need for concurrent mechanical and thermal analysis during all aspects of product design and development. Numerical modeling approaches such

as finite element analysis are widely used to estimate stresses, strains, and deformations throughout complex electronic packaging structures; and then to predict reliability and product life. However, there are many challenges to obtaining suitable accuracy in such models, including the presence of extremely complicated geometries, multiple length scales, nonlinear and timedependent material and failure behaviors that are poorly characterized, and complex loadings and boundary conditions. In addition, manufacturing variation from part to part causes further uncertainty in the results from numerical simulations. Although the accuracy of finite element models for electronic packaging continues to improve, the above constraints often cause them to be oversimplified or error prone, or to simply be impossible to analyze in an expedient manner. Thus, models most often serve to obtain relatively coarse estimates and to rank various configurations under consideration, while requiring verification by other means. Consequently, experimental testing is normally performed using one or more methods of experimental mechanics, as well as by using product qualifications performed via accelerated life testing (ALT) of product prototypes. This experimentation can either serve as a stand-alone approach during the design and prototyping process, or for guidance and verification of numerical model predictions. The ever-changing nature of electronic packaging architectures and the modeling challenges described above have made the methods of experimental mechanics valuable tools during the design process of electronic products. Accordingly, advanced experimental techniques are in high demand, and several methods of experimental solid mechanics have become widely known in the packaging industry during the past 20 years including acoustic microscopy, conventional and semiconductor strain gages, moiré interferometry, digital image correlation, and several photomechanics methods for characterization of out-of-plane displacements (warpage). In addition, members of the experimental mechanics research community have pioneered several innovative new methods for characterizing stress, strain, and displacement in electronic packages, as well as new techniques for characterizing the material behavior of packaging materials found in thin layers/films and solder joints. Finally, several specialized instruments based on these methods have been developed and commercialized, and then found their way into the modern electronic packaging laboratory. This chapter presents an overview of experimental mechanics applications to electronic packaging, ad-

Electronic Packaging Applications

dressing their critical role as tools for the design and development of electronic products. An introduction to mechanics and reliability issues in electronic packaging is first given, followed by a discussion of the role of experimental mechanics and the challenges facing the experimentalist in the packaging field. Finally, the remaining bulk of the chapter contains the presentation of a group of selected recent application areas for experimental mechanics in the electronic packaging field. These key applications are grouped by the goal of the measurement being made rather than the experimental method utilized, and include: detec-

36.1 Electronic Packaging

1017

tion of material interface delaminations; measurement of stresses, strains, and deformations in silicon chips and wafers; characterization of solder joint deformations and strains, measurement of warpage/flatness of electronic substrates, components, and MEMS; measurement of transient behavior of electronic products subject to shock and drop; and innovative characterization of the constitutive and failure behavior of electronic packaging materials. For each measurement type, the important experimental techniques are discussed and example data are given. Finally, key references are cited for further investigation by the reader.

36.1 Electronic Packaging 36.1.1 Packaging of Electronic Systems Electronic packaging serves four major functions in the performance of electronic systems: interconnection of electrical signals at several levels; distribution of power to the electronic circuits and devices; mechan-

ical and environmental protection of the components, circuits, and devices; and dissipation of the heat generated by these devices. A typical electronic packaging process hierarchy is illustrated in Fig. 36.1. In this context, packaging refers to the placement and connection of electronic, electromechanical, and electro-optical

Wafer

Chip

First-level package (single chip module)

First-level package (multichip module) Chip-on-Board (COB) Second-level package (PCB or card)

Fig. 36.1 Electronic packaging hier-

archy

Part D 36.1

Third-level package (motherboard)

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Part D

Applications

components into an enclosure/housing/product that protects the system it contains from the environment and provides easy access for routine maintenance. The packaging process starts with a chip which has been diced from a silicon wafer that was fabricated using photolithographic semiconductor fabrication processes. The chip contains electronic devices (transistors, resistors, etc.) that are interconnected in a planned manner to form integrated circuits (ICs) that perform a desired electrical function. After testing, the chip is housed in a chip carrier and small wires or solder balls are used to connect the chip electrically to the carrier. The chip carrier or component is often referred to as the first level of packaging. Next, several chip carriers are placed on a printed circuit board (PCB) or substrate (the second level of packaging), and connected together with wiring traces. Edge connectors on the circuit boards are then inserted into contacts on a back panel or module housing (the third level of packaging), which carries the higher-level connections that permit communication from one circuit board to the next. The process described above can be extended to higher packaging levels for more complex systems that involve multiple modules working together. For example, a typical packaging configuration for an on-engine electronic control module is shown in Fig. 36.2. This module is one of several single PCB units located throughout the vehicle that comprise the electrical system of the automobile. The illustrated module features flip-chip technology, where the silicon chips are directly attached to the substrate by an array of small Engine control module

Part D 36.1

Silicon die (flip chip) PCB with components

Fig. 36.2 Automotive electronics module, printed circuit board, and components

solder joints. This approach represents the ultimate in high-density assembly and packaging of chips onto a substrate. However, it is subject to a variety of reliability challenges as discussed below.

36.1.2 Electronic Packaging Failure Modes Several sources of mechanical stress and strain are present in a typical electronic packaging application such as the automotive controller shown in Fig. 36.2. For example, thermomechanical loadings will result due to temperature changes caused by powering the system on and off, or by changes in the local ambient environment. These loads are a result of nonuniform thermal expansions in the various components resulting from mismatches between the coefficients of thermal expansion of the materials comprising the assembly. Additional thermally induced stresses are produced from heat dissipated by high-power density devices during operation. Other quasistatic mechanical loadings can be transmitted to components through interaction of the electronic module with its surrounding environment. In addition, modern handheld consumer electronics as well as automotive and military hardware can be exposed to severe vibrations and/or shocks on repeated occasions and for extended periods of time. Other sources of reliability issues include extreme high or low ambient temperatures, dust, humidity, and radiation. The combination of all of the above loadings can lead to complicated states of stress, strain, and deformation within the assembly. These can cause a variety of potential failures, including fracture of the die, solder fatigue, interface delamination, encapsulant and adhesive cracking, and severing of connections such as wirebonds or PCB traces; for example, Fig. 36.3 shows an atlas of the failure modes that can be found in flip-chip assemblies such as those present in the automotive control module in Fig. 36.2. In this drawing, the solder balls have been deliberately drawn out of scale (larger than normal). The source of most reliability challenges in flip-chip assemblies is the large coefficient of thermal expansion (CTE) mismatch between the silicon chip (α = 2.6 × 10−6 1/◦ C) and glass-fiberreinforced epoxy PCB (α = 20 × 10−6 1/◦ C). When subject to temperature changes, this CTE mismatch results in uneven expansions of the high-stiffness die and substrate that can cause severe shearing deformations and strains in the solder joints. If the assembly is subjected to temperature cycling and the solder joint shearing becomes repetitive, low-cycle fatigue

Electronic Packaging Applications

36.2 Experimental Mechanics in the Field of Electronic Packaging

1019

Fig. 36.3 Electronic packaging failure modes for flip-chip assembly Die cracking

Extensive deformation of copper/low-K interconnect Copper pad

α = 2.6 ×10–6 1/°C

Soldermask

Printed circuit board Delamination

Silicon chip

Solder joint Passivation Underfill

α = 20 ×10–6 1/°C Underfill cracking

failures of the joints will quickly occur. For this reason, underfill encapsulation is typically used to distribute and minimize the solder joint strains, thus improving thermal cycling fatigue life. With the die coupled to the substrate through the underfill epoxy, the coefficient of thermal expansion mismatch between the silicon and the laminate produces a bending or curvature of the assembly (and thus the silicon die) upon changes of temperature. This leads to a greatly reduced dependence of the solder bump shear strains on the distance from the chip center (neutral point). However, several additional failure modes are introduced. For an underfilled flip chip at room temperature, high tensile normal stresses exist on the backside of the die, while compressive normal stresses are typically produced on the device side of the die containing the solder bump interconnections. These backside tensile

Solder joint fatigue

stresses can lead to fracture and are more severe for larger-area die, larger-thickness die, and die exposed to lower temperatures. The device-side die stresses can cause damage (shearing) to the integrated circuits devices and their associated interconnect structures. This is especially important for the most state-of-art chips, where multiple copper/low-k interconnect redistribution layers are present at the die surface. The remaining and most critical failure modes are related to the underfill encapsulant, namely bulk underfill cracking and delaminations at the underfill–die or underfill–soldermask interfaces. When interfacial cracks develop and propagate to the neighboring solder bumps, the previously described stress relief on the solder joints will be lost and the onset of solder joint fatigue cracking will be hastened. All of the failure modes described above are exacerbated when the package is subject to vibration or shock/drop transient loading.

36.2 Experimental Mechanics in the Field of Electronic Packaging 36.2.1 Role of Experimental Mechanics and Challenges for the Experimentalist The high stress/strain/deformations concerns and resulting failure modes in the flip-chip example discussed above are typical of most current electronics products and include delamination of material interfaces, high stresses and fracture of silicon, excessive deformations and strains in solder joints, warpage/bending of electronic assemblies and components, and response to high-rate shock/drop loadings. Such reliability concerns have motivated the important applications for experi-

Part D 36.2

Over the past several decades, experimental solid mechanics has played an important role in the highly interdisciplinary field of electronic packaging. There are several characteristics of microelectronic chips and their associated packages that have shaped the function that the discipline of experimental mechanics has played in product development, the types of experimental methods finding frequent application, and the challenges facing the experimentalist. These issues are now discussed as well as the key application areas for experimental mechanics in electronic packaging.

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Part D 36.2

mental mechanics measurements in electronic packaging. In addition, the experimentalist is commonly called upon to characterize the constitutive and failure behavior of packaging materials to aid in finite element modeling efforts. Materials of interest include solders, encapsulants, adhesives, thermal interface materials (TIM), and thin films (metallic and ceramic). All of these are applied in the assembly as thin layers or small interconnections. Thus, their behavior and properties are significantly different from the analogous bulk materials, and small-scale (microscale and nanoscale) test specimens are required for accurate characterization. The most obvious challenge to the experimentalist working on electronic packaging applications is the extremely small physical size of the samples of interest. For example, the silicon chips in flip-chip applications (Figs. 36.2 and 36.3) typically have dimensions of a few mm on a side (e.g., 5 × 5 mm), while the corresponding printed circuit board will have dimensions of many cm on a side (e.g., 10 × 10 cm). After solidification, flipchip solder joints have typical heights of 50–100 μm, which also represents the thickness of the thin underfill encapsulant between the PCB and the silicon die. Therefore, many of the measurements of interest (near silicon chips and solder joints) must be performed over regions on the mm or μm scale. A related complication is that multiple length scales (characteristics lengths ranging over several orders of magnitude) are involved in every microelectronics assembly. Thus, the mechanics and failure responses of interest in a given product are typically at several of these dimensional levels, often requiring that multiple methods be used to make measurements. A second major characteristic of electronic packaging geometries that thwarts many conventional experimental approaches is that many of the materials and structures of greatest interest are buried deep within the assembly. For example, the solder joints and the underfill layer in a flip-chip assembly are the primary locations of failure. However, they are totally inaccessible to the experimentalist. Therefore, techniques capable of interrogation of the interior of a specimen are required (ultrasonics, embedded sensors, x-ray, etc.). Most methods of experimental mechanics are applied to points on the surface of a sample. For them to be applied deep within an electronic package, approximations must be made such as cross-sectioning of the sample, so that the sample geometry being measured is not the same as the true product geometry of interest. Another approach for embedded regions of interest is to use hybrid experimental–numerical analyses, where experimental

data from the surface of the samples are used as boundary conditions for a numerical model of the structure. Other challenges to making accurate applications of experimental mechanics to electronic packaging include the complicated three-dimensional geometries of modern products and the presence of materials with poorly understood or unknown constitutive behavior. Finally, most measurements of interest must be made in harsh environments (extreme high/low temperatures, during thermal cycling, high humidity, etc.).

36.2.2 Key Application Areas and Scope of Chapter In the remaining sections of this chapter, selected applications of experimental mechanics in the electronic packaging field are presented and discussed. Due to the widespread use of experimental methods when performing research and development in the packaging field, it is impossible to cover the complete literature in the area, and it is thus necessary to limit the scope of the chapter in several ways. Firstly, the experimental techniques covered in the presented applications are limited to those of traditional experimental solid mechanics, which involve measurement of stresses, strains, and displacements (ultrasonic NDE, strain gages, moiré interferometry, photoelasticity, digital image correlation, etc.). Experimental techniques measuring other field variables such as temperature, magnetic field, electric field, moisture, etc. were considered beyond the scope of the chapter. For this reason, several important experimental methods for electronic packaging are not addressed in this work. These include thermal imaging and various methods of advanced microimaging (fine-focus x-ray, micro computerized axial tomography (micro-CAT), superconducting quantum interface device (SQUID), nuclear magnetic resonance (NMR), time domain reflectometry (TDR), etc.). The second limitation of scope relates to the levels of packaging and physical assembly sizes present in the selected examples. We have restricted discussion to applications involving electronic assembly architectures up to second-level packaging. Hence, the presented examples involve silicon chips and wafers, chip carriers, solder joints, and printed circuit boards, but do not cover larger electronic racks, chasses, cabinets, and systems. This approach parallels the bulk of the literature on packaging applications of experimental mechanics, where a large majority of the work focuses on the small length scales of microelectronic packaging. Finally, we have limited our presentation to examples involving mechan-

Electronic Packaging Applications

36.2 Experimental Mechanics in the Field of Electronic Packaging

1021

Experimental mechanics methods most commonly utilized

Experimental application

Description

Delamination detection

Detection of delamination initiation and propagation at material interfaces within the electronics assembly, especially at the interface between an encapsulant (underfill or mold compound) and the silicon chip

1. Acoustic microscopy (ultrasonic NDE)

Silicon stresses and deformations

Measurement of stress distributions in packaged silicon chips. Characterization of processing induced residual stresses and deformations in silicon wafers

1. Piezoresistive stress sensors (semiconductor strain gages) 2. Raman spectroscopy 3. Infrared photoelasticity 4. Coherent gradient sensing

Solder joint deformations and strains

Characterization of solder joint deformations and strains due to thermo-mechanical loadings caused my CTE mismatches

1. Moiré interferometry 2. Digital image correlation (2–D)

Warpage and flatness measurements

Measurement of warpage/flatness of printed circuit boards, components, silicon chips, and MEMS

1. Shadow and projection moiré 2. Twyman-Green and holographic interferometry 3. Infrared Fizeau interferometry

Transient measurements of strain and deformation

Measurements of high speed strains and deformations of electronic assemblies subjected to vibration, and shock/drop loads

1. Displacement markers 2. Strain gages 3. Digital image correlation (3–D)

Material characterization

Characterization of constitutive and failure behavior of materials found in thin layers/films and small interconnections in electronic assemblies. These include solders, underfill encapsulants, adhesives, thermal interface materials, and thin film metals and ceramics

1. Microscale and nanoscale tensile/shear testing

Fig. 36.4 Table of experimental applications and methods for electronic packaging

applications are grouped by the goal of the measurement being made rather than by the experimental method utilized. Presented applications include detection of material interface delaminations; measurement of stresses, strains, and deformations in silicon chips and wafers; characterization of solder joint deformations and strains, measurement of warpage/flatness of electronic substrates, components, and MEMS; measurement of transient behavior of electronic products subject to shock and drop; and innovative characterization of the constitutive and failure behavior of electronic packaging materials. In the following sections, the important experimental techniques are discussed for each measurement type, and data for selected examples are

Part D 36.2

ical and thermomechanical loading of the electronic assemblies. While there are certainly important packaging problems relating to other stimuli such as moisture, corrosion and chemical reactions, radiation, etc., these were considered to be outside of the scope of the current work. Given the self-imposed limitations in scope discussed above, Fig. 36.4 contains a table of the most important application areas for the practising experimentalist in electronic packaging. The corresponding stress and reliability concerns for each application area are also listed, as well as the methods of experimental solid mechanics that have been commonly utilized in the past by engineers and researchers. These key

1022

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Applications

given as well as key reference citations. It should be noted that the majority of the methods discussed in this chapter have been presented in detail in several of the earlier chapters in this Handbook. Therefore, only brief introductions are provided for most techniques.

Some of the presented methods are rather unique to electronic packaging (acoustic microscopy, embedded piezoresistive sensors, Raman spectroscopy, etc.). For these techniques, additional background material and references on the method itself are provided.

36.3 Detection of Delaminations Delamination or interfacial cracking is an important failure mode in several common packaging architectures including underfilled flip chips and molded plastic chip carriers such as the metal leadframe package (MLF), quad flat pack (QFP), and ball grid array (BGA). The delaminations of interest occur at the interface of a filled epoxy encapsulant (underfill or mold compound) with the silicon chip or metal lead frame. Once they occur, other failure modes are accelerated such as solder joint fatigue or metal trace corrosion. In addition, popcorn failures can occur when water vapor builds up at the delaminated interfaces.

ner is used to index the transducer across the sample and produce images that summarize the collected signals from the large rectangular array of point-by-point measurements. At the various material interfaces, the amount of energy that is transmitted or reflected depends on the acoustic impedances (Z) of the two media. For a single interface, the reflected and transmitted acoustic amplitudes are given by the expressions [36.10]

36.3.1 Acoustic Microscopy

where A0 is the incoming wave amplitude, Z 1 is the acoustic impedance of the first medium, and Z 2 is the acoustic impedance of the second medium. If the acoustic impedance of the first medium is less than the acoustic impedance of the second medium, a positive value will result from the reflected acoustic energy. This implies that the reflected sound wave will be in phase with the input sound wave generated from the transducer. If the acoustic impedance of the first medium is greater than that of the second medium, a negative value will result. In this case, the reflected sound wave will be out of phase with the generated sound wave. In the TS mode, the scanning acoustic microscope produces a shadowgraph image of the various layers in the package similar to an x-ray. Delaminations or voids

Part D 36.3

The primary experimental mechanics tool for study of delaminations in electronic packaging is scanning acoustic microscopy (SAM), which is a specialized version of ultrasonic nondestructive evaluation (NDE). Using various SAM techniques, the ability to detect delaminations in a variety of packaging configurations has been demonstrated including molded plastic packages [36.4, 5], ball grid arrays [36.6], ceramic substrates [36.7], stacked die packages [36.8], and flip-chip assemblies [36.9, 10]. A thorough review of the theory and application of acoustic microscopy to microsystems has been presented by Janting [36.11]. Acoustic microscopes utilize high-frequency ultrasound to examine the internal features in materials and components. Pulses of ultrasound are generated by a source transducer, and then transmitted through a liquid medium (normally water) to the electronic component and its interior. The two most common modes of operation are illustrated in Fig. 36.5. In the through scan (TS) mode, the pulse transverses the entire thickness of the sample, and is detected by a second transducer (receiver). In the pulse-echo (PE) mode, the reflections of the pulse from the various interfaces in the electronic package are collected by the same transducer as was used to generate the original pulse. In both cases, a very high-speed mechanical scan-

R = A0

Z2 − Z1 , Z1 + Z2

Pulse-echo mode Transmit and receive

T = A0

2Z 2 , Z1 + Z2

(36.1)

Through scan mode Transmit

Receive

Fig. 36.5 Modes of acoustic microscopy

Electronic Packaging Applications

Good interface Phase maintained

1023

Delaminated interface Phase reversal

Fig. 36.6 C-scan phase reversal at delaminated interfaces

Fig. 36.7 Acoustic microscopy scanning and CSAM im-

age for a delaminated QFP

0 Cycles

250 Cycles

500 Cycles

750 Cycles

1000 Cycles

1125 Cycles

1250 Cycles

1375 Cycles

1500 Cycles

1625 Cycles

1750 Cycles

1875 Cycles

2000 Cycles

2125 Cycles

2250 Cycles

2375 Cycles

2500 Cycles

2625 Cycles

2875 Cycles

3000 Cycles

Fig. 36.8 Delamination propagation in a flip-chip assembly ex-

posed to thermal cycling [36.10]

Part D 36.3

(air pockets) that are present will normally block the ultrasound from reaching the detector below the sample, so that they will appear as dark shadows in the acoustic image. The high sensitivity of the technique is due to the inability of ultrasound to traverse even a small 0.1 μm air gap. In the PE mode, the entire reflected signal that is recorded at a particular point is referred to as an A-scan. The various echoes displayed in the A-scan correspond to different interfaces in the component being examined. Normally, a single interface is studied by isolating the echo from that interface using an electronic gate that is established by knowing the distance of the interface from the transducer. This portion of the reflected pulse from a single interface is referred to as a C-scan. Based on (36.1) and the above discussion, the phase of the C-scan signal will be changed by the presence or absence of a delamination at the interface. This is illustrated for the underfill to silicon die interface in a flip chip package in Fig. 36.6. Thus, the most common approach for acoustic microscopy of electronic packages is to map out the phase of the C-scan signals for a single interface during the scanning process performed across the sample. The collected image for a particular interface formed from the C-scans across the sample is referred to as a C-mode scanning acoustic microscopy (CSAM) image. Delaminated regions are typically are identified by a single color (e.g., white) in the CSAM image, so that their presence and extent can be easily visualized. A commercial acoustic microscope operating in pulse-echo mode and scanning a typical plastic package (quad flat pack) is shown in Fig. 36.7. Also shown is the resulting CSAM image. In this case, the interface between the epoxy mold compound and the silicon chip was examined, as shown in Fig. 36.5. Delaminations (white regions) are clearly indicated near the four corners of the silicon chip. Outside of the chip area, the CSAM image reflects other features inside of the package, namely the copper lead frame. During the life of a product, delaminations can initiate and grow until catastrophic failure occurs. For example, Fig. 36.8 illustrates delamination progression in an underfilled flip-chip assembly (Figs. 36.2 and 36.3) exposed to thermal cycling from −40 to 125 ◦ C [36.10]. CSAM images of the die-to-underfill interface were recorded after various durations of cycling. At approximately 750 cycles, delamination initiates at the lower left corner of the chip. Electrical failure (solder joint fatigue) was recorded at approximately 1000 thermal cycles.

36.3 Detection of Delaminations

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Applications

36.4 Stress Measurements in Silicon Chips and Wafers Due to the central role of silicon devices and integrated circuits in microelectronics, experimental methods for characterizing the mechanics of silicon chips and wafers are of great interest. Silicon chips are often located deep within an electronics assembly, so that embedded piezoresistive sensors (semiconductor strain gages) fabricated on the chip surface are a natural tool for the experimentalist. In addition, Raman spectroscopy is popular for evaluating the stress state in silicon because it is noncontacting and nondestructive, while being applicable with fine spatial resolution using a microscope. Before final packaging, residual stresses and warpage can be induced in the silicon wafers during the semiconductor (chip) fabrication process. Both infrared photoelasticity and coherent gradient sensing have been shown to be powerful tools for understanding the mechanics of silicon wafers.

36.4.1 Piezoresistive Stress Sensors As illustrated in Fig. 36.9, piezoresistive sensors can be used to characterize the stress and strain distributions in packaged semiconductor die. The serpentine resistive sensors look remarkably like modern metal foil strain gages. However, the sensors are not mounted on the surface of the chip like conventional strain gages. They are conveniently fabricated into the surface of the die using microelectronic technology. Therefore, they are embedded sensors that are capable of providing nonintrusive measurements of silicon surface stress and strain states inside electronic packages. In conductors such as silicon that exhibit a strong piezoresistive effect; the electrical resistivity changes dramatically when the material is subjected to stress/strain, which leads to measurable resistance changes in the rosette elements. The magnitudes of Wafer

Test chip

Part D 36.4

Sensor rosette

Package

3-D stress state

Fig. 36.9 Piezoresistive stress sensor concept

the resistivity-induced resistance changes are orders of magnitude greater than those caused by dimensional changes, which are normally neglected. On-chip piezoresistive sensors in electronic packaging are usually referred to as stress sensors (rather than semiconductor strain gages), because their response is normally calibrated with configurations having known stress states such as four-point beam bending or hydrostatic pressure. However, they could also be calibrated in terms of strain if desired, or strains can be calculated from the measured stresses using the well-known linear elastic stress–strain relations and material properties for silicon. If the piezoresistive sensors are characterized over a wide temperature range, thermally induced stresses can be measured. Finally, a full-field mapping of the stress distribution over the surface of a die can be obtained using specially designed test chips that incorporate a tightly packed array of sensor rosette and on-chip multiplexing (Fig. 36.10). The piezoresistive behavior of semiconductors such as silicon has been studied extensively for many years, with the first studies being performed by Bridgeman [36.12] and Smith [36.13] in the 1950s. Early silicon transducer applications were pioneered at Bell Labs by Mason and Thurston [36.14], while Edwards and co-workers [36.15, 16] at Texas Instruments made the first applications of on-chip piezoresistive sensors within plastic packages in the early 1980s. Many potential applications exist for piezoresistive sensors in the microelectronic packaging industry including qualifying of manufacturing processes, guiding material selection, and evaluating reliability. An extensive review of piezoresistive sensor theory, rosette design, calibration techniques, and modern electronic packaging applications has been given by Suhling and Jaeger [36.17]. Sensor theory and its relation to semiconductor physics has been reviewed by Sweet [36.18]. The first detailed derivation of the sensor theory for embedded piezoresistive sensors used in electronic packaging was performed by Bittle et al. [36.19]. In that work, the equation for the normalized resistance change of an arbitrarily oriented sensor on a general silicon wafer surface (Fig. 36.11) was related to the applied stress components σij and the temperature change ΔT . For the most common (100) and (111) crystal orientation silicon wafer planes used in modern microelectronics, the basic sensor equations become [36.19]:

Electronic Packaging Applications

•

(100) surface   ΔR π11 + π12 + π44  = σ11 R 2    π11 + π12 − π44  + σ22 cos2 φ 2   π11 + π12 − π44  σ11 + 2    π11 + π12 + π44  + σ22 sin2 φ 2   + (π11 − π12 )σ12 sin 2φ + π12 σ33 + (α1 ΔT + α2 ΔT 2 + . . . ) ,

•

1025

Shear stress sensors in corner

Normal stress sensor array

Shear stress sensor array

(36.2)

(111) surface ΔR     + B2 σ22 + B3 σ33 = B1 σ11 R √  2  cos φ + 2 2(B2 − B3 )σ23     + B2 σ11 + B1 σ22 + B3 σ33 √   sin2 φ − 2 2(B2 − B3 )σ23   √   sin 2φ + 2 2(B2 − B3 )σ13 + (B1 − B2 )σ12   2 (36.3) + α1 ΔT + α2 ΔT + . . . ,

where x1 , x2 , x3 is the wafer coordinate system shown in Fig. 36.11, φ is the angle between the x1 -axis and the resistor orientation, π11 , π12 , π44 are the three piezoresistive coefficients of silicon (material constants), and α1 , α2 , . . . are the temperature coefficients of resistance of silicon (material constants). The B j constants in (36.3) are a set of combined piezoresistive coefficients given by

Fig. 36.10 Stress test chip with piezoresistive sensor arrays

general resistance change expression in (36.3) is dependent on all six of the unique stress components at a point. Therefore, sensor rosettes can be developed using (111) silicon that can measure the complete threedimensional state of stress at points on the surface of a die [36.17, 19]. Such a rosette is pictured in Fig. 36.9, and consists of two sets of 0◦ –45◦ –90◦ –135◦ oriented sensors made from p-type and n-type doped silicon. This sensor design on (111) silicon represents one of the few tools in experimental mechanics capable of extracting the complete state of stress at a point within a structure. Stress extraction relations are established for a particular rosette through application of either (36.2) or (36.3) to each element in the rosette, and then inverting the equations. The resulting expressions give each stress component in terms of the measured normalized resistance changes of the various sensors ΔRi /Ri , the temperature change ΔT , and the silicon sensor material constants (piezoresistive coefficients and temperature coefficients of resistance). Such equations for the most popular rosettes on (100) and (111) silicon x'2

n

φ

x'1

Fig. 36.11 General sili-

con wafer and arbitrarily oriented piezoresistive sensor

Part D 36.4

π11 + π12 + π44 , 2 π11 + 5π12 − π44 , B2 = 6 π11 + 2π12 − π44 (36.4) . B3 = 3 From (36.2), the resistance change of an in-plane sensor fabricated on (100) silicon is observed to depend  , σ  , σ  , σ  ) and the on four components of stress (σ11 22 33 12 orientation of the sensor. Rosettes on this surface are capable of measuring these four stress components, but cannot characterize the out-of-plane (interfacial) shear  and σ  . The magnitudes of these interfastresses σ13 23 cial shear stresses are often significant for encapsulated chips, where a mold compound or underfill encapsulant is in direct contact (bonded to) the die device surface. For this reason, it is important to note that the B1 =

36.4 Stress Measurements in Silicon Chips and Wafers

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Applications

Fig. 36.12 Flip-chip test assembly and stress test chip σ '11 –σ'22 (MPa) 60 40

Site 2 Board 11 Underfill: UF1

x'2 s2

20

x'1

0 –20 –40

Part D 36.4

–60 –80

Cool down

Cure at T = 165 °C 0

200

400

600

800

1000

1200

1400

Time (s)

Fig. 36.13 Flip-chip die stress variation during underfill cur-

ing [36.22]

are given in [36.17]. With clever choices of the sensor orientations in a rosette, it is possible to eliminate the temperature change ΔT in some or all of the stress extraction equations, so that the stress components can be calculated directly from the resistance change measurements without the need to know the temperature change. Such measurements are referred to as temperature compensated. This is a particularly important attribute for making accurate measurements with a rosette, given the large errors which can be introduced into non-temperature-compensated stress sensor data when the temperature change is not precisely known [36.20, 21]. Test chips incorporating piezoresistive stress sensors have been used in a wide variety of ways to evaluate assembly and packaging technologies [36.17]. For example, the flip chip on laminate geometry shown in Fig. 36.3 has been explored by Rahim and coworkers [36.10, 22]. The test board and (111) silicon stress test chip utilized in those studies are pictured in Fig. 36.12. For each test assembly the transient sensor resistances were first monitored during the cure cycle of the underfill encapsulant (8 min at 165 ◦ C). After final assembly was completed, the sensor resistances were also measured at room temperature, and then during repeated thermal cycling between −40 to +125 ◦ C. Using the measured resistance change data from each step of the assembly procedure and thermal cycling testing, the die stress variations were easily calculated. Figure 36.13 shows the variation of the in-plane normal stress difference with time during curing of one of the assemblies. In this plot, the stress was measured at the rosette located at site 2 (on the die boundary at the midpoint of one of the die sides). During the first 8 min (480 sec) of exposure to the 165 ◦ C temperature in the cure oven, the assembly temperature gradually rises. It is seen that this temperature change is accompanied by small changes in the stresses on the die surface contacting the underfill. These are likely due mostly to the encapsulant shrinkage during cure and other complicated changes in the polymeric underfill material as it hardens. Upon completion of the hold at 165 ◦ C, the boards were removed from the oven and allowed to cool on a flat table (chip side facing up) in a roomtemperature environment. As seen in Fig. 36.13, the majority of the final assembly die stresses are built up during the cooling of the flip-chip assembly after cure, where the underfill encapsulant is fully hardened and can provide a significant stiffness to cause the flip-chip assembly to bend. The final room-temperature stress values were found for several different underfill ma-

Electronic Packaging Applications

Stress (MPa) 25

|σ'12|

20

15 Delamination ocurred and destroyed rosette

10

5

τinterfacial 0

0

200

400

600

800 1000 1200 1400 1600 1800 2000

Thermal cycles (– 40 to 125 °C)

Fig. 36.14 Flip-chip die stress variation during thermal cy-

cling [36.10]

terials, allowing them to be ranked for their level of induced die stress. After underfill cure and cool-down, the final assemblies were exposed to thermal cycling from −40 to +125 ◦ C in an environmental chamber. Such reliability qualification testing is often carried out to evaluate the capability of the flip chip on laminate assemblies and underfill encapsulants to survive harsh environments. The thermal cycling tests were performed in a staged fashion with several different rounds of testing. Each stage consisted of either 125 or 250 cycles. After each increment of cycling was completed, room-temperature stress measurements were made on all of the test boards. In addition, CSAM images of the die-to-underfill interface were also recorded for each flip-chip sample. In this way, both the die stress and delamination histories of the samples were recorded as a function of the

36.4 Stress Measurements in Silicon Chips and Wafers

1027

number of thermal cycles. Figure 36.8 illustrates the delamination growth at the underfill-to-die passivation interface measured in one of the flip-chip assemblies (board 22) at several levels of thermal cycling. The delaminations at the die device surface are seen to initiate at approximately 750 thermal cycles at the corner where the underfill was dispensed and where the measured high in-plane and interfacial shear stresses were observed. At a little over 1000 thermal cycles, the delamination front had passed through the sensor rosette at site S9, nearest to the lower left-hand corner of the chip. The measured shear stress histories at site S9 for this assembly are shown in Fig. 36.14. From these data, it can be seen that the interfacial shear stress remains fairly constant until delamination occurs. At the point of delamination at the sensor site, the interfacial shear stress becomes approximately zero since the tractions between the underfill and die are released. On the contrary, the in-plane shear stress is observed to have changed dramatically, even before delaminations were detected using CSAM. The value was found first to suddenly increase, and then decrease gradually until delamination occurred. At the point of delamination, either the sensors themselves or their associated solder joints were damaged, so that the rosette no longer functioned.

36.4.2 Raman Spectroscopy When monochromatic light is scattered from an object, most photons are of the same frequency as the incident light. However, a small number of photons will interact inelastically with the lattice vibrations (phonons) of the material, and will have a different frequency than the incident light. This interaction is called Raman scattering, and can be detected by careful observation of

Computer Monitor Detector

Camera

Sample Argon laser

XY-stage

Fig. 36.15 Micro-Raman spectrometer for stress measurement

Part D 36.4

Microscope

Spectrometer

1028

Part D

Applications

the spectrum of the scattered light. The majority of Raman photons are excited to a higher frequency level (Stokes–Raman scattering). A typical (micro-)Raman spectrometer is shown in Fig. 36.15. Laser light is focused on the sample through a microscope, and the back-scattered radiation is collected and analyzed in the pre-monochromator and spectrometer. The resulting spectrum contains information on the Raman signal of the sample. Stress and strain in the crystalline material affect the frequency of the material phonons, and therefore influence the position of the Raman peak in the recorded spectrum. By observing the shifts in the Raman peak, the stresses on the surface of the observed material can be experimentally characterized. For the (100) silicon surface, the stress sensitivity of the Raman peak is well understood, and many studies have been performed over the past 30 years to characterize the effects of various semiconductor fabrication and packaging processes on the stresses in silicon chips. The literature on Raman stress measurements in silicon has been reviewed by De Wolf [36.23] for semiconductor processing, and Chen and De Wolf [36.24] for packaging. Figure 36.16 shows a typical Raman spectrum for crystalline silicon illuminated with an argon-ion laser. The sharp peak on the left side of the spectrum is a Rayleigh scattered plasma line from the laser, while the peak on the right side of the spectrum is the Raman (Stokes) peak for silicon at a frequency of ω = 521 cm−1 . As indicated on the graph, mechanical stress in the silicon will shift the peak. By recording the Raman shifts at different positions on the sample, a mapping of the stress distribution can be made. The relation between the stress state at the point and the resulting Raman frequency shift is very complicated, and all six stress components affect the position of the

Raman peak for a general stress state. In addition, the theoretical relationship between the Raman shift and the stress components depends on the wavelength and polarization of the input radiation, the angle of the input radiation relative to the silicon crystallographic coordinate system, and a set of Raman material constants. For normally incident light on a (100) silicon chip surface in plane stress, the equation becomes [36.24] σ11 + σ22 ≈ −518Δω (MPa) ,

(36.5)

cm−1 ,

where Δω has units of and x1 = (100) and x2 = (010) are the crystallographic directions of silicon. Because only a single shift of the spectrum is observed, the Raman spectroscopy method typically is sensitive to a combined stress, as demonstrated in (36.5). In modern Raman microscopes using argon-ion lasers, the spot size of the laser is typically less than 1 μm in diameter, spatial positioning can be made to the nearest 0.1 μm, and the penetration depth of the incident laser into the silicon is 0.8 μm. The spectral resolution of a Raman spectrometer is often estimated to be ±0.02 cm−1 . Using (36.5), this corresponds to a stress measurement resolution of about 10 MPa. With such high spatial resolution, it is possible to map out the stress distributions around silicon devices on a processed die; for example, Fig. 36.17 illustrates the stresses along a line on the surface of a chip that transverses several oxide filled isolation trenches. Such trenches are required to electrically isolate closely spaced devices (transistors), but are known to induce large stress in the silicon substrate [36.25]. Steep gradients are observed in the stress distribution near the SiO2 Si Compressive

Intensity (arb. units) Tensile stress Δω < 0

800

Compressive stress Δω> 0

600 Plasma line

Si Raman peak

Stress

400

Part D 36.4

Process 1 Process 2

200 460

480

500

520

540

560 1 Raman frequency ω = (cm–1) λ

Fig. 36.16 Raman spectrum for silicon [36.24]

Tensile –10

0

10

20 30 Position (μm)

Fig. 36.17 Raman spectroscopy study of stresses near

oxide-filled trenches in silicon

Electronic Packaging Applications

Metal lines

Wire bond Si

Ag

Cu

36.4 Stress Measurements in Silicon Chips and Wafers

with appropriate infrared light sources and intensity detectors. However, silicon is optically and mechanically anisotropic (m3m cubic crystal symmetry), which leads to a much more complicated set of equations when applying the method. One ramification of this anisotropy is that there is a difference between the isoclinic angle identifying the fast and slow directions of infrared trans-

(MPa) 150

Czochralski

MPa 9

100

50

8

50

100

7

0

150

6

–50

200

–100

250

–150

300

Fig. 36.18 Raman spectroscopy study of stresses near

350

a wirebond pad [36.24]

400

5 4 3 2 1 100

boundaries of the oxide regions. In another study, the stresses induced by silicon wafer back-grinding have been measured using Raman spectroscopy [36.26]. Figure 36.18 shows a Raman-measured stress distribution in a silicon chip bonded to a copper substrate [36.24]. In this case, the silicon chip was cross-sectioned and the incident radiation impinged the side of the die. It is easy to distinguish the stresses introduced by both the bonding to the copper substrate and the wirebond on the top of the chip. Such investigations allow for determination of the effect of different assembly processes on the packaging induced stresses. In other studies, the Raman method has been used to characterize stresses in chips packaged in ball grid arrays [36.27], power semiconductor packaging [36.28], and MEMS cantilevers [36.29].

200

300

400

Edge defined film

MPa

50

14

100

12

150

10

200

8

250

6

300 4 350 2

400 100

200

300

400

Conventional cast

MPa 10

36.4.3 Infrared Photoelasticity

50

9

100

8 7

150

6 200

5

250

4

300

3

350

2

400

1 100

200

300

400

0

Fig. 36.19 Photoelastic stress distributions in silicon

wafers

Part D 36.4

Silicon wafers contain residual stresses due to a variety of reasons, including the thermal gradients that occur during crystal growth and processing of the integrated circuit chips. Stresses can induce warpage or may lead to failure of the processed chips during subsequent packaging steps. Infrared photoelasticity has proved to be a powerful method for stress measurement in silicon wafers, strips, and chips. Silicon is transparent to infrared radiation over the wavelength range of approximately λ = 1100–1500 nm. Thus, photoelastic methods based on plane and circular polariscopes can be utilized

1029

1030

Part D

Applications

mission, and the angle to the principal stress directions of the material. Also, the stress–optic coefficient is orientation dependent in most silicon crystal planes, and must be calculated from the isoclinic angle at the point. Early studies by Lederhandler [36.30] and DeNicola and Tauber [36.31] have showed the potential of using birefringence measurements to understand the stresses in rather thick (5–7 mm) silicon samples. Gamarts and co-workers [36.32], Niitsu et al. [36.33], and Fukuzawa and Yamada [36.34] have used scanning infrared polariscopes for interrogating the photoelastic properties of silicon wafers and strips. The main disadvantage of their scanning systems is the use of a fine-diameter infrared laser and single-point detector. This requires that each wafer be mounted on a translating stage and scanned point by point to generate a full-field view. More recently, Danyluk and co-workers [36.35, 36] have developed a full-field infrared polariscope that employs phase-stepping techniques and fringe multiplication methods to increase stress resolution and allow the stress distribution to be characterized over large areas; for example, Fig. 36.19 illustrates typical residual stress mappings of the principal stress difference that they have obtained for silicon wafers for photovoltaic cells that have been grown with various methods including Czochralski crystal growth, edgedefined film sheet forming, and conventional casting. The pattern of residual stress is closely related to that of the crystal structure, with bands of high stress following the features of the crystal. In addition, the stresses in the edge-defined film technique are notably higher.

Wafer

y

Beam splitter

They have also applied their system to characterize residual stresses in silicon solar cells [36.37]. Wong et al. [36.38] have also used a full-field approach to measure stress distributions caused by thin-film structures deposited onto the wafer surface. In their study, the wafer was cross-sectioned, and the stress measurements were made through the thickness of the wafer by employing incident radiation parallel to the wafer surface. As mentioned above, the photoelastic theory for silicon is considerably more involved than the conventional equations for isotropic materials, and a set of linear anisotropic piezo-optical constitutive relations must be utilized. The first measurements of the stress–optic coefficients of silicon were made by Giardini [36.39]. Liang et al. have developed the detailed equations needed to measure the stresses in (100) and (111) silicon wafers for illumination and observation orientations in the plane of the wafer [36.40]. For illumination and observation perpendicular to the wafer, the appropriate theoretical expressions for (100) and (111) silicon can be found in the work of Gamarts et al. [36.32] and Danyluk and co-workers [36.35, 36]. In addition, [36.36] contains a careful set of calibration experiments for the stress–optic coefficients using modern high-quality silicon crystals.

36.4.4 Coherent Gradient Sensing Warpages or deviations from flatness are often experienced in silicon wafers during the various steps in the semiconductor (chip) fabrication process. These

y x

x z

Filtering lens Collimated laser beam

Filter plane

Part D 36.4

Grating G2 Grating G1

Camera

Fig. 36.20 Shearing interferometer for coherent gradient sensing (CGS)

Electronic Packaging Applications

36.5 Solder Joint Deformations and Strains

1031

beam passes through two parallel gratings G1 and G2 of pitch p, which are separated by a distance δ. An interference pattern is then generated from the grating arrangement and is subsequently imaged onto a chargecoupled device (CCD) imaging sensor. The fringes of the interference pattern are contours of the constant slope whose values are governed by the relation ∂w ∂x

∂w ∂y

Fig. 36.21 Silicon wafer slope contours (fringes) measured by CGS [36.41]

can be locked into the wafer due to thermomechanical residual stresses, leading to a variety of problems in future processing steps. Such effects are of increased concern due to several trends in the semiconductor industry including the continuing increases in wafer diameter to accommodate improvements in batch processing throughput, the decreases in wafer thickness via back-grinding and other methods to support reduction in packaged chip size, and the use of sophisticated interconnect structures that involve Damascene processes with chemical–mechanical polishing (CMP). Coherent gradient sensing (CGS) is a method of optical shearing interferometry that can measure slopes or gradients of the out-of-plane displacement of a sample when used in reflection mode [36.42]. A schematic of the experimental setup is shown in Fig. 36.20. A coherent collimated laser beam is directed to the wafer and reflected off of the curved wafer surface. The reflected

Np ∂w = , ∂y δ

N = 0, ±1, ±2, . . . ,

(36.6)

where w is the z-displacement of the wafer surface and N is the fringe order. A second set of contours for the other slope can be obtained by rotating the gratings (or the specimen) by 90◦ : Mp ∂w (36.7) = , M = 0, ±1, ±2, . . . . ∂x δ Rosakis and co-workers have applied the CGS method to silicon wafers to extract slopes and curvatures [36.41, 43]; for example, Fig. 36.21 illustrates contours of constant slope for a processed silicon wafer of diameter 150 mm. In this case, the process of applying a thin tungsten film was the source of the majority of the observed warpage. By differentiating the slopes from the processed fringe patterns, the normal and twisting curvatures were calculated and then shown to correlate well with analytical predictions. Boye et al. [36.44] have used CGS in a similar manner to investigate silicon wafer deformations during Damascene processing. Curvatures were calculated from the measured slope distributions, and then stresses were estimated based on a plate-theory-based analytical model.

36.5 Solder Joint Deformations and Strains to characterizing strains in solder joints. Both techniques require flat surfaces in their most conventional forms, so that cross-sectioning of the sample is performed. Analysis is then performed on the exposed surface that was formerly in the interior of the electronic assembly. Therefore, these methods only provide approximate measurements of the actual displacements in the package since the sample being measured is not the same as the true product geometry of interest. In particular, the visualized surface is a traction-free surface due to the cross-sectioning process. Thus, the points being viewed in the sectioned solder joints are in states of plane stress, whereas the points in the interior of actual joints are subjected to full three-dimensional stress states.

Part D 36.5

Perhaps the most well-known and often investigated mechanics issue for electronic packaging is solder joint reliability. As discussed earlier, thermomechanical loadings and nonuniform thermal expansions result in most packaging architectures due to temperature changes and mismatches between the coefficients of thermal expansion of the materials comprising the assembly. Solder joints often experience severe shearing deformations and strains due to their role and position as interconnections between the low-expansion silicon chip and high-expansion PCB substrate. The full-field optical methods of moiré interferometry and digital image correlation are well suited for measuring twodimensional (in-plane) thermomechanical deformations of electronic packaging, and have been widely applied

1032

Part D

Applications

36.5.1 Moiré Interferometry Moiré interferometry is a powerful method for measuring in-plane deformations (u, v) in the directions parallel to the surface of the body. A detailed description of the method is presented in Chap. 22, and a comprehensive treatment of the theory and application of the technique to engineering problems has been given by Post et al. [36.46]. It is used extensively in the microelectronics industry for characterizing thermally induced deformations of electronic packages, and a recent review article summarizing the state of the art in interferometric moiré applications to packaging has been published by Han [36.47]. Thermal deformations are most easily analyzed using room-temperature observations. This requires the specimen grating to be applied at an elevated temperature, and then the package is allowed to cool to room temperature before it is observed in a moiré interferometer. With this approach, the thermomechanical deformations incurred by the temperature change are locked into the grating and recorded at room temperature [36.45–48]. For real-time thermal loading and fringe processing at elevated temperatures, an environmental chamber or local heater must be used and positioned adjacent to the moiré interferometer [36.49– 52]. As shown in Fig. 22.12, a specimen in an environmental chamber must be connected rigidly to the interferometer to avoid transmission of vibrations from the air circulating in the chamber. The connecting rods do not contact the chamber due to the use of compliant baffles. The first applications of moiré interferometry to electron packaging were performed by Voloshin et al. [36.49, 50], who measured thermally induced deformations within sectioned plastic packages (e.g., DIP and QFP) used for memory devices. This work concentrated on the mismatches occurring between the

deformations in the polymeric mold compound and silicon chip. The epoxy encapsulated silicon chips were subjected to various uniform temperature changes above and below the reference temperature at which the specimen grating was formed. Early applications of moiré interferometry to solder joint interconnections were performed at IBM by Guo and co-workers [36.45, 48]. Using a ceramic ball grid array (CBGA) package mounted on a laminate substrate (Fig. 36.22), they demonstrated in dramatic fashion the dependence of the solder joint shear strains on the distance from the package center (neutral point). The horizontal and vertical in-plane displacement contours for the CBGA package are shown in Fig. 36.23. In this application, a reference grating frequency of 2400 lines/mm was used, resulting in a displacement sensitivity of 0.417 μm per fringe order. The specimen grating was applied at T = 82 ◦ C, and then the sample was cooled ΔT = −60 ◦ C to room temperature (T = 22 ◦ C). Because of the large CTE mismatch between the ceramic chip carrier (α = 6.5 × 10−6 1/◦ C) and the laminate PCB (α = 19 × 10−6 1/◦ C), severe shearing deformations resulted in the solder joints. The solder joint strains can be extracted from the u, v fringe patterns using the linear strain–displacement relations. A plot of the solder joint shear strain versus ball position is shown in Fig. 36.24. In this graph, the strains calculated from the moiré fringes are referred to as the nominal shear strains. In addition, approximate solder strains calculated from the relative displacements of the ceramic and PCB, and the analytical distance to neutral point (DNP) formula are also shown. The a) Horizontal displacement contours (u)

b) Vertical displacement contours (v) y

Heat sink

Part D 36.5

x Multilayer ceramic module

z

Printed circuit board

Fig. 36.22 Ceramic ball grid array cross-section used for moiré interferometry

Fig. 36.23 Moiré interferometry fringe patterns for a CBGA subjected to temperature change [36.45]

Electronic Packaging Applications

Relative displacement |u | (μm) 10

Nominal shear strain |γxy | (%) 0.5

y x

8 n = –9

6

n=0

0.4

n=9

DNP formula Relative displacement Nominal shear strain

0.3

4

0.2

2

0.1

0 –10

–8

–6

–4

–2

1033

0

2

0 4 6 8 10 Index of solder ball n

Fig. 36.24 Solder joint shear strains in the CBGA assembly Solder bump and underfill

Chip Substrate

y, v

Solder ball PCB

1.37 mm

x, u

Fig. 36.25 Flip-chip BGA and moiré interferometry fringe patterns [36.51]

by using an extension of the standard moiré technique called microscopic moiré interferometry. As described in Chap. 22, the microscopic moiré procedure uses an immersion interferometer and fringe multiplication via phase stepping to increase the basic displacement sensitivity. In addition, the specimen is viewed through a microscope objective to increase the spatial resolution and visualize small features on the sample. Figure 36.26 shows a diagram of the microscopic moiré interferometer developed by Han, and Fig. 36.27 illustrates recorded fringe patterns for flip-chip solder joints [36.52]. In this application, the fringe sensitivity was increased by nearly a factor of 10–52 nm/fringe. The measured deformation contours illustrate high sol-

Part D 36.5

DNP formula assumes unconstrained deformations of the ceramic and PCB, and predicts a linear relationship between the solder joint shear strain and the distance from the center of the package (neutral point). It is observed that the true solder strains are smaller than those yielded by the approximate calculations. In addition, the shear strains are seen to increase monotonically from the center to outside of the package, as expected by the DNP formula. However, the response is not linear. The maximum shear strains occur at the most exterior solder joint, which will be the one of greatest reliability concern for solder joint fatigue during thermal cycling. Real-time thermal deformation measurements have also been performed on BGA packages using moiré interferometry by Han et al. [36.51, 52]; for example, fringe patterns for the u, v displacements of a PBGA component are presented as a function of temperature in Fig. 22.13. After performing several thermal cycles, significant plastic and creep deformations were observed in the solder joints. Similar moiré displacement contours for a flipchip ball grid array (FC-BGA) package exposed to ΔT = −60 ◦ C are shown in Fig. 36.25 [36.51]. In this case, the thin printed circuit boards above and below the large BGA solder joints have comparable expansion coefficients. Only the silicon chip has a relatively low expansion coefficient, and it is buffered from the BGA solder balls by the BGA interposer board. Based on a long history of using the DNP approximation for estimating solder strains, engineers in the electronics industry often assume that the exterior solder joints (furthest from the center of the assembly) will be the least reliable. However, this has often proved to be incorrect for modern packaging architectures, and moiré interferometry has repeatedly provided experimental demonstration of this fact. Using the fringe patterns in Fig. 36.25, one can see that the most densely spaced fringes are present in the circled solder balls, directly underneath the so-called die shadow. By processing the fringe data, and extracting the shear strains, it was shown that the highest strains are also in this joint, making it the least reliable during thermal cycling exposures [36.51]. From the fringe patterns in Fig. 36.25, it is also observed that little or no fringe data can be obtained for the very small flip-chip solder joints directly underneath the silicon chip. Although the strains can be extremely high in such joints, the displacements are still extremely small due to the tiny size of the region of interest. As developed by Han and co-workers [36.52, 53], flip-chip solder ball deformations and strains can be explored

36.5 Solder Joint Deformations and Strains

1034

Part D

Applications

positioned to explore deformations and strains in future generations of high-density interconnects.

S: Optical fiber from laser light source T: Translation device for fringe shifting

CCD video camera

36.5.2 Digital Image Correlation Digital image correlation (DIC) techniques generate full-field deformation and shape measurements from images of an object stored in digital form. Image analysis is performed to correlate patterns in the images (e.g., lines, grids, dots, and random speckle patterns). The most typical version of the method makes the use of optical images of random speckle patterns on a flat (two-dimensional) specimen surface, and involves a comparison (correlation) of subregions throughout the undeformed and deformed images to obtain a full field of deformation measurements. The specimen speckle patterns can be obtained using coherent laser illumination or by using white-light illumination when a high-contrast speckle pattern has been applied to the sample. Fundamental continuum mechanics concepts are applied to the deformation fields, and numerical algorithms have been developed for matching subsets throughout each image. The end result is a dense set of full-field two-dimensional displacement measurements. The displacement field can then be differentiated to yield the in-plane strains.

Collimator Objective lens Specimen

T S

Loading frame

Rotary table

Immersion interferometer

x, y traverse

Fig. 36.26 Microscopic moiré interferometer

der joint shear strains for the flip-chip assembly without underfill, and negligible solder joint strains when underfill was employed. With the increased resolution and sensitivity of the microscopic moiré method, it is well

No encapsulation Flip chip bump

Underfill encapsulation

Chip

5 mm

Substrate

0

200

Encapsulant

400 μm

Chip Thermal loading of ΔT = –60°C

γxayve = 0.1%

γxayve = 1% PCB β=2

Part D 36.5

After 5 thermal cycles from – 40 °C to 80 °C

104 nm/fringe

β=4

γxayve = 1.3%

52 nm/fringe

γxayve = 0.1%

Fig. 36.27 Microscopic moiré β=1

208 nm/fringe

β=4

52 nm/fringe

fringe patterns for flip-chip solder joints [36.52]

Electronic Packaging Applications

y

Silicon – 0.001 Epoxy underfill

x

36.5 Solder Joint Deformations and Strains

Silicon

εy

1035

γxy

– 0.001

Silicon chip

0.005

0.000

0.001 PCB

Soldermask

Solder bumps

– 0.001

Flip-chip assembly

0.010 FR- 4 board

FR- 4 board

Fig. 36.28 Flip-chip solder joint strain distributions characterized by DIC [36.54]

In addition to optical images, DIC methods have been applied to digital data from a variety of other imaging/contouring sources, including scanning electron microscopy (SEM) and scanning probe microscopy (SPM). Three-dimensional correlation techniques that require stereo vision (two digital cameras) have also been developed. These methods can be used to measure three-dimensional deformations, and avoid the errors present in two-dimensional (2-D) DIC measurements due to out-of-plane motions that cause magnification SAC solder joint

changes. Overviews of the DIC method and associated theoretical developments are given in Chap. 20. DIC techniques have been applied to electronic packaging configurations by various groups, and the method continues to increase in popularity as turn-key systems become available in the marketplace. Typically, a cross-sectional sample of the package that contains solder joints and/or other materials of interest within the assembly is prepared. Images are then recorded at various times (e.g., during a temperature change). VoGrain boundaries

150 μm

150 μm

Displacement

u (µm)

εVM (%) 3

von Mises strain

0.4 2.5

0.2

2

0

1

–0.4 150 μm 150 μm

–0.6

Fig. 36.29 SAC solder joint and associated DIC deformation and strain contours [36.55]

0.5

Part D 36.5

1.5

–0.2

1036

Part D

Applications

gel et al. [36.54, 56] have made several investigations concentrating on deformations and strains of flip-chip solder joints. In their work, the before and after images of the solder joints were made in an SEM. Figure 36.28 illustrates solder joint images with superimposed strain contours calculated from the two-dimensional deformation fields extracted with DIC for a temperature change of ΔT = +100 ◦ C [36.54]. Their work has emphasized generating solder joint experimental displacement and strain data for correlation with numerical finite element simulations. An overview of DIC applications in microsystems is given in [36.57]. Park et al. [36.55] have explored displacement and strain distributions in lead-free Sn–Ag–Cu (SAC) solder joints used in BGA components. Using polarized-light microscopy, they have identified the grain boundaries in the joint, and correlated their positions with unusual features in the deformation and strain fields

induced by temperature change. For example, Fig. 36.29 shows photographs of an SAC solder joint and its grain structure, as well as the horizontal displacement and von Mises strain distributions characterized using DIC. Both mechanical fields contain strong features and peaks that correlate directly to the solder joint grain structure. Kelly et al. [36.58] have performed a detailed comparison of moiré interferometry and DIC measurements for several flip-chip solder joint packaging geometries. Their overall conclusion was that there was a strong correlation of the results from the two methods, and that when properly applied, the DIC method is capable of achieving accuracy levels equal to the most advanced moiré interferometry techniques. Relative to moiré methods, DIC has several advantages including quicker specimen preparation, shortened data-processing times, and fewer restrictions in harsh thermal environments.

36.6 Warpage and Flatness Measurements for Substrates, Components, and MEMS

Part D 36.6

Warpage is a common problem in electronic packaging. In most packaging architectures, manufacturing steps occur at elevated temperatures (e.g., solder joint reflow and plastic encapsulation), when the assembly is nearly flat. As cooling takes place, out-of-plane deformations (bending) can occur due to CTE mismatches or unbalanced constructions. Several whole-field optical methods of experimental mechanics have been utilized for warpage measurements in microelectronic components, often as a function of temperature. These include holographic interferometry, Twyman–Green interferometry, shadow moiré, and infrared Fizeau interferometry. In the most common incarnations of these techniques, the sample (PCB, component, chip, etc.) is typically flat or nearly flat, and it is both illuminated and viewed in the direction normal to its surface. The obtained fringe patterns are contours of out-of-plane displacement. The various methods offer different capabilities, particularly in terms of the amount of surface preparation required and the fringe sensitivity to out-ofplane deformation.

36.6.1 Holographic Interferometry and Twyman–Green Interferometry Holographic interferometry (Chap. 24) and Twyman– Green interferometry [36.47] require illumination of the

sample by coherent (laser) light. For normal illumination and viewing, the out-of-plane deformation fringes satisfy w=

Nλ , 2

(36.8)

where w is the warpage displacement, N is the fringe order, and λ is the wavelength of the laser light employed. For common He–Ne lasers with λ = 633 nm, the contour increment for the fringe pattern is λ/2 = 0.316 μm. Holographic interferometry has been applied by several groups to measure deformations in microsystems [36.59–63]. Recent overviews of these efforts have been given by Pryputniewicz [36.61], Furlong [36.62], and Osten and Ferraro [36.63]. Holography has been particularly popular for looking at the deformations of MEMS and MOEMS devices using microscope based interferometry systems; for example, Fig. 36.30 shows holographic interferometry fringes for the normal deformations (deviation from flatness) of a MEMS gear assembly [36.61]. The diameter of the small gear is about 50 μm, and the system is designed to operate at 360 000 rpm. The fringe patterns were obtained for several positions of the gears when the mechanism was being operated, allowing the tilt of the gears to be plotted as a function of gear position. The maximum tilting

Electronic Packaging Applications

36.6 Warpage and Flatness Measurements for Substrates, Components, and MEMS

Device

1037

Window Window adhesive Ceramic substrate

Die attachment adhesive

–10 °C

25 °C room temperature N=0

Fig. 36.30 Holographic interferometry contours for tilt of gears in a MEMS micromachine [36.61]

36.6.2 Shadow Moiré Shadow moiré has been widely applied to measure relatively large out-of-plane displacements (Chap. 21). In this method, a reference line grating is located in front

100 °C

150 °C

Fig. 36.31 Twyman–Green interferometric warpage con-

tours for bonded silicon chip [36.64]

of the specimen, and white-light illumination is used to cast a shadow of the grating on the sample. The moiré fringe pattern formed by the interference of the reference grating and its shadow represents a contour map of displacements normal to the grating plane. For normal viewing, the out-of-plane deformation fringes satisfy w=

Np , tan α

(36.9)

where N is the fringe order, p is the pitch of the grating, and α is the illumination angle measured from the direction normal to the grating. Typically, α = 45◦ is utilized so that w = N p. Diffraction effects limit the minimum pitch of the ruling that can be used to cast high-contrast shadows, and practical sensitivities for high-quality fringes are in the range of 100–125 μm per fringe. Fractional fringe enhancements can be obtained using phase-shifting or multiplication techniques.

Part D 36.6

deformation was found to be approximately 1.8 μm. Movies of the fringe shifts in time were also recorded. Twyman–Green interferometry is based on the Michelson interferometer, and replaces one of the mirrors in the system with an initially flat section of an electronic package with a mirror-like finish. Because of the mirror surface requirement, applications of Twyman–Green interferometry to measure out-ofplane deformations in packaging have been restricted to measurements on polished silicon chips [36.47, 51, 64]. Figure 36.31 shows fringe patterns of the warping deformations of one quadrant of a silicon chip subjected to temperature change [36.64]. In this application, the chip was glued to a ceramic substrate using a die attachment adhesive. Curing of the adhesive occurred at high temperature, leading to little warpage at 150 ◦ C. However, the deformations of the chip increased dramatically as the temperature decreased, due to the CTE mismatch between the silicon chip and the ceramic substrate.

1038

Part D

Applications

T = 25 °C (RT) Initial fringes

T = 130 °C

T = 25 °C (RT) Final fringes

Fig. 36.32 Shadow moiré deformation contours for a warped HDI substrate [36.67]

Due to its lower sensitivity relative to coherent optical methods (e.g., holographic interferometry), shadow moiré is best suited for measurement of the relatively large warpage deformations in printed circuit boards and other electronic substrates [36.65–67]; for example, Ume and co-workers [36.66] have developed a shadow moiré measurement system for measuring PCB warpage during thermal excursions typical of the electronics assembly process. They have applied their system to a variety of electronic substrates including the high-density interconnect (HDI) board shown in Fig. 36.32 [36.67]. The dimensions of the boards were 75 × 75 × 0.575 mm, and the moiré fringe pattern contour increment was 125 μm. HDI boards are inherently thinner than their conventional counterparts, so they are more susceptible to thermally induced warpage. It is observed that the warpage was negligible at T = 130 ◦ C,

and much larger at room temperature (before and after heating). Additional examples of the use of shadow moiré for measuring warpage in packaging are presented in Chap. 21.

36.6.3 Infrared Fizeau Interferometry Han and co-workers have applied an infrared Fizeau interferometry technique that provides fringe sensitivity for out-of-plane deformations between the submicron fringe sensitivities of holographic interferometry and Twyman–Green interferometry, and the 100 μm fringe sensitivity of shadow moiré [36.68–70]. Using illumination with far-infrared light from a CO2 laser (λ = 10.6 μm), their approach provides a whole-field map of surface topography with a basic measurement sensitivity of 5.31 μm per fringe contour [36.70]. Rel-

12 mm

31 mm

Part D 36.6

T = 22 °C

T = 100 °C

T = 150 °C

Fig. 36.33 Infrared Fizeau interferometry contours for a warped flip-chip BGA component [36.68]

Electronic Packaging Applications

atively rough surfaces can be characterized by the technique due to the longer wavelength of radiation being used for illumination. Since the infrared Fizeau method can be applied to surfaces that are optically rough using visible light, it is well suited for measuring warpage deformations of first-level chip carriers, which deform more than individual chips but less than larger PCBs; for example, Fig. 36.33 illustrates fringe patterns obtained for the top surface warpage of a 31 × 31 mm flip-chip BGA component [36.68]. The Fizeau method was implemented using a computer-controlled environmental chamber for real-time measurement. The package was exposed to temperatures of 22 ◦ C (room temperature),

36.7 Transient Behavior of Electronics During Shock/Drop

1039

and elevated temperatures of 100 ◦ C (operating temperature) and 150 ◦ C (underfill curing temperature). At the 150 ◦ C curing/assembly temperature, the package is nearly flat with few fringes. The large CTE mismatch between the chip and the chip carrier substrate causes the package to warp extensively as the temperature is decreased. When this type of component is used for microprocessor applications, a heat sink is typically attached to the top of the component using a thermal interface material. The warpage deformations present at the operating temperature of processor (≈100 ◦ C) will cause the efficiency of the heat conduction from the die to the heat sink to be reduced.

36.7 Transient Behavior of Electronics During Shock/Drop Cellular phones and other handheld electronic products are often exposed to severe shock/drop loadings. In addition, automotive and military hardware can be subjected to impact loadings on repeated occasions, and high levels of vibration for extended periods of time. Thus, characterization of transient deformations and strains in electronic products has become a key area of interest in the past few years. The most commonly employed experimental methods involve the use of high-speed digital video systems to visualize mode shapes of deformed printed circuit boards and to track the deformations of selected reference points. In addition, full-field transient measurements are now being made using three-dimensional digital image correlation.

36.7.1 Reference Points with High-Speed Video

PCB and reference points (front view)

PCB during drop (side view)

1 Angle 2 3 4 5 6

Fig. 36.34 PCB drop specimen with deformation reference points [36.71]

Part D 36.7

Transient deformations and rotations of electronic assemblies have been measured by recording high-speed digital video of drop events with simultaneous image tracking of surface artifacts such as reference points [36.71]; for example, Fig. 36.34 shows a test board with displacement markers being subjected to a controlled drop. The reference points placed at the edge of the printed circuit assembly serve to track surface artifacts. Optical sampling during the drop event was achieved using a high-speed digital video camera operating at 40 000 frames per second. During the measurements, care was exercised to keep the assembly in the field of focus during the shock impact. One of the locations on the printed circuit assembly was specified as

the reference point, and a reference edge was also chosen for rigid-body rotation measurements. The recorded horizontal displacement versus time responses of selected reference points are shown in Fig. 36.35, as well as the time dependence of the rotation of the line joining these points. It is easily observed that very rapidly changing deformations occur once the impact begins (at approximately t = 10 ms), with several bending curvature changes taking place during the first 20 ms after impact. The presented technique can also be used to ensure repeatability of the drop event (e.g., orientation of the sample and velocity at impact). Repeatability is critical since small variations in the drop orientation can produce vastly different transient dynamic board responses. Figure 36.36 demonstrates excellent repeatability of the relative displacement measurements

1040

Part D

Applications

a) Pos (m)

Fig. 36.35 (a) Deformation versus

Position versus time

0.01

0.008

Point 2

.0006 Point 3

0.004 0.002

time plots during a drop event for selected reference points. (b) PCB angle versus time plot during a drop event [36.71]

Point 4

0 – 0.002

Point 5

– 0.004 – 0.006

0

0.01

0.02

0.03

0.04

Time (s)

b) Angle (deg)

Angle versus time

6 4 2 0

–2 –4 –6

0

0.01

0.02

0.03

0.04

Time (s)

for a single reference point during eight successive shock events. The board-level transient mode shapes from high-speed video recordings were also correlated with explicit finite element model predictions of the transient deformation behavior [36.71, 72], as shown in Fig. 36.37. Excellent correlations were obtained at several time steps. Reliability predictions for the component solder joints can also be made using a submodeling procedure [36.72].

Relative displacement (m) 0.002

0.001

0

–0.001 Drop 1 Drop 2 Drop 3 Drop 4 Drop 5 Drop 6 Drop 7 Drop 8

–0.002

–0.003

–0.004

36.7.2 Strain Gages and Digital Image Correlation

–0.005 0

0.0016

0.0032

0.0047

0.0063

0.0079

0.0095

Part D 36.7

Time (s)

Fig. 36.36 Repeatability of transient deformation measurements for several drop events [36.71]

DIC methods have also recently been applied to measure full-field transient deformations of populated printed circuit board assemblies subjected to shock/drop loading [36.73, 74]. Due to the complicated deformation fields in the PCB and the presence of rigid-body motion, three-dimensional correlation techniques were utilized with stereo high-speed digital video cameras. In the investigation of Lall et al. [36.73], random speckle

Electronic Packaging Applications

t = 2.4 ms von Mises stress (Pa) +6.598e+07 +5.784e+07 +5.302e+07 +4.820e+07 +4.338e+07 +3.856e+07 +3.374e+07 +2.892e+07 +2.410e+07 +1.928e+07 +1.446e+07 +9.639e+06 +4.820e+06 +0.000e+00

t = 4.5 ms

36.8 Mechanical Characterization of Packaging Materials

Fig. 36.37 PCB transient mode shapes from high-speed video and explicit finite element analysis (FEA) simulations [36.71, 72]

a)

Strain (εy) 1140 1000 860 720 580 440 300 160 20 –120 –260 –400 –540 –680 –820 –960 –1100

y

von Mises stress (Pa) +5.784e+07 +5.302e+07 +4.820e+07 +4.338e+07 +3.856e+07 +3.374e+07 +2.892e+07 +2.410e+07 +1.928e+07 +1.446e+07 +9.639e+06 +4.820e+06 +0.000e+00

1041

x

b) εy (microstrain)

PCB strain at selected gage site

3000 2000 1000 0 –1000

t = 6 ms von Mises stress (Pa) +5.784e+07 +5.302e+07 +4.820e+07 +4.338e+07 +3.856e+07 +3.374e+07 +2.892e+07 +2.410e+07 +1.928e+07 +1.446e+07 +9.639e+06 +4.820e+06 +0.000e+00

patterns were pre-applied to the test boards before testing. Figure 36.38 shows example full-field contours of the vertical normal strain superimposed on one of the test boards. The strains were calculated from the recorded time histories of the displacement fields using the strain–displacement relations. The strain data from DIC were also compared to high-speed strain measure-

–2000 –3000

DIC Strain gage

– 4000 –5000 – 0.005

0

0.005

0.01

0.015

0.02 Time (s)

Fig. 36.38 (a) Transient strain contours from processing of DIC displacement data. (b) Correlation of the transient strain response from DIC and strain gages [36.73]

ments at discrete locations in the electronic assembly recorded using strain gages. An example correlation of the transient strain history at one of the gage sites is also given in Fig. 36.38. Good agreement was observed throughout the time history of the drop/shock event, showing the potential of image correlation methods for characterizing high-speed dynamic loading of electronic packaging.

Typical electronic packages contain many materials with poorly understood mechanical behavior including solders, encapsulants, adhesives, thermal interface materials (TIM), and thin films (metallic and ceramic). All of these are applied in the assembly as thin layers

or small interconnections. Hence, their constitutive response and failure properties are significantly different from bulk materials, and small-scale (microscale and nanoscale) test specimens and loading fixtures are required for accurate characterization. Members of the

Part D 36.8

36.8 Mechanical Characterization of Packaging Materials

1042

Part D

Applications

experimental mechanics community are routinely involved in such testing and a variety of specialized methods have been developed. Due to the extent of the literature in this area, a review of all of these methods is beyond the scope of this chapter. The reader is referred to Chap. 30 for an in-depth presentation of me-

chanical testing techniques for thin films and MEMS materials at the microscale and nanoscale. A review of mechanical test methods for solders can be found in the paper by Ma et al. [36.75], while techniques for underfill encapsulants have been reviewed by Islam and co-workers [36.76].

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36.75

36.76

Inspection of Microsystems, ed. by W. Osten (Taylor Francis, New York 2007) pp. 325–350 M. Variyam, B. Han: DMD module calibration using interferometry, TI Tech. J. 18, 1–8 (2001) B. Han, Y. Guo, H.C. Choi: Out-of-plane displacement measurement of printed circuit board by shadow moiré with variable sensitivity, Proc. 1993 ASME International Electronics Packaging Conference, Vol. 4-1 (ASME 1993) pp. 179–185 M.R. Stiteler, I.C. Ume, B. Leutz: In-process board warpage measurement in a lab scale wave soldering oven, IEEE Trans. Compon. Packag. Technol. A 19(4), 562–569 (1996) G.J. Petriccione, I.C. Ume: Warpage studies of HDI test vehicles during various thermal profiling, IEEE Trans. Adv. Packag. 22(4), 624–637 (1999) B. Han: Optical measurement of flip-chip package warpage and its effect on thermal interfaces, Electron. Cool. 9(1), 10–16 (2003) K. Verma, D. Columbus, B. Han: Development of real time/variable sensitivity warpage measurement technique and its application to plastic ball grid array package, IEEE Trans. Electron. Packag. Manuf. 22(1), 63–70 (1999) K. Verma, B. Han: Warpage measurement on dielectric rough surfaces of microelectronics devices by far infrared fizeau interferometry, J. Electron. Packag. 122(3), 227–232 (2000) P. Lall, D. Panchagade, Y. Liu, W. Johnson, J.C. Suhling: Models for reliability prediction of fine-pitch BGAs and CSPs in shock and drop-impact, IEEE Trans. Compon. Packag. Technol. 29(3), 464–474 (2006) P. Lall, S. Gupte, P. Choudhary, J. Suhling: Solderjoint reliability in electronics under shock and vibration using explicit finite element submodeling, IEEE Trans. Electron. Packag. Manuf. 30(1), 74–83 (2007) P. Lall, D. Panchagade, D. Iyengar, S. Shantaram, J. Suhling, H. Schrier: High speed digital image correlation for transient-shock reliability of electronics, Proc. 57th IEEE Electronic Components and Technology Conference (2007) pp. 924–939 S. Park, C. Shah, J. Kwak, C. Jang, J. Pitarresi, T. Park, S. Jang: Transient dynamic simulation and full-field test validation for a slim-PCB of mobile phone under drop/impact, Proc. 57th IEEE Electronic Components and Technology Conference (2007) pp. 914–923 H. Ma, J.C. Suhling, P. Lall, M.J. Bozack: Effects of aging on the stress-strain and creep behaviors of lead free solders, Proc. ITherm 2006 (IEEE 2006) pp. 961–976 M.S. Islam, J.C. Suhling, P. Lall: Measurement of the temperature dependent constitutive behavior of underfill encapsulants, IEEE Trans. Compon. Packag. Technol. 28(3), 467–476 (2005)

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A.6

Active Materials by Guruswami Ravichandran

The author gratefully acknowledges the Army Research Office and the National Science Foundation for their support, which provided the stimulus for his research in the area of active materials. He thanks his colleague Professor K. Bhattacharya for stimulating discussions and continued collaboration. He also thanks his collaborators at Caltech in this area, E. Burcsu, D. Shilo, R. Zhang, C. Franck, and S. Kramer for their contributions to his understanding of the subject. A.7

Biological Soft Tissues by Jay D. Humphrey

I wish to thank Professor W. N. Sharpe (Johns Hopkins University) for inviting this review, for I feel that the experimental mechanics community has much to contribute to the continuing advancement of biomechanics. It is also a pleasure to acknowledge a few of the many agencies that support biomechanics research in general and my study of this fascinating field in particular: the American Heart Association, the National Institutes of Health, the National Science Foundation, the Texas Advanced Technology Program, and the Whitaker Foundation. B.12 Bonded Electrical Resistance Strain Gages by Robert B. Watson

The inestimable technical contributions, editorial comments and suggestions, and encouragement by Dr. C. C. Perry are gratefully acknowledged. Support from Vishay Micro-Measurements with resources and kind permission for use of literature is greatly appreciated. A special thanks is extended to Dr. Felix Zandman for many spirited and helpful discussions concerning the fundamental nature of strain gage performance. An irredeemable debt of gratitude is owed to Dr. Daniel Post for introducing the author to strain gages, and to Mr. Jim Dorsey for mentoring the author in strain gage technology. B.14 Optical Fiber Strain Gages by Chris S. Baldwin

The author would like to thank Omnisens for providing permission to use graphics and information

Acknowl.

Acknowledgements

regarding Brillouin measurement techniques. The author would also like to thank all the scientists and engineers pursuing fiber optic sensing. Since the writing of this chapter, new fiber optic strain measurement techniques have been developed and publicized. Continual improvements and developments of fiber optic sensing techniques will allow for the expanded use of the technology in many application areas in the near future. B.17 Atomic Force Microscopy in Solid Mechanics by Ioannis Chasiotis

The author would like to thank his graduate students who have co-authored the referenced publications, and Mr. Scott Maclaren for providing some AFM micrographs for this Chapter. The support by the Air Force Office of Scientific Research (AFOSR) through grant F49620-03-1-0080 with Dr. B. L. Lee as the program manager, and by the National Science Foundation (NSF) under grant CMS-0515111 is acknowledged for part of the work of this author, which is referenced in this Chapter. C.20 Digital Image Correlation for Shape and Deformation Measurements by Michael A. Sutton

The author would like to thank Dr. Hubert Schreier, Dr. Stephen R. McNeill, Dr. Junhui Yan and Dr. Dorian Garcia for their assistance in completing this manuscript. In addition, the support of (a) Dr. Charles E. Harris, Dr. Robert S. Piascik and Dr. James C. Newman, Jr. at NASA Langley Research Center, (b) Dr. Oscar Dillon, Dr. Clifford Astill, and Dr. Albert S. Kobayashi, former NSF Solid Mechanics and Materials Program Directors, (c) Dr. Julius Dasch at NASA Headquarters, (d) Dr. Bruce LaMattina at the Army Research Office, (e) Dr. Kumar Jatta at the Air Force Research Laboratory, (f) Dr. Kenneth Chong through NSF CMS0201345, and (g) the late Dr. Bruce Fink at the Army Research Laboratory is gratefully acknowledged. Also, the support provided by Correlated Solutions, Incorporated through granting access to their commercial software for our internal use is deeply appreciated. Through the unwavering technical and financial assistance of all these individuals and organizations, the potential of image correlation methods is now being

1046

Acknowledgements

Acknowl.

realized. Finally, the support of my advisor, Prof. Emeritus Charles E. Taylor, and his wife, Nikki Taylor, as well as the support of my wife, Elizabeth Ann Severns, and my children, Michelle Mary Katherine Sutton Spigner and Elizabeth Marie Rosalie Sutton, require special mention, for it is with their continual support over the past three decades that this work has been possible. C.21 Geometric Moiré by Bongtae Han, Daniel Post

We acknowledge and thank Prof. Peter G. Ifju for his contributions to [1.1, 2], and Dr. C.-W. Han for the research published in his Ph. D. Thesis [1.3] and related technical papers [1.13–15].

C.26 Thermoelastic Stress Analysis by Richard J. Greene, Eann A. Patterson, Robert E. Rowlands

The authors wish to thank Ms S. J. Lin and Professor Y. M. Shkel, University of Wisconsin, Madison, WI, B. Boyce and J. Lesniak of Stress Photonics, Inc., Madison, WI, and Dr. S. Quinn, University of Southampton, UK for informative discussions, the US Air Force Research Laboratory, QinetiQ Plc., Rolls-Royce Plc., The University of Sheffield for the release of experimental data, the Society of Experimental Mechanics for permission to reproduce Table 26.1 and Elsevier for permission to reproduce Fig. 26.6. C.28 X-Ray Stress Analysis by Jonathan D. Almer, Robert A. Winholtz

C.24 Holography by Ryszard J. Pryputniewicz

This work was supported by the NEST Program at WPI-ME/CHSLT. The author gratefully acknowledges support from all sponsors and thanks them for their permissions to present the results of their projects in this chapter. C.25 Photoelasticity

The authors wish to gratefully acknowledge the late Professor Jerome B. Cohen and his significant contributions to their experience in this field. They further wish to thank Drs. D. Haeffner and J. Bernier, and Prof. C. Noyan for assistance with the manuscript and helpful discussions. One of the authors (JA) acknowledges support of the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract DE-AC02-06CH11357.

by Krishnamurthi Ramesh

The author wishes to acknowledge Macmillan Publishers (Fig. 25.2), Tata McGraw Hill (Fig. 25.8), Elsevier limited (Fig. 25.13), ASME (Fig. 25.14) and Blackwell publishing (Figs. 25.32a and b) for their consent to reproduce the figures that have been published in their journals/books and Stress photonics for Figs. 25.32c and d from their product brochure. Excerpts of the book Digital photoelasticity - Advanced techniques and applications are included in this chapter with the kind permission of Springer, Berlin. The author also acknowledges IIT Madras for having given permission to use selected animations developed for the e-Book on Experimental Stress Analysis to be provided in the accompanying CD of this Handbook. Part of the results reported in this chapter are obtained from several projects funded by the Structures Panel of Aeronautical Research and Development Board of India while the author was a faculty at IIT Kanpur and the IITM-ISRO cell projects while at IIT Madras. Last but not the least, the author wishes to acknowledge the Society for Experimental Mechanics and Springer for having given permission to reproduce the figures from their books and journals.

D.32 Implantable Biomedical Devices and Biologically Inspired Materials by Hugh Bruck

The writing of this chapter was made possible through a Fulbright Scholar award administered by the US– Israel Educational Foundation, and the National Science Foundation through grant EEC0315425 and the Office of Naval Research award number N000140710391. Contributions were also made by Michael Peterson of the University of Maine, James J. Evans of the University of Reading, Dan Cole of the University of Maryland, Eric Brown of Los Alamos National Laboratory, Jane Grande-Allen of Rice University, Arkady Voloshin of Lehigh University, Krishnaswamy RaviChandar of the University of Texas-Austin, and Debra Wright-Charlesworth of Michigan Technological University. D.35 Structural Testing Applications by Ashok Kumar Ghosh

I would like to thank a team of investigators, William E. Luecke, J. David McColskey, Chris McCowan, Tom Siewert, Stephen Banovic, Tim Foecke, Richard Fields,

Acknowledgements

buquerque, NM for Contributing Case study 3 and sharing their experience during the development of a lightweight automobile airbag from inception through innovation to engineering development. This case study also illustrates the close ties between structural testing and numerical simulation and the importance of engineering economics in the overall development of a marketable product. They have demonstrated the power of simulation. In the absence of standardized test specifications, they formulated their own test procedures and validated with simulated output. I would like to thank the reviewers and proofreaders Dr. Maggie Griffin and Holy Chamberlin. I would like to thank my wife, Pritha, for her understanding, encouragement, and patience during the preparation of this book chapter.

Acknowl.

and Frank Gayle from the National Institute of Standard and Technology (NIST), Washington, DC for contributing case study 1 to this chapter on structural testing. This forensic investigation illustrates how structural testing can be very challenging and how the information generated from these tests can play a crucial role in the overall goal of investigating the sequence of events that caused the fall of the World Trade Center buildings. Any structural failure is a very quick phenomenon where a sequence of events takes place. When a number of loading environment is involved, the problem can be very complex. William and his team have performed a systematic investigation to overcome these challenges. I would like to thank Kenneth W. Gwinn and James M. Nelsen of Sandia National Laboratories, Al-

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About the Authors

Chapter C.28

Argonne National Laboratory Argonne, IL, USA [email protected]

Jonathan Almer received the PhD degree in 1998 from Northwestern University. After a postdoc in Linkoping, Sweden, he joined the Advanced Photon Source at Argonne National Laboratory in 2000, where he is currently working as a physicist. His research interests include materials analysis using wide- and small-angle x-ray scattering, particularly performed in~situ under thermal and mechanical loads.

Archie A.T. Andonian

Chapter D.29

The Goodyear Tire & Rubber Co. Akron, OH, USA [email protected]

Archie A.T. Andonian has received the PhD degree in Engineering Science and Mechanics from Virginia Polytechnic Institute and S.U. in 1978. He joined Goodyear Research in 1984 after teaching for 5 years at the University of Illinois. He is currently a Senior R & D Associate and his research interests are in the general areas of experimental stress analysis, optical methods, fracture mechanics, composite materials, and tire mechanics. He has published more than 250 research papers and has numerous trade secrets proprietary to Goodyear to his name. He teaches a graduatelevel course, accredited by Akron University, at Goodyear Institute of Technology. He is a long-time member of the Society for Experimental Mechanics. He has served the society as Application Committee Chair, Technical Activities Council Chair, Executive Board member, and SEM President for 2007–2008.

David F. Bahr

Chapter B.16

Washington State University Mechanical and Materials Engineering Pullman, WA, USA [email protected]

Dave Bahr received the PhD degree from the University of Minnesota in 1997. He has been at Washington State University since 1997 and is currently a Professor in Mechanical and Materials Engineering. He received the US Presidential Early Career Award for Scientists and Engineers in 2000 and is active in the area of nanomechanical behavior and experimental deign and testing of micro-electromechanical systems (MEMS).

Chris S. Baldwin

Chapter B.14

Aither Engineering, Inc. Lanham, MD, USA [email protected]

Dr. Baldwin currently serves as Technical Director with Aither Engineering, Inc. He has over 10 years of experience developing and integrating fiber-optic sensors and systems for various applications. Projects include fiber-optic-based acoustic sensors, fiber-optic accelerometers, embedded strain sensors for composite materials, multipoint strain and temperature sensor systems, and shape sensing systems.

Stephen M. Belkoff

Chapter D.31

Johns Hopkins University International Center for Orthopaedic Advancement, Department of Orthopaedic Surgery, Bayview Medical Center Baltimore, MD, USA [email protected]

Stephen Belkoff graduated from Michigan State University in Applied Mechanics (Biomechanics) under the mentorship of Roger Haut in 1990. Prior to joining the faculty at Johns Hopkins University, he was at the University of Maryland. He has taught mechanical engineering courses and conducted orthopaedic research with a primary focus on osteoporosis and fracture fixation for almost two decades.

Authors

Jonathan D. Almer

1050

About the Authors

Authors

Hugh Bruck

Chapter D.32

University of Maryland Department of Mechanical Engineering College Park, MD, USA [email protected]

Dr. Bruck received the PhD degree in Materials Science from the California Institute of Technology in 1995. He is the recipient of the ONR Young Investigator Award, the Fulbright Scholar Award, and the A.J. Durelli Innovative Researcher Award from the SEM. He is author or co-author on over 80 technical publications on materials processing and characterization involving digital image correlation, interferometry, scanning probe microscopy, functionally graded materials, bioinspired structures, and smart structures.

Ioannis Chasiotis

Chapter B.17

University of Illinois at Urbana-Champaign Aerospace Engineering Urbana, IL, USA [email protected]

Ioannis Chasiotis received the PhD degree in Aeronautics from the California Institute of Technology in 2002. He is a member of the faculty at the Department of Aerospace Engineering at the University of Illinois at Urbana-Champaign. His research focuses on the experimental deformation and failure mechanics of thin films, micro- and nano-electromechanical systems (MEMS/NEMS), and nanostructured materials. He is a recipient of an NSF CAREER award in 2008, an ONR Young Investigator Award in 2007, a Xerox Award for Faculty Research in 2007, and the Founder’s Prize from the American Academy of Mechanics in 2000.

Gary Cloud

Chapter C.18

Michigan State University Mechanical Engineering Department East Lansing, MI, USA [email protected]

Gary Cloud is University Distinguished Professor of Mechanical Engineering and Director of the Composite Vehicle Research Center at Michigan State University. He is a Registered PE, Chartered Scientist, Chartered Physicist, Fellow of both the SEM and the Institute of Physics, and is recipient of numerous awards for teaching and research. His research involves development and applications of optical techniques in experimental mechanics.

Wendy C. Crone

Chapter A.9

University of Wisconsin Department of Engineering Physics Madison, WI, USA [email protected]

Professor Crone is an accomplished researcher in the area of experimental mechanics, with expertise in improving fundamental understanding of mechanical response of materials, enhancing material behavior through surface modification and nanostructuring, and developing new applications and devices. She is a Fellow of the University of Wisconsin–Madison Teaching Academy and was granted a CAREER Award by the National Science Foundation.

James W. Dally

Chapter A.11

University of Maryland Knoxville, TN, USA [email protected]

James W. Dally obtained the BS and MS degrees, both in Mechanical Engineering, from the Carnegie Institute of Technology. He earned a Doctoral degree in Mechanics from the Illinois Institute of Technology. Currently he is a Glenn L. Martin Professor of Engineering at the University of Maryland, College Park. He is a fellow of the American Society for Mechanical Engineers, Society for Experimental Mechanics, and the American Academy of Mechanics. He was appointed as an honorary member of the Society for Experimental Mechanics in 1983 and elected to the National Academy of Engineering in 1984. Professor Dally has co-authored several textbooks, has written over 200 scientific papers, and holds five patents.

James F. Doyle

Chapter A.10

Purdue University School of Aeronautics & Astronautics West Lafayette, IN, USA [email protected]

Professor James F. Doyle received the PhD degree in Theoretical and Applied Mechanics from the University of Illinois in 1977. He joined Purdue University the same year as an Assistant Professor and is currently a Professor of Aeronautics and Astronautics.

About the Authors

Chapter A.3

University of Ljubljana Center for Experimental Mechanics Lubljana, Slovenia [email protected]

Igor Emri has made major experimental and theoretical contributions to the understanding of the effect of thermomechanical loading on the time-dependent behavior of polymers in the nonequilibrium solid state, and in the process of their solidification. He is a Member of the Russian Academy of Engineering (1996), the Russian Academy of Natural Sciences (1997), the Slovenian Academy of Engineering (1998), the European Academy of Sciences and Arts (2006), and an Associate Member of the Slovenian Academy of Sciences and Arts (2005).

Yimin Gan

Chapter C.23

Universität GH Kassel Fachbereich 15 – Maschinenbau Kassel, Germany [email protected]

Yimin Gan received the BSc degree from Shanghai University of Technology and Science, China, in 1993. From 1993 to 1998 he was an engineer in the field of noise and vibration in the construction of vehicles and Diesel engines at Shanghai Automobile Industry Company (SAIC). In 2002 and 2007 he received the Diploma and PhD degree in Mechanical Engineering from the University of Kassel in Germany, respectively. Currently he is the leader of the Metrology and Developement Department at Vibtec GmbH, Germany. Prior to joining Vibtec GmbH, he was employed in the Laboratory for Photoelasticity, Holography, and Shearography at the Department of Machine Elements and Construction of the University of Kassel.

Ashok Kumar Ghosh

Chapter D.35

New Mexico Tech Mechanical Engineering and Civil Engineering Socorro, NM, USA [email protected]

Dr. Ashok Kumar Ghosh is an Assistant Professor in Mechanical and Civil Engineering at New Mexico Institute of Mining and Technology. His areas of special interest include the macro behavior of composites, biomechanics, finite element analysis, structural health monitoring, restoration construction materials, and project management. He has completed more than 15 industry-sponsored projects of which the World Bank funded 3 projects. He has been awarded two Indian patents. He has published more than 35 research papers.

Richard J. Greene

Chapter C.26

The University of Sheffield Department of Mechanical Engineering Sheffield, UK [email protected]

Dr. Richard John Greene received the PhD degree in experimental mechanics from The University of Sheffield, UK, in 2003 and is currently a lecturer in solid mechanics at the same institution. His professional interests include thermoelastic stress analysis, thermal nondestructive evaluation (NDE), digital image correlation, and photoelastic stress analysis, with particular emphasis on their use in aerospace and biomechanical applications.

Bongtae Han

Chapter C.22

University of Maryland Mechanical Engineering Department College Park, MD, USA [email protected]

Bongtae Han received the PhD degree in Engineering Mechanics from Virginia Tech in 1991. He is currently a Professor of the Mechanical Engineering Department of the University of Maryland at College Park. His research interest is centered on design optimization of microelectronics devices for enhanced mechanical reliability using various experimental techniques for full-field deformation measurements. He is responsible for development of portable moiré systems (PEMI and SM-NT) and holds related patents. He was a recipient of the 2002 SEM Brewer Award for his contributions to experimental characterization of microelectronics devices. He is a Fellow of the SEM and the ASME.

Authors

Igor Emri

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1052

About the Authors

Chapter D.30

Pennsylvania State University Department of Mechanical Engineering University Park, PA, USA [email protected]

Aman Haque received the PhD degree in Mechanical Engineering from the University of Illinois at Urbana-Champaign in 2002. He then joined the Department of Mechanical and Nuclear Engineering at Pennsylvania State University. His research interests are in multiphysics of nanoscale materials and interfaces, nanofabrication, and miniaturization of experimental techniques. He has published over 30 journal papers in these areas.

Authors

M. Amanul Haque

Craig S. Hartley

Chapter A.2

El Arroyo Enterprises LLC Sedona, AZ, USA [email protected]

Dr. Hartley holds degrees in Metallurgical Engineering from Rensselaer Polytechnic Institute and the Ohio State University. After a career in academia and government, he retired to Sedona, AZ, where he continues his professional activities as a consultant in materials research and education. Dr. Hartley is Emeritus Professor of Mechanical Engineering at Florida Atlantic University and Program Manager Emeritus in the Air Force Research Laboratory. He is a Fellow of the American Association for the Advancement of Science, ASM International, and ASME. Dr. Hartley’s principal research areas are in the mechanics and mechanical behavior of metallic materials. He has made contributions in the areas of dislocation theory and the mechanics of metal deformation processing.

Roger C. Haut

Chapter D.31

Michigan State University College of Osteopathic Medicine, Orthopaedic Biomechanics Laboratories East Lansing, MI, USA [email protected]

Roger C. Haut is a University Distinguished Professor at Michigan State University in the Colleges of Engineering and Osteopathic Medicine. He is the Director of the Orthopaedic Biomechanics Laboratories in the College of Osteopathic Medicine. His research in soft-tissue biomechanics deals primarily with the mechanisms of joint trauma, and the development of methods of disease intervention and prevention.

Jay D. Humphrey

Chapter A.7

Texas A&M University Department of Biomedical Engineering College Station, TX, USA [email protected]

Jay D. Humphrey received the PhD degree from The Georgia Institute of Technology in Engineering Science and Mechanics and completed a postdoctoral fellowship in Cardiovascular Science at The Johns Hopkins University. He is currently Professor of Biomedical Engineering at Texas A&M University. He has authored a graduate textbook (Cardiovascular Solid Mechanics), co-authored an undergraduate textbook (An Introduction to Biomechanics

Peter G. Ifju

Chapter A.4

University of Florida Mechanical and Aerospace Engineering Gainesville, FL, USA [email protected]

Dr. Peter Ifju is a Professor in the Department of Mechanical and Aerospace Engineering at the University of Florida (UF). Before arriving at UF in 1993, Dr. Ifju performed a Postdoc at NASA Langley Research Center. He received the PhD degree in Materials Engineering Science from Virginia Polytechnic Institute and State University in 1992, the MS degree in Engineering Science and Mechanics in 1989, and the BS degree in Civil Engineering in 1987. He is an expert in the areas of experimental stress analysis, optical methods (moiré interferometry, luminescent photoelastic coatings), composite materials, and micro air vehicles.

About the Authors

Chapter A.3

California Institute of Technology-GALCIT 105-50 Pasadena, CA, USA [email protected]

Wolfgang G. Knauss, von Kàrmàn Professor of Aeronautics and Applied Mechanics at the California Institute of Technology, has been on the faculty there since 1965. His mostly experimental work is devoted to understanding the mechanics of timedependent behavior of materials and, in particular, to the fracture of polymeric materials so as to enable prediction of the long-term failure of structures made from or incorporating time-dependent materials. Dr. Knauss has also been a Visiting Professor at several distinguished foreign universities and a consultant to many companies and agencies. Dr. Knauss received all his academic degrees, including the PhD, from the California Institute of Technology and is a fellow of the ASME, the Society for Experimental Mechanics, the American Academy of Mechanics, and the Institute for the Advancement of Engineering.

Albert S. Kobayashi

Chapter A.1

University of Washington Department of Mechanical Engineering Seattle, Washington, USA [email protected]

Dr. Albert S. Kobayashi has been Professor Emeritus in the Department of Mechanical Engineering, University of Washington, since June 1997. Dr. Kobayashi received the BE degree in l947 from the University of Tokyo, the MS degree in Mechanical Engineering in l952 from the University of Washington, and the PhD degree in l958 from Illinois Institute of Technology. He is a member of the National Academy of Engineers, a Fellow of the ASME, Honorary Life Member of the Society for Experimental Mechanics, and Member of the American Academy of Mechanics. He was the President of SEM for 1989–1990. His publications, which exceed 500, cover the fields of experimental stress analysis, finite element analysis, and biomechanics in addition to his main interest in fracture mechanics. He was elected to the Mechanical Engineering Hall of Fame of the University of Washington in 2006.

Sridhar Krishnaswamy

Chapter C.27

Northwestern University Center for Quality Engineering & Failure Prevention Evanston, IL, USA [email protected]

Sridhar Krishnaswamy obtained the PhD degree in Aeronautics from the California Institute of Technology in 1989. He has been on the faculty of Mechanical Engineering at Northwestern University since 1990. Professor Krishnaswamy is actively involved in the areas of nondestructive materials characterization, optical metrology, and structural health monitoring where he and his co-workers have developed several photoacoustic methods. He is a Fellow of the ASME and a Member of SPIE. He is currently the Director of the Center for Quality Engineering and Failure Prevention at Northwestern University.

Yuri F. Kudryavtsev

Chapter B.15

Integrity Testing Laboratory Inc. PreStress Engineering Division Markham, Ontario, Canada [email protected]

Dr. Yuri F. Kudryavtsev obtained the MS degree in Mechanical Engineering from the National Technical University (KPI), Kiev, Ukraine in 1977 and the PhD degree from the Paton Welding Institute of the Ukrainian Academy of Sciences in 1984. Dr. Kudryavtsev is a recognized authority in fatigue of welded elements and residual stress analysis. He is a Delegate of Canada for the Commission XIII "Fatigue behavior of welded components and structures" of the International Institute of Welding (IIW) and a member of a number of professional societies.

Pradeep Lall

Chapter D.36

Auburn University Department of Mechanical Engineering Center for Advanced Vehicle Electronics Auburn, AL, USA [email protected]

Pradeep Lall is the Thomas Walter Professor with the Department of Mechanical Engineering and Associate Director of the NSF Center for Advanced Vehicle Electronics at Auburn University. He received the MS and PhD degrees from the University of Maryland and the MBA from Kellogg School of Management. He has published extensively in the area of electronic packaging with emphasis on modeling and predictive techniques.

Authors

Wolfgang G. Knauss

1053

1054

About the Authors

Chapter D.34

University of Texas Aerospace Engineering and Engineering Mechanics Austin, TX, USA [email protected]

Kenneth M. Liechti is a Professor of Aerospace Engineering and Engineering Mechanics at the University of Texas at Austin, where he has been since 1982. Dr. Liechti’s research deals with the mechanics of adhesion and friction over a range of length and time scales. Applications range from primary structural adhesive joints and composite materials to microelectronics devices and micro-electromechanical systems (MEMS). He is coauthor with Marc Bedford of the book Mechanics of Materials. He is a Fellow of the Society for Experimental Mechanics, the American Society of Mechanical Engineers, the Adhesion Society, and the American Academy of Mechanics.

Authors

Kenneth M. Liechti

Hongbing Lu

Chapter A.3

Oklahoma State University School of Mechanical and Aerospace Engineering Stillwater, OK, USA [email protected]

Dr. Hongbing Lu is a Professor of Mechanical and Aerospace Engineering at Oklahoma State University. He received the MS degree in Engineering Mechanics from Tsinghua University, China in 1988, the MS degree in Solid Mechanics at Huazhong University of Science and Technology in 1986, and the PhD degree in Aeronautics from Caltech in 1997. He is recepient of the NSF Career award in 2000. His research is primarily on the mechanics of time-dependent materials, including such materials as polymers, biomaterials, and porous nanostructured crosslinked aerogels.

Ian McEnteggart

Chapter B.13

Instron Buckinghamshire, UK

Ian McEnteggart has a physics degree from Birmingham University and works for Instron in the UK. He has been involved in developing both contacting and noncontacting extensometers for use in materials testing. He is currently responsible for electromechanical testing machine operations in Europe and is actively involved with the development of international standards for materials testing and extensometer calibration.

Dylan J. Morris

Chapter B.16

National Institute of Standards and Technology Materials Science and Engineering Laboratory Gaithersburg, MD, USA [email protected]

Dylan Morris received the PhD degree from the University of Minnesota in 2004. He was at Washington State University from 2004 to 2007 working in nanomechanics and MEMS technologies. Dylan is now Project Leader for Nanoindentation Measurements and Standards in the Materials Science and Engineering Laboratory at the National Institute of Standards and Technology.

Sia Nemat-Nasser

Chapter A.8

University of California Department of Mechanical and Aerospace Engineering La Jolla, CA, USA [email protected]

Sia Nemat-Nasser is a Distinguished Professor of Mechanics of Materials and Director of the Center of Excellence for Advanced Materials at the University of California (UC) San Diego. He is Founding Director of UC San Diego’s Materials Science and Engineering Graduate Program and is recipient of numerous Awards and Medals. He is a Member of NAE, an Honorary Member of World Innovation Foundation, and an Honorary Member of ASME. In 2008 the ASME Materials Division established ’The Sia Nemat-Nasser Early Career Medal’ to recognize research excellence by young investigators. Dr. Nemat-Nasser’s current research is on multifunctional composites with tunable electromagnetic functionality, thermal management, self-healing, and self-sensing; polyelectrolytes composites as soft actuators/sensors; shape-memory alloys; advanced metals; ceramics; elastomers; and granular materials.

About the Authors

Chapter C.19

Universität Stuttgart Institut für Technische Optik Stuttgart, Germany [email protected]

Wolfgang Osten received the BS degree from the University of Jena in 1979 and the PhD degree from the Martin-Luther-University Halle– Wittenberg in 1983. In 1991 he joined the Bremen Institute of Applied Beam Technology (BIAS) where he established and directed the Department of Optical 3-D Metrology. Since September 2002 he has been a Full Professor at the University of Stuttgart and Director of the Institute for Applied Optics. His research is focused on new concepts for industrial inspection and metrology by combining modern principles of optical metrology, sensor technology. and image processing. Special attention is paid to the development of resolution-enhanced technologies for the investigation of micro- and nanostructures.

Eann A. Patterson

Chapter C.26

Michigan State University Department of Mechanical Engineering East Lansing, MI, USA [email protected]

Dr. Patterson’s research interests include computational biomechanics, experimental fracture mechanics, and the application of experimental mechanics in the aerospace industry. He was elected a Fellow of the SEM in 2007 and is a Fellow of the Institution of Mechanical Engineers, London. He is Chair of Mechanical Engineering at Michigan State University.

Daniel Post

Chapter C.22

Virginia Polytechnic Institute and State University (Virginia Tech) Department of Engineering Science and Mechanics Blacksburg, VA, USA [email protected]

Daniel Post received the PhD degree in Theoretical and Applied Mechanics in 1957 from the University of Illinois. He served in government, industry, and academia prior to retirement from Virginia Polytechnic Institute and State University. He was instrumental in the development of experimental techniques of stress and strain analysis throughout his career, and their extensive use in solid mechanics and materials science. His developments spanned the fields of electrical strain gages, optical interferometry, photoelasticity, holography, moiré, and moiré interferometry. He completed a comprehensive book (High Sensitivity Moiré) together with coauthors B. Han and P.G. Ifju.

Ryszard J. Pryputniewicz

Chapter C.24

Worcester Polytechnic Institute NEST – NanoEngineering, Science, and Technology CHSLT – Center for Holographic Studies and Laser Micro-Mechatronics Worcester, MA, USA [email protected]

Ryszard J. (Rich) Pryputniewicz, educated both in Poland and in the USA, is the K.G. Merriam Professor of Mechanical Engineering as well as Professor of Electrical and Computer Engineering, and, since 1978, founding Director of the Center for Holographic Studies and Laser micro-mechaTronics (CHSLT) at Worcester Polytechnic Institute (WPI) in Worcester, MA. In his work, he emphasizes unification of analytical, computational, and experimental solutions (ACES) methodologies. He is a Registered Professional Engineer (PE), Fellow of the SPIE and the SEM, and Chairman of the Education Committee of the IEEE Nanotechnology Council. He has over 350 publications to his name and has organized over 100 conferences, symposia, and workshops. Rich was appointed as a professor in several countries in Europe and Asia and has received numerous awards including the 2002 ASME International Award and the 2004 Sigma Xi Senior Faculty Research Award.

Authors

Wolfgang Osten

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1056

About the Authors

Chapter D.33

Johns Hopkins University Department of Mechanical Engineering Baltimore, MD, USA [email protected]

Professor K.T. Ramesh’s research interests include nanostructured materials, high-strain-rate behavior, dynamic failure of materials, biomechanics, and planetary impact. He received his doctorate from Brown University in 1987 and did his postdoctoral fellowship at University of California San Diego. He joined the Mechanical Engineering Faculty at the Johns Hopkins University in 1988, was Department Chair from 1999–2002, and became the Director of the Center for Advanced Metallic and Ceramic Systems in 2001.

Authors

Kaliat T. Ramesh

Krishnamurthi Ramesh

Chapter C.25

Indian Institute of Technology Madras Department of Applied Mechanics Chennai, India [email protected]

Professor Ramesh is currently heading the Department of Applied Mechanics at IIT Madras. His areas of interest are digital photoelasticity, fracture mechanics, and educational technology. He has developed an e-book on Engineering Fracture Mechanics published by IIT Madras, which mimics the classroom environment. He is a Fellow of the Indian National Academy of Engineering and a member of several National and International professional societies such as the SEM, BSSM, OSI, and ISTAM.

Krishnaswamy Ravi-Chandar

Chapter A.5

University of Texas at Austin Austin, TX, USA [email protected]

Dr. Ravi-Chandar is interested in the characterization of deformation and failure of brittle and ductile materials subjected to extreme loads in short durations. The main objectives of this work are to develop an understanding of the mechanisms of deformations and to develop quantitative models for use in engineering design.

Guruswami Ravichandran

Chapter A.6

California Institute of Technology Graduate Aeronautical Laboratories Pasadena, CA, USA [email protected]

Guruswami Ravichandran is the John E. Goode, Jr. Professor of Aeronautics and Mechanical Engineering at the California Institute of Technology. He received the PhD degree in Solid Mechanics and Structures from Brown University. He is a Fellow of the ASME and the recipient of a Presidential Young Investigator award from the NSF and B.J. Lazan award from the SEM.

Robert E. Rowlands

Chapter C.26

University of Wisconsin Department of Mechanical Engineering Madison, WI, USA [email protected]

Professor Rowlands received the BASc degree in Mechanical Engineering from the University of British Columbia, Vancouver, Canada and the PhD (1967) degree in Theoretical and Applied Mechanics from the University of Illinois, Urbana, IL. He was affiliated with the IIT Research Institute, Chicago, IL from 1967 to 1974 and has been at the University of Wisconsin, Madison, WI since 1974. He has one patent and over 100 publications in experimental mechanics. He is a Fellow of ASME and the Society for Experimental Mechanics. His honors include the Hetenyi (1970, 1976) and Frocht (1987) Awards of the Society for Experimental Mechanics (1982). He is a Registered Professional Engineer and consults to industry, including as a legal expert witness.

About the Authors

Chapter D.30

University of Illinois at Urbana-Champaign Micro and Nanotechnology Laboratory, 2101D Mechanical Engineering Laboratory Urbana, IL, USA [email protected]

Taher Saif received the BS and MS degrees in Civil Engineering from Bangladesh University of Engineering and Technology and Washington State University, respectively, in 1984 and 1986. He obtained the PhD degree in Theoretical and Applied Mechanics from Cornell University in 1993. Currently, he is a Professor in the Department of Mechanical Science and Engineering at the University of Illinois at Urbana-Champaign. His current research includes mechanosensitivity of single living cells, and electro-thermo-mechanical behavior of nanograined metals.

Jeffrey C. Suhling

Chapter D.36

Auburn University Department of Mechanical Engineering Auburn, AL, USA [email protected]

Jeffrey C. Suhling is the Quina Distinguished Professor with the Department of Mechanical Engineering at Auburn University, where he also serves as Director of the NSF Center for Advanced Vehicle Electronics (CAVE). He received the PhD degree in Engineering Mechanics from the University of Wisconsin-Madison. His research concerns studies of reliability, mechanics, and materials issues for modern electronic packaging.

Michael A. Sutton

Chapter C.20

University of South Carolina Center for Mechanics, Materials and NDE Department of Mechanical Engineering Columbia, SC, USA [email protected]

Michael A. Sutton is a Carolina Distinguished Professor in the Department of Mechanical Engineering at the University of South Carolina. A Fellow of both the Society for Experimental Mechanics and the American Society for Mechanical Engineers and a past President of SEM, Prof. Sutton has received numerous honors for both his computer vision developments and applications in solid mechanics and his contributions in ductile fracture mechanics. His current areas of research are experimental and analytical fracture mechanics, 2-D and 3-D computer vision, and numerical methods, with recent emphasis on noncontacting measurements in biological materials.

Robert B. Watson

Chapter B.12

Vishay Micro-Measurements Sensors Engineering Department Raleigh, NC, USA [email protected]

Robert Watson received the BS degree in Engineering Science and Mechanics from Virginia Polytechnic Institute in 1980. He then joined the engineering department at Vishay Micro-Measurements, where he presently serves as Senior Manager in the Sensors Engineering R&D. His professional career has been devoted to the development, production, and understanding of electrical resistance strain gages. He has served as Chairman of the SEM Technical Committee on Strain Gages and of ASTM Subcommittee E28.14 on Strain Gages.

Robert A. Winholtz

Chapter C.28

University of Missouri Department of Mechanical and Aerospace Engineering Columbia, MO, USA [email protected]

Robert "Andy" Winholtz received the PhD degree in 1991 from Northwestern University. He joined the University of Missouri in 1991 where he is currently an Associate Professor in Mechanical and Aerospace Engineering and a Senior Research Scientist at the Research Reactor Center (MURR). Research interests include the use of neutrons and x-rays to study materials.

Authors

Taher Saif

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List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXVII

Part A Solid Mechanics Topics 3 4 4 6 7 9 11 11 12 12 12 13 13 14 14

2 Materials Science for the Experimental Mechanist Craig S. Hartley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Structure of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Atomic Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Classification of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Atomic Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Equilibrium and Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Observation and Characterization of Structure . . . . . . . . . . . . . . 2.2 Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Continuum Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Equilibrium Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Dissipative Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Transport Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Measurement Principles for Material Properties . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 21 22 28 31 33 34 35 38 43 46 47

3 Mechanics of Polymers: Viscoelasticity Wolfgang G. Knauss, Igor Emri, Hongbing Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The Building Blocks of the Theory of Viscoelasticity . . . . . . . .

49 49 50

Detailed Cont.

1 Analytical Mechanics of Solids Albert S. Kobayashi, Satya N. Atluri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Elementary Theories of Material Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Viscoplasticity and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Boundary Value Problems in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Basic Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Plane Theory of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Basic Field Equations for the State of Plane Strain . . . . . . . . . 1.2.4 Basic Field Equations for the State of Plane Stress . . . . . . . . . 1.2.5 Infinite Plate with a Circular Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Point Load on a Semi-Infinite Plate . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2

Detailed Cont.

Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 A Simple Linear Concept: Response to a Step-Function Input . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Specific Constitutive Responses (Isotropic Solids) . . . . . . . . . . . 3.2.3 Mathematical Representation of the Relaxation and Creep Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 General Constitutive Law for Linear and Isotropic Solid: Poisson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Spectral and Functional Representations . . . . . . . . . . . . . . . . . . . . 3.2.6 Special Stress or Strain Histories Related to Material Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7 Dissipation Under Cyclical Deformation . . . . . . . . . . . . . . . . . . . . . . 3.2.8 Temperature Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.9 The Effect of Pressure on Viscoelastic Behavior of Rubbery Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.10 The Effect of Moisture and Solvents on Viscoelastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Measurements and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Laboratory Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Volumetric (Bulk) Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 The CEM Measuring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Nano/Microindentation for Measurements of Viscoelastic Properties of Small Amounts of Material . . . . 3.3.5 Photoviscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Nonlinearly Viscoelastic Material Characterization . . . . . . . . . . . . . . . . . . . . 3.4.1 Visual Assessment of Nonlinear Behavior . . . . . . . . . . . . . . . . . . . 3.4.2 Characterization of Nonlinearly Viscoelastic Behavior Under Biaxial Stress States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known . . 3.6.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Composite Materials Peter G. Ifju . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Strain Gage Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Transverse Sensitivity Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Error Due to Gage Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Temperature Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Self-Heating Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Additional Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Material Property Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Tension Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Compression Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Shear Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Single-Geometry Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 53 53 55 55 56 63 63 68 69 69 70 71 74 76 84 84 85 86 89 90 91 92

97 98 98 99 100 101 101 102 103 103 105 106

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4.3

5 Fracture Mechanics Krishnaswamy Ravi-Chandar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fracture Mechanics Based on Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Linearly Elastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Asymptotic Analysis of the Elastic Crack Tip Stress Field . . . . 5.2.2 Irwin’s Plastic Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Relationship Between Stress Analysis and Energy Balance – The J-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Fracture Criterion in LEFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Elastic–Plastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Dugdale–Barenblatt Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Elastic–Plastic Crack Tip Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Fracture Criterion in EPFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 General Cohesive Zone Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Damage Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Dynamic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Dynamic Crack Initiation Toughness . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Dynamic Crack Growth Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Dynamic Crack Arrest Toughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Subcritical Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Lateral Shearing Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Strain Gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 108 109 111 111 111 113 114 114 114 116 117 117 118 118 119 120 120 121 121

125 126 128 128 129 130 131 132 132 134 135 136 136 137 138 139 139 140 140 141 143 147 151

Detailed Cont.

Micromechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 In Situ Strain Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Fiber–Matrix Interface Characterization . . . . . . . . . . . . . . . . . . . . . 4.3.3 Nanoscale Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Self-Healing Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Interlaminar Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Mode I Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Mode II Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Edge Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Textile Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Documentation of Surface Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Strain Gage Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Edge Effects in Textile Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Residual Stresses in Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Composite Sectioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Hole-Drilling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Strain Gage Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Laminate Warpage Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.5 The Cure Reference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Future Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.6.5 Method of Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.6 Measurement of Crack Opening Displacement . . . . . . . . . . . . . . 5.6.7 Measurement of Crack Position and Speed . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Detailed Cont.

6 Active Materials Guruswami Ravichandran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Mechanisms of Active Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Mechanics in the Analysis, Design, and Testing of Active Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Piezoelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Ferroelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Electrostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Domain Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152 153 155 156

159 159 160 160 161 162 162 163 163 165 166 166 167 167

7 Biological Soft Tissues Jay D. Humphrey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Constitutive Formulations – Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Traditional Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Poroelasticity and Mixture Descriptions . . . . . . . . . . . . . . . . . . . . . 7.2.4 Muscle Activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Thermomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Growth and Remodeling – A New Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Early Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Kinematic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Constrained Mixture Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 171 172 172 176 176 177 177 178 178 179 180 182 182 183

8 Electrochemomechanics of Ionic Polymer–Metal Composites Sia Nemat-Nasser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Microstructure and Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Cluster Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 188 188 189 191

Detailed Contents

8.2

191 191 192 192 192 193 194 195 195 196 197 197 197 198 199 199

9 A Brief Introduction to MEMS and NEMS Wendy C. Crone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 MEMS/NEMS Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Common MEMS/NEMS Materials and Their Properties . . . . . . . . . . . . . . . . . . 9.3.1 Silicon-Based Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Other Hard Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Polymeric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Active Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.6 Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.7 Micromachining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.8 Hard Fabrication Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.9 Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.10 Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.11 Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Bulk Micromachining versus Surface Micromachining . . . . . . . . . . . . . . . . 9.5 Wafer Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Soft Fabrication Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Other NEMS Fabrication Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Experimental Mechanics Applied to MEMS/NEMS . . . . . . . . . . . . . . . . . . . . . . 9.8 The Influence of Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 Basic Device Characterization Techniques . . . . . . . . . . . . . . . . . . . 9.8.2 Residual Stresses in Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.3 Wafer Bond Integrity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.4 Adhesion and Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Mechanics Issues in MEMS/NEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.1 Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 206 206 207 208 208 208 209 209 210 211 211 211 212 213 214 215 215 216 217 217 218 219 220 220 221 221

Detailed Cont.

Stiffness Versus Solvation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The Stress Field in the Backbone Polymer . . . . . . . . . . . . . . . . . . . 8.2.2 Pressure in Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Membrane Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 IPMC Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Voltage-Induced Cation Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Equilibrium Cation Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Nanomechanics of Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Cluster Pressure Change Due to Cation Migration . . . . . . . . . . . 8.4.2 Cluster Solvent Uptake Due to Cation Migration . . . . . . . . . . . . . 8.4.3 Voltage-Induced Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Evaluation of Basic Physical Properties . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Potential Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1063

1064

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Detailed Cont.

9.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

224 225

10 Hybrid Methods James F. Doyle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Basic Theory of Inverse Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Partially Specified Problems and Experimental Mechanics 10.1.2 Origin of Ill-Conditioning in Inverse Problems . . . . . . . . . . . . . . 10.1.3 Minimizing Principle with Regularization . . . . . . . . . . . . . . . . . . . 10.2 Parameter Identification Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Sensitivity Response Method (SRM) . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Experimental Data Study I: Measuring Dynamic Properties 10.2.3 Experimental Data Study II: Measuring Effective BCs . . . . . . . 10.2.4 Synthetic Data Study I: Dynamic Crack Propagation . . . . . . . . 10.3 Force Identification Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Sensitivity Response Method for Static Problems . . . . . . . . . . . 10.3.2 Generalization for Transient Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Experimental Data Study I: Double-Exposure Holography . 10.3.4 Experimental Data Study II: One-Sided Hopkinson Bar . . . . 10.4 Some Nonlinear Force Identification Problems . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Nonlinear Data Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Nonlinear Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Nonlinear Space–Time Deconvolution . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Experimental Data Study I: Stress Analysis Around a Hole 10.4.5 Experimental Data Study II: Photoelastic Analysis of Cracks 10.4.6 Synthetic Data Study I: Elastic–Plastic Projectile Impact . . . 10.4.7 Synthetic Data Study II: Multiple Loads on a Truss Structure 10.4.8 Experimental Data Study III: Dynamic Photoelasticity . . . . . . 10.5 Discussion of Parameterizing the Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Parameterized Loadings and Subdomains . . . . . . . . . . . . . . . . . . 10.5.2 Unknowns Parameterized Through a Second Model . . . . . . . . 10.5.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 231 231 232 234 235 235 237 239 240 240 241 242 243 244 246 246 247 248 250 251 252 253 254 255 255 256 257 257

11 Statistical Analysis of Experimental Data James W. Dally . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Characterizing Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Graphical Representations of the Distribution . . . . . . . . . . . . . . 11.1.2 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Statistical Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Confidence Intervals for Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Comparison of Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Statistical Safety Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Statistical Conditioning of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259 260 260 261 262 263 263 265 267 270 271 272

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11.7

Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1 Linear Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 Multivariate Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.3 Field Applications of Least-Square Methods . . . . . . . . . . . . . . . . 11.8 Chi-Square Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1065

272 272 274 275 277 278 279

Part B Contact Methods 283 284 284 285 285 286 287 291 291 292 294 295 296 297 302 303 306 310 312 325 325 327 328 331 332 332

13 Extensometers Ian McEnteggart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 General Characteristics of Extensometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 Temperature Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.5 Operating Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 336 336 336 336 336 337

Detailed Cont.

12 Bonded Electrical Resistance Strain Gages Robert B. Watson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Standardized Strain-Gage Test Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Strain and Its Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Strain-Gage Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Elementary Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 The Potentiometer Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 The Wheatstone Bridge Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 The Bonded Foil Strain Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Gage Factor – A Practical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Strains from Every Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 Gage Factor – The Manufacturer’s Value . . . . . . . . . . . . . . . . . . . . 12.4.5 Transverse Sensitivity Error – Numerical Examples . . . . . . . . . 12.4.6 The Influence of Temperature Changes . . . . . . . . . . . . . . . . . . . . . . 12.4.7 Control of Foil Strain Gage Thermal Output . . . . . . . . . . . . . . . . . . 12.4.8 Foil Strain Gages and the Wheatstone Bridge . . . . . . . . . . . . . . . 12.4.9 Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.10 Gage Selection Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.11 Specific Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Semiconductor Strain Gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Strain Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.3 Temperature Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.4 Special Circuit Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.5 Installation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Detailed Cont.

13.1.6 Contact Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.7 Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.8 Response Time/Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.9 Kinematics and Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Transducer Types and Signal Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Strain-Gaged Flexures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 LVDTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.3 Potentiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 Capacitance Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.5 Linear Incremental Encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.6 Electronics and Signal Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Ambient-Temperature Contacting Extensometers . . . . . . . . . . . . . . . . . . . . . 13.3.1 Clip-On Axial Extensometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Other Types of Clip-On Extensometers . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Long-Travel (Elastomeric) Extensometers . . . . . . . . . . . . . . . . . . . . 13.3.4 Automatic Extensometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 High-Temperature Contacting Extensometers . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Longitudinal-Type High-Temperature Extensometers . . . . . . 13.4.2 Side-Loading High-Temperature Extensometers . . . . . . . . . . . 13.5 Noncontact Extensometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 Optical Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Servo Follower Type Optical Extensometers . . . . . . . . . . . . . . . . . 13.5.3 Scanning Laser Extensometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.4 Video Extensometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.5 Other Noncontact Optical Extensometers . . . . . . . . . . . . . . . . . . . . 13.5.6 Noncontact Extensometers for High-Temperature Testing . 13.6 Contacting versus Noncontacting Extensometers . . . . . . . . . . . . . . . . . . . . . . 13.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 337 337 337 337 337 337 338 338 338 338 338 338 340 340 341 341 341 341 343 343 344 344 344 344 345 345 346 346

14 Optical Fiber Strain Gages Chris S. Baldwin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Optical Fiber Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Guiding Principals for Optical Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Types of Optical Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 General Fiber Optic Sensing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Strain Sensing System Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Basic Fiber Optic Sensing Definitions . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Advantages of Fiber Optic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Limitations of Fiber Optic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.5 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.6 Introduction to Strain–Optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Two-Beam Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Strain–Optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Optical Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 Mach–Zehnder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 348 348 349 351 351 351 352 353 354 354 354 355 355 356 357

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15 Residual Stress Yuri F. Kudryavtsev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Importance of Residual Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Definition of Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2 Origin of Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 Residual Stress Management: Measurement, Fatigue Analysis, and Beneficial Redistribution . . . . . . . . . . . . 15.2 Residual Stress Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Destructive Techniques for Residual Stress Measurement . . 15.2.2 Nondestructive Techniques for Residual Stress Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Ultrasonic Method for Residual Stress Measurement . . . . . . . 15.3 Residual Stress in Fatigue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Residual Stress Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357 357 358 358 359 359 359 361 361 362 363 364 366 367 367 367 367 368 368 368 369

371 371 372 372 373 373 373 375 377 381 383 386 386

16 Nanoindentation: Localized Probes of Mechanical Behavior

of Materials David F. Bahr, Dylan J. Morris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Hardness Testing: Macroscopic Beginnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Spherical Impression Tests: Brinell and Meyers . . . . . . . . . . . . . 16.1.2 Measurements of Depth to Extract Rockwell Hardness . . . . . 16.1.3 Pyramidal Geometries for Smaller Scales: Vickers Hardness

389 389 390 390 391

Detailed Cont.

14.3.5 Michelson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.6 Fabry–Pérot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.7 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.8 Interrogation of Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Brillouin Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Strain Sensing Using Brillouin Scattering . . . . . . . . . . . . . . . . . . . . 14.5 Fiber Bragg Grating Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1 Fabrication Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Fiber Bragg Grating Optical Response . . . . . . . . . . . . . . . . . . . . . . . . 14.5.3 Strain Sensing Using FBG Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.4 Serial Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.5 Interrogation of FBG Sensors, Wavelength Detection . . . . . . . 14.5.6 Other Grating Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Applications of Fiber Optic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Marine Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 Oil and Gas Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.3 Wind Power Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.4 Civil Structural Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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16.2

Detailed Cont.

Extraction of Basic Materials Properties from Instrumented Indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 General Behavior of Depth Sensing Indentation . . . . . . . . . . . . 16.2.2 Area Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3 Assessment of Properties During the Entire Loading Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Plastic Deformation at Indentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 The Spherical Cavity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Analysis of Slip Around Indentations . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Measurement of Fracture Using Indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Fracture Around Vickers Impressions . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2 Fracture Observations During Instrumented Indentation . . 16.5 Probing Small Volumes to Determine Fundamental Deformation Mechanisms . . . . . . . . . . . . . . . . 16.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Atomic Force Microscopy in Solid Mechanics Ioannis Chasiotis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Tip–Sample Force Interactions in Scanning Force Microscopy . . . . . . . . 17.2 Instrumentation for Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 AFM Cantilever and Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Calibration of Cantilever Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.3 Tip Imaging Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4 Piezoelectric Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.5 PZT Actuator Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Imaging Modes by an Atomic Force Microscope . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Contact AFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Non-Contact and Intermittent Contact AFM . . . . . . . . . . . . . . . . . 17.3.3 Phase Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.4 Atomic Resolution by an AFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Quantitative Measurements in Solid Mechanics with an AFM . . . . . . . . 17.4.1 Force-Displacement Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Full Field Strain Measurements by an AFM . . . . . . . . . . . . . . . . . . 17.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

392 392 394 395 396 397 398 399 399 400 402 404 404

409 411 412 414 415 417 419 420 423 423 425 430 431 432 432 434 438 439 440

Part C Noncontact Methods 18 Basics of Optics Gary Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Nature and Description of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.1 What Is Light? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.2 How Is Light Described? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.3 The Quantum Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447 448 448 448 448

Detailed Contents

448 449 449 449 449 450 450 451 451 451 452 452 452 453 453 453 453 453 454 455 455 456 458 459 461 462 464 464 465 466 467 467 467 468 468 468 470 470 470 471 471 472 472 472 474

Detailed Cont.

18.1.4 Electromagnetic Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.5 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.7 The Harmonic Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Interference of Light Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 The Problem and the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Collinear Interference of Two Waves . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Path Length and the Generic Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Index of Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Optical Path Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Path Length Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.4 A Generic Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.5 A Few Important Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.6 Whole-Field Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.7 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Oblique Interference and Fringe Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.1 Oblique Interference of Two Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Fringes, Fringe Orders, and Fringe Patterns . . . . . . . . . . . . . . . . . 18.5 Classical Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.1 Lloyd’s Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.2 Newton’s Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.3 Young’s Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5.4 Michelson Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Colored Interferometry Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6.1 A Thought Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Optical Doppler Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.1 The Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.2 Theory of the Doppler Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . 18.7.3 Measurement of Doppler Frequency Shift . . . . . . . . . . . . . . . . . . . 18.7.4 The Moving Fringe Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.5 Bias Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.6 Doppler Shift for Reflected Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7.7 Application Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.8 The Diffraction Problem and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.8.1 Examples of Diffraction of Light Waves . . . . . . . . . . . . . . . . . . . . . . 18.8.2 The Diffraction Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.8.3 History of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9 Complex Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9.1 Wave Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9.2 Scalar Complex Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9.3 Intensity or Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10 Fraunhofer Solution of the Diffraction Problem . . . . . . . . . . . . . . . . . . . . . . . 18.10.1 Solution of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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18.11 Diffraction at a Clear Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.11.1 Problem and Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.11.2 Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.11.3 Numerical Examples and Observations . . . . . . . . . . . . . . . . . . . . . . 18.12 Fourier Optical Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.12.1 The Transform Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.12.2 Optical Fourier Processing or Spatial Filtering . . . . . . . . . . . . . . . 18.12.3 Illustrative Thought Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.12.4 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.13 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Detailed Cont.

19 Digital Image Processing for Optical Metrology Wolfgang Osten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 Basics of Digital Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.1 Components and Processing Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.2 Basic Methods of Digital Image Processing . . . . . . . . . . . . . . . . . . 19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Intensity Models in Optical Metrology . . . . . . . . . . . . . . . . . . . . . . . 19.2.2 Modeling of the Image Formation Process in Holographic Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.3 Computer Simulation of Holographic Fringe Patterns . . . . . . 19.2.4 Techniques for the Digital Reconstruction of Phase Distributions from Fringe Patterns . . . . . . . . . . . . . . . . . 19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.1 The Technology of Optical Nondestructive Testing (ONDT) . . 19.3.2 Direct and Indirect Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.3 Fault Detection in HNDT Using an Active Recognition Strategy . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

474 474 475 476 476 476 477 478 478 479 479

481 483 483 484 485 485 487 491 492 545 547 549 551 557

20 Digital Image Correlation for Shape

and Deformation Measurements Michael A. Sutton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1

20.2 20.3

20.4

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.1 Two-Dimensional Digital Image Correlation (2-D DIC) . . . . . . 20.1.2 Three-Dimensional Digital Image Correlation (3-D DIC) . . . . Essential Concepts in Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . Pinhole Projection Imaging Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Image Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2 Camera Calibration for Pinhole Model Parameter Estimation . . . . . . . . . . . . . . . . . . . . 20.3.3 Image-Based Objective Function for Camera Calibration . . Image Digitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

565 566 567 568 568 569 571 572 572 573 573

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20.5

573 575 575 577 577 579 579 580 580 580 581 581 582 583 583 584 585 585 586 587 587 587 588 589 589 590 591 592 593 593 594 594 595 596 597 597 599

Detailed Cont.

Intensity Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Subset-Based Image Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Pattern Development and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 Two-Dimensional Image Correlation (2-D DIC) . . . . . . . . . . . . . . . . . . . . . . . . . 20.8.1 2-D Camera Calibration with Image-Based Optimization . . 20.8.2 Object Displacement and Strain Measurements . . . . . . . . . . . . . 20.8.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9 Three-Dimensional Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9.1 Camera Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9.2 Image Correlation for Cross-Camera Matching . . . . . . . . . . . . . . 20.9.3 3-D Position Measurement with Unconstrained Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9.4 3-D Position Measurement with Constrained Cross-Camera Image Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.10 Two-Dimensional Application: Heterogeneous Material Property Measurements . . . . . . . . . . . . . . . . . . . . . . 20.10.1 Experimental Setup and Specimen Geometry . . . . . . . . . . . . . . . 20.10.2 Single-Camera Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.10.3 Camera Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.10.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.10.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.11 Three-Dimensional Application: Tension Torsion Loading of Flawed Specimen . . . . . . . . . . . . . . . . . . . . . . . . . 20.11.1 Experimental Setup with 3-D Imaging System . . . . . . . . . . . . . . 20.11.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.11.3 Calibration of Camera System for Tension–Torsion Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.11.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.11.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12 Three-Dimensional Measurements – Impact Tension Torsion Loading of Single-Edge-Cracked Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.2 3-D High-Speed Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.3 Camera Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.12.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.13 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.14 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Detailed Cont.

21 Geometric Moiré Bongtae Han, Daniel Post . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Basic Features of Moiré . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Gratings, Fringes, and Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Intensity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Multiplicative and Additive Intensities . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Moiré Fringes as Parametric Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 In-Plane Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Fringe Formation by Pure Rotation and Extension . . . . . . . . . 21.2.2 Physical Concepts: Absolute Displacements . . . . . . . . . . . . . . . . . 21.2.3 Experimental Demonstration: Relative Displacements . . . . . 21.2.4 Fringe Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Out-Of-Plane Displacements: Shadow Moiré . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Shadow Moiré . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Projection Moiré . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT) . . . . . . . . . . . 21.4.1 The Talbot Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.2 Fringe Contrast Versus Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.3 Dynamic Range and Talbot Distance . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.4 Parameters for High-Sensitivity Measurements . . . . . . . . . . . . . 21.4.5 Implementation of SM-NT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Increased Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5.1 Optical/Digital Fringe Multiplication (O/DFM) . . . . . . . . . . . . . . . . 21.5.2 The Phase-Stepping (or Quasiheterodyne) Method . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

601 601 602 602 603 605 607 607 607 609 610 611 611 615 617 617 617 618 619 620 621 623 624 624 626

22 Moiré Interferometry Daniel Post, Bongtae Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Current Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.1 Specimen Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Optical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.3 The Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.4 Fringe Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.5 Strain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Important Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.1 Physical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.2 Theoretical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.3 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.4 Insensitivity to Out-of-Plane Deformation . . . . . . . . . . . . . . . . . . 22.2.5 Accidental Rigid-Body Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.6 Carrier Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.7 Loading: Mechanical, Thermal, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.8 Bithermal Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.9 Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.10 Data Enhancement/Phase Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.11 Microscopic Moiré Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .

627 630 630 632 632 633 634 634 634 635 635 635 637 637 638 638 640 640 643

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22.3

Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3.1 Strain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3.2 Replication of Deformed Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Characterization of Moiré Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Moiré Interferometry in the Microelectronics Industry . . . . . . . . . . . . . . . . 22.5.1 Temperature-Dependent Deformation . . . . . . . . . . . . . . . . . . . . . . 22.5.2 Hygroscopic Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5.3 Standard Qualification Test of Optoelectronics Package . . . . 22.5.4 Micromechanics Studies by Microscopic Moiré Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24 Holography Ryszard J. Pryputniewicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1 Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.1 Hologram Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Fundamentals of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.1 Recording of Holograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.2 Reconstructing a Hologram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.3 Properties of Holograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 Techniques of Hologram Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.1 Optoelectronic Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.2 Quantitative Interpretation of Holograms . . . . . . . . . . . . . . . . . . . 24.4 Representative Applications of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4.1 Determination of the Absolute Shape of Objects . . . . . . . . . . . . 24.4.2 Determination of Time-Dependent Thermomechanical Deformation Due to Operational Loads . . . . . . . . . . . . . . . . . . . . . . 24.4.3 Determination of the Operational Characteristics of MEMS 24.5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

644 644 644 645 646 646 648 650 651 652

655 655 655 656 658 658 660 662 665 668 668 670 672 672

675 676 676 677 677 678 679 679 681 683 685 685 687 688 696 697

Detailed Cont.

23 Speckle Methods Yimin Gan, Wolfgang Steinchen (deceased) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1 Laser Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1.1 Laser Speckle Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1.2 Some Properties of Speckles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Speckle Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.1 Speckle Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.2 Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.3 Shearography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.4 Quantitative Evaluation (SI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.1 NDT/NDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.2 Strain Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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25 Photoelasticity Krishnamurthi Ramesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1.2 Birefringence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1.3 Retardation Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Transmission Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.1 Physical Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.2 Stress–Optic Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.3 Fringe Contours in a Plane Polariscope . . . . . . . . . . . . . . . . . . . . . . 25.2.4 Jones Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.5 Ordering of Isoclinics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.6 Ordering of Isochromatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.7 Calibration of Model Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Variants of Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.1 Three-Dimensional Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.2 Dynamic Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.3 Reflection Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.4 Photo-orthotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.5 Photoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4 Digital Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.1 Fringe Multiplication and Fringe Thinning . . . . . . . . . . . . . . . . . . 25.4.2 Phase Shifting in Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.3 Ambiguity in Phase Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.4 Evaluation of Isoclinics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.5 Unwrapping Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.6 Color Image Processing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4.7 Digital Polariscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.5 Fusion of Digital Photoelasticity Rapid Prototyping and Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.6 Interpretation of Photoelasticity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.7 Stress Separation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.7.1 Shear Difference Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.7.2 Three-Dimensional Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 25.7.3 Reflection Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.8 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.9 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Thermoelastic Stress Analysis Richard J. Greene, Eann A. Patterson, Robert E. Rowlands . . . . . . . . . . . . . . . . . . . 26.1 History and Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3 Test Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

701 704 704 704 705 705 705 706 706 707 708 709 709 710 710 714 715 717 718 719 719 720 723 725 727 729 730 732 734 735 735 736 736 737 737 738 740

743 744 745 747 749

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27 Photoacoustic Characterization of Materials Sridhar Krishnaswamy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.1 Elastic Wave Propagation in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.1.1 Plane Waves in Unbounded Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.1.2 Elastic Waves on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.1.3 Guided Elastic Waves in Layered Media . . . . . . . . . . . . . . . . . . . . . . 27.1.4 Material Parameters Characterizable Using Elastic Waves . . 27.2 Photoacoustic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2.1 Photoacoustic Generation: Some Experimental Results . . . . 27.2.2 Photoacoustic Generation: Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2.3 Practical Considerations: Lasers for Photoacoustic Generation . . . . . . . . . . . . . . . . . . . . . . . . . 27.3 Optical Detection of Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3.1 Ultrasonic Modulation of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3.2 Optical Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3.3 Practical Considerations: Systems for Optical Detection of Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.4 Applications of Photoacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.4.1 Photoacoustic Methods for Nondestructive Imaging of Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.4.2 Photoacoustic Methods for Materials Characterization . . . . . 27.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

749 749 750 751 751 751 752 752 752 753 753 753 755 756 757 757 758 759 759 760 762 763

769 770 771 772 774 776 777 777 780 783 783 783 785 789 789 789 793 798 798

Detailed Cont.

Experimental Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.1 Background Infrared Radiation Shielding . . . . . . . . . . . . . . . . . . . 26.5.2 Edge Effects and Motion Compensation . . . . . . . . . . . . . . . . . . . . . 26.5.3 Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.4 Reference Signal Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.5 Infrared Image Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.6 Adiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.7 Thermoelastic Data Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.8 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.1 Isotropic Structural Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.2 Orthotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.3 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.4 Experimental Stress Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.5 Residual Stress Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.6 Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.7 Elevated Temperature Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6.8 Variable-Amplitude Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.A Analytical Foundation of Thermoelastic Stress Analysis . . . . . . . . . . . . . . . 26.B List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Detailed Cont.

28 X-Ray Stress Analysis Jonathan D. Almer, Robert A. Winholtz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1 Relevant Properties of X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.1 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.2 X-Ray Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1.3 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.1 Measurement Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.2 Biaxial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.3 Triaxial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.4 Determination of Diffraction Peak Positions . . . . . . . . . . . . . . . . 28.3 Micromechanics of Multiphase Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.1 Macrostresses and Microstresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.2 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.3 Diffraction Elastic Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.1 Conventional X-Ray Diffractometers . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.2 Special-Purpose Stress Diffractometers . . . . . . . . . . . . . . . . . . . . . . 28.4.3 X-Ray Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.4.4 Synchrotron and Neutron Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5 Experimental Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5.1 Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5.2 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5.3 Sample-Related Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.6 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.6.1 Biaxial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.6.2 Triaxial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.6.3 Oscillatory Data Not Applicable to the Classic Model . . . . . . . . 28.6.4 Synchrotron Example: Nondestructive, Depth-Resolved Stress . . . . . . . . . . . . . . . . . . . . . . 28.6.5 Emerging Techniques and Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.8 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

801 802 802 802 803 804 805 805 805 806 807 807 808 808 809 809 810 810 810 810 811 811 812 813 813 814 815 815 816 817 818 818

Part D Applications 29 Optical Methods Archie A.T. Andonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.1 Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.1.1 Transmission Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.1.2 Reflection Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 Electronic Speckle Pattern Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2.1 Calculation of Hole-Drilling Residual Stresses . . . . . . . . . . . . . . . 29.2.2 Quantification of Dynamic 3-D Deformations and Brake Squeal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

823 824 824 827 828 828 829

Detailed Contents

29.3 29.4 29.5

30 Mechanical Testing at the Micro/Nanoscale M. Amanul Haque, Taher Saif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1 Evolution of Micro/Nanomechanical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 Novel Materials and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 Micro/Nanomechanical Testing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.2 Challenges in Micro/Nanomechanical Testing . . . . . . . . . . . . . . . 30.3.3 Micro/Nanomechanical Testing Tools . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.4 Nontraditional (MEMS) Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.5 AFM-Based Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.6 Nanoindentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.7 Other Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4 Biomaterial Testing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.5 Discussions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Experimental Methods in Biological Tissue Testing Stephen M. Belkoff, Roger C. Haut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 General Precautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Connective Tissue Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Experimental Methods on Ligaments and Tendons . . . . . . . . . . . . . . . . . . . 31.3.1 Measurement of Cross-Sectional Area . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Determination of Initial Lengths and Strain Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 Gripping Issues in the Mechanical Testing of Ligaments and Tendons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 Preconditioning of Ligaments and Tendons . . . . . . . . . . . . . . . . . 31.3.5 Temperature and Hydration Effects on the Mechanical Properties of Ligaments and Tendons . . . . . . . . . . . . . . . . . . . . . . . 31.3.6 Rate of Loading and Viscoelastic Considerations . . . . . . . . . . . .

830 831 832 832 832 833 834 834 834 835 836 836

839 840 841 842 842 843 844 847 851 853 856 856 859 862 862

871 871 872 873 873 873 874 874 875 875

Detailed Cont.

Shearography and Digital Shearography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Point Laser Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5.1 Measurement of Strains at High Temperature . . . . . . . . . . . . . . . 29.5.2 High-Speed Spin Pit Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5.3 Measurement of Deformations in Microelectronics Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6 Laser Doppler Vibrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.6.1 Laser Doppler Vibrometry in the Automotive Industry . . . . . . 29.6.2 Vibration Analysis of Electron Projection Lithography Masks . . . . . . . . . . . . . . . . . . . 29.7 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.8 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1077

1078

Detailed Contents

31.4

Experimental Methods in the Mechanical Testing of Articular Cartilage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.1 Articular Cartilage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 Tensile Testing of Articular Cartilage . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.3 Confined Compression Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.4 Unconfined Compression Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.5 Indentation Tests of Articular Cartilage . . . . . . . . . . . . . . . . . . . . . .

876 876 876 876 877 877

Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5.1 Bone Specimen Preparation and Testing Considerations . . . 31.5.2 Whole Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5.3 Constructs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5.4 Testing Surrogates for Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5.5 Outcome Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

878 878 879 880 880 880

Skin Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6.2 In Vivo Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6.3 In Vitro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

883 883 883 883 884

31.5

Detailed Cont.

31.6

32 Implantable Biomedical Devices

and Biologically Inspired Materials Hugh Bruck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

891

32.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Experimental Mechanics Challenges . . . . . . . . . . . . . . . . . . . . . . . . .

892 892 893

32.2

Implantable Biomedical Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.2 Brief Description of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.3 Prosthetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.4 Biomechanical Fixation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.5 Deployable Stents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

899 899 899 900 904 907

32.3

Biologically Inspired Materials and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.2 Brief Description of Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.3 Functionally Graded Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.4 Self-Healing Polymers/ Polymer Composites . . . . . . . . . . . . . . . . 32.3.5 Active Materials and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.6 Biologically Inspired Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

909 909 910 911 915 917 921

32.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4.1 State-of-the-Art for Experimental Mechanics . . . . . . . . . . . . . . 32.4.2 Future Experimental Mechanics Research Issues . . . . . . . . . . . .

923 923 924

32.5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

924 924

Detailed Contents

33 High Rates and Impact Experiments Kaliat T. Ramesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1079

929

High Strain Rate Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.1 Split-Hopkinson or Kolsky Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.2 Extensions and Modifications of Kolsky Bars . . . . . . . . . . . . . . . . 33.1.3 The Miniaturized Kolsky Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.4 High Strain Rate Pressure-Shear Plate Impact . . . . . . . . . . . . . .

930 931 937 941 942

33.2

Wave Propagation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2.1 Plate Impact Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

945 946

33.3

Taylor Impact Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

949

33.4

Dynamic Failure Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.1 Void Growth and Spallation Experiments . . . . . . . . . . . . . . . . . . . 33.4.2 Shear Band Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.3 Expanding Ring Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.4 Dynamic Fracture Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.5 Charpy Impact Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

949 950 952 952 953 953

33.5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

953 954

34 Delamination Mechanics Kenneth M. Liechti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

961

34.1

Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.1 Interface Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.2 Crack Growth Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.3 Delamination in Sandwiched Layers . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.4 Crack Nucleation from Bimaterial Corners . . . . . . . . . . . . . . . . . . . 34.1.5 Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

962 962 963 964 966 968

34.2

Delamination Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2.1 Dynamic Interface Fractureand Coherent Gradient Sensing 34.2.2 Dynamic Interface Fracture and Photoelasticity . . . . . . . . . . . . 34.2.3 Dynamic Matrix Cracking and Coherent Gradient Sensing . 34.2.4 Quasistatic Interface Fracture and Photoelasticity . . . . . . . . . . 34.2.5 Quasistatic Interface Fracture and Crack Opening Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2.6 Quasistatic Interface Fracture and Moiré Interferometry . . . 34.2.7 Crack Nucleation from Bimaterial Corners and Moiré Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2.8 Bimaterial Corner Singularities and Photoelasticity . . . . . . . . 34.2.9 Crack Growth in Bonded Joints and Speckle . . . . . . . . . . . . . . . . 34.2.10 Thin-Film Delamination and Out-of-Plane Displacements

968 969 969 970 970

34.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

980 980

971 977 977 978 978 979

Detailed Cont.

33.1

1080

Detailed Contents

Detailed Cont.

35 Structural Testing Applications Ashok Kumar Ghosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.1 Past, Present, and Future of Structural Testing . . . . . . . . . . . . . . . . . . . . . . . . 35.2 Management Approach to Structural Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2.1 Phase 1 – Planning and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2.2 Phase 2 – Test Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2.3 Phase 3 – Execution and Documentation . . . . . . . . . . . . . . . . . . . 35.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.3.1 Models for High-Rate and High-Temperature Steel Behavior in the NIST World Trade Center Investigation . . . . . 35.3.2 Testing of Concrete Highway Bridges – A World Bank Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.3.3 A New Design for a Lightweight Automotive Airbag . . . . . . . . 35.4 Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Electronic Packaging Applications Jeffrey C. Suhling, Pradeep Lall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.1 Electronic Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.1.1 Packaging of Electronic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.1.2 Electronic Packaging Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 36.2 Experimental Mechanics in the Field of Electronic Packaging . . . . . . . . 36.2.1 Role of Experimental Mechanics and Challenges for the Experimentalist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.2.2 Key Application Areas and Scope of Chapter . . . . . . . . . . . . . . . . 36.3 Detection of Delaminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3.1 Acoustic Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4 Stress Measurements in Silicon Chips and Wafers . . . . . . . . . . . . . . . . . . . . . 36.4.1 Piezoresistive Stress Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4.2 Raman Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4.3 Infrared Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4.4 Coherent Gradient Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.5 Solder Joint Deformations and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.5.1 Moiré Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.5.2 Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.6 Warpage and Flatness Measurements for Substrates, Components, and MEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.6.1 Holographic Interferometry and Twyman–Green Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . 36.6.2 Shadow Moiré . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.6.3 Infrared Fizeau Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.7 Transient Behavior of Electronics During Shock/Drop . . . . . . . . . . . . . . . . . . 36.7.1 Reference Points with High-Speed Video . . . . . . . . . . . . . . . . . . . 36.7.2 Strain Gages and Digital Image Correlation . . . . . . . . . . . . . . . . . 36.8 Mechanical Characterization of Packaging Materials . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Detailed Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detailed Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1081

1045 1049 1059 1083

Detailed Cont.

1083

Subject Index

2-hydroxyethyl methacrylate (HEMA) 209 3-D digital image correlation

589

A

B background intensity 491, 515 backing 292 balanced mode 290 ball grid array (BGA) 1022 ballast gage 287 ballast resistor 286 ballistic impact 929 basic fringe pattern 552 beam-splitter (BS) 678 beam-splitter (DBS) – directional 683 beat frequency 466 bending moment 1006 Bessel fringe – J0 fringe 681 biaxial analysis 805 biaxial loading 295 biaxial stress measurement 813 bilinear 574

bimaterial constant 962 bimaterial corner 966, 977, 978 binomial 263 biological – beam 1012 – material 438 – sample 410 biomaterial 817 biomechanics – definition 182 – history 169 biomolecular recognition device 222 birefringence 702, 704, 973 black body 748, 751 blast and explosive load 994 blister 980 bloodborne pathogens 871 body force 734 body-centered cubic (bcc) 24 Bogen tripod 586 Boltzmann’s constant 745 bonded joint 978 bottom-up strategy 551 boule 325 boundary condition 734 boundary layer 194, 199 boundary, initial, and loading (BIL) 682 box-kernel 498 Bragg grating (FBG) – fiber 361, 906 Bragg wavelength 363 branch cut algorithm 727 bridge 1002 bright field 708 Brillouin frequency shift 359 Brillouin scattering 359 Brinell hardness number (BHN) 390 British Standards Institute (BSI) 102 bromoundecyltrichlorosilane (BrUTS) 974 buckyball 210 bulge 980 bulk wave – flaw imaging 789 bundle adjustment 573 burst pressure 1009

Subject Index

absolute phase measurement 495, 532 absolute shape 682 AC measurement 413 accelerated life testing (ALT) 1016 accelerometer 693 accident 1009 accuracy and resolution 990 accurate clear epoxy solid (ACES) 733 active material 209 additive intensity 603, 604 additive moiré 605 adhesion 220, 221, 961 – measurement technique 220 – prevention of 220 adhesive fracture 961, 962 adiabatic response 747 adiabatic shear band 952 adiabaticity 752 advanced photon source (APS) 810 advancing side 586 AFM – /DIC method 436 – /DIC strain measurement 435 – cantilever 411, 414 – contact 411, 413, 423 – instrumentation 409 – metrology 411 – probe 412, 414 Airy disc 475 aluminum 208 ambiguity 708, 724 American Association of State Highway Transportation Officials (AASHTO) 991 American Bureau of Shipping (ABS) 367 American Institute of Steel Construction (AISC) 991 American Railway Engineering Association (AREA) 991 American Society for Testing and Materials (ASTM) 102, 284, 991, 998, 1001

amorphous material 817 amorphous silicon 207 amplification 610 amplitude 708 amplitude grating 601 amplitude modulation 427 analog camera 567 analog-to-digital converter 412 analytical, computational, and experimental solutions (ACES) 688 analyzer 706, 712 animation of experimental data 688 anisotropic 97 anisotropy – elastic 804 aperture effect 618, 619 applied strain 293 articulated truck 1002 assembly stress 704 assembly stress/residual stress 703 Association Française de National (AFNOR) 102 atomic force microscopy (AFM) 216, 220, 397, 409–411, 846 atomic resolution 412, 424, 431 atomic structure 18, 40 attractive force 412 automotive airbag 987

1084

Subject Index

C

Subject Index

calcium-fluoride-coated window 758 calculation of the form 278 calibration 748, 996 – of a photoelastic material 709 – of cantilever stiffness 415 – of the coating material 716 camera calibration 572, 594 camera calibration procedure 572 camera coordinate system 569 cantilever – dynamics 411 – oscillation amplitude 429 – spring constant 414 – thermal fluctuation 416 – tip 417 – tip artifact 411 capacitance 194, 199 capacitive coupling 286 capillary adhesion 433 capillary force 212, 433 carbon nanotube (CNT) 107, 210, 920 carrier-frequency method 492, 512 cast polyimide 311 cedip 746 celanese compression fixture 104 cell 438, 439 central tendency 260 ceramic ball grid array (CBGA) 646, 1032 ceramic matrix composite (CMC) 22, 909 challenge 1002 characteristic – function 485 – parameter 713 – relation 932 – retardation 712 – rotation 712 charge density 194 charge-coupled device (CCD) 567, 745, 752 Charpy impact 953 Chauvenet’s method 272 chemical shrinkage 117 chemical vapor deposition (CVD) 208, 210, 211, 214 chemical–mechanical polishing (CMP) 1031 chirped grating 367 chi-square (χ 2 ) 263 – statistic 277 – test 277

Christoffel equation 771 chromium 208 circular polariscope 708 circularly polarized 704 civil structural monitoring 368 classification of structural testing 991 cleveland method 415 closed-loop system 995 C-mode scanning acoustic microscopy (CSAM) 1023 coating 751, 782 coating and thin film 809 coefficient of hygroscopic swelling (CHS) 648 coefficient of thermal expansion (CTE) 117, 622, 757, 1016, 1018 coefficient of variation 263 coherent gradient sensing (CGS) 150, 969, 970, 1030, 1031 coherent imaging 487 coherent optical data processing 478 cohesive fracture 961 cohesive zone model 968 cohesive-volumetric finite elements (CVFE) 914 cold contact time 938 collinear interference 450 collision avoidance radar (CAR) 687 color code 709, 729 color stepping 725 colored interference fringe 462 common-path interferometer 461 compact disc (CD) 417 compatibility condition 734 complementary metal–oxide semiconductor (CMOS) 567 complementary pattern 624 complex amplitude 470, 471 complex stress intensity factor 962 compliance 127 composite – laminate 753, 755, 756 – material 97 Composite Materials Technical Division (CMTD) 98 compression testing 103 compressor blade 759 computer-aided design (CAD) 685 computer-aided engineering (CAE) 685 computer-aided manufacturing (CAM) 685 concentrated load 275

concrete 1002 concurrent engineering 685 condition experimental data 272 conductive – polymer (CP) 920 confidence interval 268 confidence level 268 constant stress 293 constantan 291 constant-voltage excitation 285 constitutive formulation 171 constitutive response 934 constrained matching 585 contact 961 – AFM 413, 423 – stress 275, 704 continuous stiffness 396 continuous stiffness module (CSM) 81, 82 continuous time-average hologram interferometry 680 continuous wave (CW) 680 continuum concept 569 contour 702 – map 601 contourable plastics 715 contours of displacement/strain/stress 701 convolution kernel 498 coordinate measuring machines (CMMs) 685 coordinate system 569 coordinates 580 copper–nickel alloy 291 core testing 1006 correction factor 716 correlation 273 correlation coefficient 273, 274 corrosion test 1006 cosinusoidal fringe 680 cost-effective 1009 covariance matrix 543 cover test 1005 crack 1007 – closure 755 – growth 437, 438 – growth criteria 963 – initiation 438 – nucleation 966, 977 – opening displacement (COD) 145, 972 – opening interferometry (COI) 145, 971–973 – propagation 113, 438

Subject Index

D damage 930, 1003 dark field 708 data – acquisition 996 – correlation 753 – filtering 753 – format 753 – transmission 996 DC measurement 413 dead load 993 deep reactive-ion etching (DRIE) 210, 213, 848 deflection prediction 1006 deformation – elastic 292 deformation measurement 566, 586, 589 deformed image 572, 586 degree of freedom 268, 277

delamination 108, 111, 112, 961, 979 delta rosette 322 Deltatherm 746 dependent 274 dependent variable 272 deployment 1009 Derjaguin–Muller–Toporov (DMT) 429 description of light 448 desktop Kolsky bar 941 destructive/failure test 991 detector – infrared 745 Deutsches Institut für Normung (DIN) 102 deviation ratio 272 diametric loading 364 diamond 208 – method 287 – pyramid hardness (DPH) 391 diamond-like carbon (DLC) 436, 795 dicyclopentadiene (DCPD) 111 difference between two means 270 diffraction – at aperture 474 – by aperture 469 – by grid 469 – elastic constant (DEC) 805, 808 – problem 468 – theory 473 – theory history 470 digital – photoelasticity 737 – speckle correlation (DSC) 658 – speckle pattern shearing interferometry (DSPSI) 830 – speckle photography (DSP) 658 – speckle-pattern interferometry (DSPI) 517 digital fringe multiplication 719 digital fringe thinning 719 digital holography 517 digital image – correlation (DIC) 141, 220, 434, 566, 660, 845, 1034 – processing (DIP) 719 digital micromirror device (DMD) 221 – and adhesion 221 – and creep 221 digital photoelasticity 704 dilatometry 316 dipole 199

dip-pen nanolithography (DPN) 216 direct impact 930 direct problem 549 direction of illumination and observation 681 directional – beam-splitter (DBS) 683 directional coupler (DC) 681 directional filter 502 disaster 997 dislocation cross-slip 398 dispersion 273, 935 displacement 489 displacement vector 536 dissipative 34, 38, 39, 43 distance to neutral point (DNP) 646, 1032 distributed temperature sensing (DTS) 367 distribution 260 distribution function 263 division of quantity 278 dodecyltrichlorosilane (DTS) 974 Doppler – effect 464 – interferometry 466 – picture velocimetry (DPV) 468 – shift 465 – shift for reflected light 467 Doppler–Fizeau 464 double cantilever beam (DCB) 111, 126, 240 double refraction 704 double-exposure hologram interferometry 680 dropweight 931 Dugdale–Barenblatt model 132 dummy compensation 101 dummy gage 301 dynamic 1009 – energy release rate 138 – failure 930 – fracture 953 – interface fracture 969 – matrix cracking 970 – mechanical analysis (DMA) 84 – photoelasticity 703, 735 – range 619, 620, 623 – stress intensity factor 138 dynamic/vibration load 994 Dynatup 593

Subject Index

crack path – selection 964 – separation 965 – stability 965 crack tip 275 – deformation 438 – opening displacement (CTOD) 133 cracking 112 Cranz Schardin camera 714 creep 221, 930, 1001 – crack growth 140 crit