Recent Advances in Experimental Mechanics. In Honor of Isaac M. Daniel

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Recent Advances in Experimental Mechanics

Table of Contents Editor’s Preface

v

Biography of Isaac M. Daniel

xv

List of Publications by Isaac M. Daniel

xxi

List of Contributors

xlv

1. Mechanical Behavior of Materials Hopkinson Techniques for Dynamic Triaxial Compression Tests J. Rome, J. Isaacs and S. Nemat-Nasser

3

Asymptotic Scaling of Gradient Theory of Micro-Scale Plasticity of Metals

13

The Role of Pressure in the Behavior of Time-Dependent Materials T. Prodan and I. Emri

19

High Strain Rate Testing of Sandwich Core Materials M. Vural and G. Ravichandran

31

Development of a Shear Test for Low Modulus Foam Materials A. K. Roy and J. D. Camping

43

Indentation of a PVC Cellular Foam E. E. Gdoutos and J. L. A bot

55

Nanomanipulation and Characterization of Individual Carbon Nanotubes R. S. Ruoff, M.-F. Yu and H. Rohrs

65

Quasi-Static and Dynamic Torsion Testing of Ceramic Micro and Nano-Structured Coatings Using Speckle Photography F. Barthelat, K. Malukhin and H. Espinosa

75

2. Composite Materials Measured Response: State Variables for Composite Materials K. Reifsnider and M. Pastor

87

On the Modeling of the Mechanical Properties of Composite Materials at High Strain Rates J. R. Vinson and S. Xiao

99

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TABLE OF CONTENTS

Damage Quantification in Metal Matrix Composites G. Z. Voyiadjis, A. R. Venson, and R. K. Abu-Alrub

109

Study of Damage in Particulate Composites C. A. Sciammarella and F. M. Sciammarella

121

Hygric Characterization of Composite Laminates S.-C. Wooh and C.-L Tsai

133

Pneumatic Behavior of Composite Materials C.-L. Tsai and Y.-S. Tsai

145

The Effect of Specimen Size on the Compressive Strength of Carbon Fibre-Epoxy Laminates C. Soutis and J. Lee

153

Interfacial Strength and Toughness Characterization Using a Novel Test Specimen G. P. Tandon, R. Y. Kim and V. T. Bechel

163

A Model for the Accurate Prediction of the Residual Strength after Damage Due to Impact and Erosion of FRPs G. C. Papanicolaou, G. Samoilis, S. Giannis, N.-M. Barkoula and J. Karger-Kocsis

175

3. Fracture and Fatigue The Origin and Inception of Fatigue in Steel – A Probabilistic Model S. A. Guralnick and J. Mohammadi

187

Fatigue Damage Tolerant Analysis Using the Fatigue Damage Map C. A. Rodopoulos and J. R. Yates

197

Crack Growth Behavior and SIF’s as Observed by Optical Methods C. W. Smith

209

A Model for Failure Initiation in Ductile Materials J. Zuo, M. A. Sutton and X. Deng

217

Crack Paths in Adhesive Bonds L. Banks-Sills and J. Schwartz

225

Experimental Determination of Fracture Parameters for Predicting Crack Growth in Viscoelastic Polymers D. H. Allen and J. J. Williams

235

Failure of Spot Weld: A Competition Between Crack Mechanics and Plastic Collapse Y. J. Chao

245

TABLE OF CONTENTS

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Investigating the Effects of Specimen Thickness and Pressure on the Crack Growth Behavior of a Particulate Composite Material C. T. Liu

257

Dynamic Fracture Experiments Using Point Impact D. Rittel

267

Experimental and Numerical Investigation of Shear-Dominated Intersonic Crack Growth and Friction in Unidirectional Composites A. J. Rosakis, C. Yu, M. Ortiz, D. Coker and A. Pandolfi

275

4. Optical Methods Moiré Interferometry - Past, Present and Future D. Post

291

Optical Fibre Bragg Grating Sensors in Experimental Mechanics of Composite Laminated Plates J. Botsis, F. Bosia, M. Facchini, Th. Gmür

303

Optoelectronic Displacement Measurement Method for Rotating Disks C. E. Bakis, B. J. Haldeman and R. P. Emerson

315

Deformation Measurement of Sheet Metal Forming Using Photogrammetry K. Andresen

325

Fracture Processes of Quasi-Brittle Materials Studied with Digital Image Correlation J. S. Lawler and S. P. Shah

335

On the Use of Different Wavelengths to Digitally Determine the Isochromatic Fringe Order T. Y. Chen, Y. C. Chou, H. L. Lee and S. H. Tsao

345

The Application of Speckle Metrology to Heart Mechanics G. R. Gaudette, E. U. Azeloglu, J. Todaro, L. Keene, I. B. Krukenkamp and F. P. Chiang

353

5. Non-Destructive Evaluation Line Focus Acoustic Microscopy for Thin-Film Measurements Z. Guo and J. D. Achenbach

367

Recent Advances in Acoustography-Based NDE J. S. Sandhu and H. Wang

381

Experimental Limitations to Guided Wave Generation in Elastic Materials D. A. Sotiropoulos and E. Babatsouli

389

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TABLE OF CONTENTS

Nondestructive Testing Using Shearography M.Y.Y. Hung

397

Theoretical and Experimental Study of Laser-Ultrasonic Signal Characteristics Enhanced by Wetting the Surface S.-C. Wooh

409

Defect Detection by the Scattering Analysis of Flexural Waves P. Fromme and M. B. Sayir

421

Evaluation of Fiber Waviness in Thick Composites by Ultrasonic Test H.-J. Chun

433

Development of a Dry-Contact Ultrasonic Technique and its Application to NDE of IC Packages H. Tohmyoh and M. Saka

443

6. Neutron Diffraction and Synchrotron Radiation Methods High-Resolution Neutron Diffraction Techniques for Strain Scanning P. Mikula, M. Vrana, P. Lukas and V. Wagner

457

Draft Standard for the Measurement of Residual Stresses by Neutron Diffraction G. A. Webster, A. G. Youtsos, C. Ohms and R. C. Wimpory

467

Microstresses Determined by Neutron Diffraction and Self-Consistent Model A. Baczmanski, C. Braham, R. Levy-Tubiana, A. Lodini and K. Wierzbanowski

477

Residual Stress Measurements at the Metal/Ceramic Interface Using Modelling of Neutron Diffraction Spectrometer A. Carrado, J.-M. Sprauel, L. Barrallier and A. Lodini

487

Elastoplastic Deformation of Two Phase Steels Studied by Neutron Diffraction and Self-Consistent Modelling M. R. Daymond, H. G. Priesmeyer and A. M. Korsunsky

495

Residual Stresses and Elastic Constants in Thermal Deposits T. Gnäupel-Herold, H. J. Prask and F. S. Biancaniello

507

Neutron Diffraction Assisted Residual Stress Analysis in Welded Structures C. Ohms and A. G. Youtsos

515

TABLE OF CONTENTS

Synchrotron Radiation In-Situ Analyses of AA 6061 + During Tensile Deformation at Ambient and Elevated Temperature A. Pyzalla, B. Reetz, A. Jacques, J.-P. Feiereisen, O. Ferry and T. Buslaps

xi

527

7. Hybrid Methods Reflections on the Importance of Experimental Results to all Mechanicists, Especially Theoreticians R. M. Jones

537

Mixed Numerical-Experimental Techniques : Past, Present and Future A. H. Cardon, H. Sol, W. P. De Wilde, J. De Visscher, K. Hoes and D. Dinescu

551

A Moiré-FE Method for Internal CTOA Determination J. H. Jackson and A. S. Kobayashi

561

Patterns of Modern Experimental Mechanics J. T. Pindera

571

Inverse Methods in Experimental Mechanics J. F. Doyle

585

Complex Stiffness Identification by Inverse Methods H. Sol and W. P. De Wilde

595

Considerations of a Flutter Prediction Methodology Using a Combined Analytical-Experimental Procedure P. Marzocca, L. Librescu and W. A. Silva

609

Displacement-Based Smoothing Hybrid Finite-Element Representation for Stress Analyzing Perforated Composites K. Y. He and R. E. Rowlands

619

8. Composite Structures Future Experimental Methods Needed to Verify Composite Life-Cycle Simulations C. C. Chamis and L. Minnetyan

631

Experimental Observations on the Delamination Behavior in Composite Structures G. A. Kardomateas, V. La Saponara and G. J. Simitses

645

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TABLE OF CONTENTS

Direct Identification of Elastic Properties of Composite Structures A Wave-Controlled Impact Approach S. Abrate

661

Static Behaviour of Pre-Stressed Polymer Composite Sandwich Beams R. A. W. Mines, Q. M. Li, R. S. Birch, R. Rigby, M. Al-Khalil and A. Tanner

671

On Debond Failure of Foam Core Sandwich L. A. Carlsson

683

Core Crush Mechanisms and Solutions in the Manufacturing of Sandwich Structures H. M. Hsiao, S. M. Lee and R. A. Buyny

689

Displacement Fields Around a Circular Hole in Composite Laminates S. M. Chern and M. E. Tuttle

701

9. Structural Testing and Analysis Recent Advances in Long-Term Monitoring of Bridges J. R. Casas

715

An Experimental Mechanics Approach to Structural Health Monitoring for Civil Aircraft E. W. O'Brien

727

Smart Structures Application to Airworthiness and Repair R. Jones, I. H. McKenzie, S. Galea and S. Pitt

737

Experimental Investigation of Shrinkage Strains in Multilayered Stereolithography Parts D. E. Karalekas

749

Suppression of Dimpling In Sheet Metal Parts Formed on Discrete Tooling R. C. Schwarz, J. Nardiello and J. M. Papazian

757

Effect of Loading Rate and Geometry Variation on the Dynamic Shear Strength of Adhesive Lap Joints V. Srivastava, V. Parameswaran, A. Shukla and D. Morgan

769

Author Index

781

Subject Index

797

Editor’s Preface

This book contains 71 papers presented at the symposium on “Recent Advances in Experimental Mechanics” which was organized in honor of Professor Isaac M. Daniel. The symposium took place at Virginia Polytechnic Institute and State University on June 23-28, 2002, in conjunction with the 14th US National Congress of Applied Mechanics. The book is a tribute to Isaac Daniel, a pioneer of experimental mechanics and composite materials, in recognition of his continuous, original, diversified and outstanding contributions for half a century. The book consists of invited papers written by leading experts in the field. It contains original contributions concerning the latest developments in experimental mechanics. It covers a wide range of subjects, including optical methods of stress analysis (photoelasticity, moiré, etc.), composite materials, sandwich construction, fracture mechanics, fatigue and damage, nondestructive evaluation, dynamic problems, fiber optic sensors, speckle metrology, digital image processing, nanotechnology, neutron diffraction and synchrotron radiation methods. The papers are arranged in the following nine sections: Mechanical characterization of material behavior, composite materials, fracture and fatigue, optical methods, nondestructive evaluation, neutron diffraction and synchrotron radiation methods, hybrid methods, composite structures, and structural testing and analysis. The first section on mechanical characterization and material behavior contains eight papers dealing with dynamic tests using Hopkinson bar techniques, gradient theory of micro-scale plasticity, time-dependent materials, foam materials, carbon nanotubes and nanostructured coatings. The second section on composite materials contains nine papers dealing with state variables, properties at high strain rates, metal matrix composites, particulate composites, hygric characterization, pneumatic behavior, compressive strength, interfacial strength and toughness, and residual strength after damage due to impact and erosion. The third section on fracture and fatigue contains ten papers dealing with a probabilistic model of fatigue, damage tolerance analysis, crack growth, failure initiation, crack paths in adhesive bonds, failure of spot welds, and dynamic and intersonic crack growth. The fourth section on optical methods contains seven papers dealing with moiré interferometry, optical fiber sensors, displacement measurements based on optoelectronics and photogrammetry, fracture processes using digital image correlation, determination of fringe order using different wavelengths, and heart mechanics problems by means of speckle metrology. The fifth section on nondestructive evaluation contains eight papers dealing with acoustic microscopy, acoustography, waves in elastic materials, shearography, laser-ultrasonics, scattering of flexural waves, and ultrasonic applications. The sixth section on neutron diffraction

Biography of Isaac M. Daniel Isaac M. Daniel was born on October 7, 1933 in Thessaloniki (Salonica), Greece, second of four children of Mordochai Daniel and Bella (Modiano) Daniel. Thessaloniki, the second largest city in Greece, is an ancient city named by Alexander the Great for his sister. It is also near Aristotle’s birthplace. His family, for the most part, traces its origins to Spain from where they were expelled in 1492 for religious reasons, although it also has roots in a community that lived in the area for over 2000 years. His father was born and raised in the nearby town of Veria (Veroia) also an ancient city. His mother’s family came from Livorno, Italy and is distantly related to the painter Amedeo Modigliani. Thessaloniki, originally the center of Alexander’s Macedonia became successively part of the Roman, Byzantine and Ottoman empires until 1912 when it was incorporated in Greece. Sometimes when talking about his family’s background, Isaac says that his mother was born in Turkey, raised in Greece as an Italian citizen, attended an Italian school where she learned French, but at home she spoke Ladino (a medieval Spanish dialect with admixtures of Hebrew, Portuguese, Italian, Greek, French and Turkish). Before the Second World War, Isaac’s family moved to Veria (about 45 miles west of Thessaloniki) which at the time had a population of around 15,000. He entered the local Jewish school and attended the Synagogue that the apostle Paul visited some 2000 years ago. Isaac’s father was a cabinet maker and occasionally worked as a grain dealer. His peaceful life came to a halt with the German invasion and its consequences. He still remembers all the horrible details of that period. The roundup of the Jewish community came suddenly, but his family miraculously escaped and hid in their basement. They lived in that basement, just like Ann Frank, for about a month until they were found out, arrested and put in the Baron Hirsch transit camp in Thessaloniki. They were scheduled to be shipped within days to the death camps of Auschwitz, when another miracle occurred. They were put on somebody’s list, the list of Guelfo Zamboni, the Italian consul in Thessaloniki. Through his intervention, 280 inmates, with direct or remote connection to Italian citizenship, out of more than 55,000 who passed through that camp, were released and taken to Athens which was then under Italian control. After Italy surrendered to the allies, life in Athens during the remaining war years became turbulent and full of dangerous adventures and more miraculous and narrow escapes. After the liberation on October 12, 1944, the troubles were not completely over, as they were caught in the middle of a brief but bloody civil war. Finally, they managed to return to their home in Veria and tried to start rebuilding their broken lives. They had lost over fifty of their relatives and most of their friends. In 1945 Isaac Daniel enrolled in one of the public schools in Veria and then entered, after an entrance examination, the Gymnasium (High School). He has very

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BIOGRAPHY OF ISAAC M. DANIEL

good memories of his high school years. He was very popular and had many friends. The school followed a classical curriculum which included among others, six years of ancient Greek, and four years of Latin and French. In the senior year the students put on a play and they raised enough funds for a class trip to southern Greece and the islands. It was Isaac’s first and only attempt as a playright. He had no favorite or less favorite subjects. He was equally at ease with Math and Science as with history and literature. He wrote poetry and had one poem published. He was selected to give the valedictory address at his graduation. He was told that a couple of his teachers had tears in their eyes during the speech. Graduation from high school was the end of a care-free period and the beginning of a new phase of career planning and preparation. The high school principal who counseled all the graduating students individually told Isaac that he had no special advice to give him, that he would do well in whatever career he followed. It was a choice among Literature, Medicine and Engineering. He chose the latter because it seemed to be the most challenging at the time. The entrance exams for the National Technical University in Athens (the only technical school in the country at that time) were the toughest of all. The best graduating students from all high schools would take a year off and enter a preparatory school before attempting the entrance exams. Upon graduation from high school Isaac Daniel attended a preparatory school for less than a month and he spent the rest of the summer studying by himself. He took the exams in the fall and it was one of the greatest days of his life when he heard that he was successful. He entered the School of Civil Engineering at the National Technical University (Metsovion Polytechnion). They had classes from 5 to 8 hours a day, six days a week. He finished two years and had enrolled for the third year when he immigrated with his family to the United States. The family settled in Chicago in 1955 and Isaac transferred to the Illinois Institute of Technology. It was a bit of a cultural shock. He was more advanced in technical and math subjects, but he had to take English literature and other liberal arts courses. He was a straight A student and graduated with distinction, first in his class as well as the entire Engineering Division of IIT in 1957. He received a Fellowship from the Chicago Bridge and Iron Co. and entered graduate school at IIT. IIT had an illustrious tradition in Mechanics with names like Enrico Volterra, Bill Ramberg, Eli Sternberg, Daniel Drucker and others associated with it before Isaac came. He had great teachers like August Durelli, Max Frocht, Lloyd Donnell and Phil Hodge. Durelli was his first mentor who introduced him to the field of Experimental Mechanics. Durelli was a scientist, philosopher and humanist. Frocht taught him precision and quality of presentation, especially in the case of photoelastic fringe patterns. Donnell impressed him by demonstrating the thought processes and approximations involved in developing complex relations. Hodge was clear and methodical in his lectures on continuum mechanics and plasticity. In addition to the technical knowledge, he learned one wise life guideline: "what is not worth doing is not worth doing well.” As he was finishing his Master’s degree studies, Durelli offered Isaac a job at the Armour Research Foundation (later IIT Research Institute) the research affiliate of IIT, where he became deeply involved in Experimental Stress Analysis. At that time, the


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late 50’s and early 60’s, the IIT campus was a hotbed of activity in experimental stress analysis. Albert Kobayashi and Pericles Theocaris had both worked in Durelli’s laboratory. He was fortunate to have supervisors and colleagues like Jim Dally, Bill Riley, Cesar Sciammarella, Vince Parks, Bob Rowlands and Ted Liber. The biggest compliment that Durelli paid Isaac, when he ran the first dynamic photoelasticity test and showed him the results, was a quotation from El Cid: “leurs coups d’essai veulent des coups de maître” (their trial strokes are master strokes). His first technical presentation, based on his Master’s thesis, was at the ASME meeting in New Orleans in 1960. The work at Armour (IITRI) was exciting. Isaac Daniel worked on stress analysis problems using photoelastic, moiré and strain gage methods. He worked on fracture of brittle materials and published work on fracture probabilities with N. A. Weil. He worked on creep of rubberlike materials and was impressed to verify experimentally that the equilibrium modulus of these materials is proportional to the absolute temperature as predicted. He became interested in viscoelasticity and this led him to write a proposal to the Air Force for the development of dynamic photoviscoelasticity and application to the study of stress waves in earth media. This led him to his Ph.D. research which he did while working full-time at IITRI. The research was reviewed by a committee including K. H. Chu, Jim Dally, Phil Hodge and Frank Essenburg of IIT and IITRI. In the course of this research he received valuable advice from Y. H. Pao of Cornell who was visiting at IITRI for one summer. He developed a theory of photothermoviscoelasticity and applied it to wave propagation problems. In 1964, Isaac received his Ph.D. He continued research in the areas of wave propagation and rock mechanics and started applying experimental stress analysis methods to the emerging technology of composite materials. Initially, he studied the micromechanics of composites by means of photoelasticity. In the middle 60’s there was a government thrust to develop advanced composites. Texaco had developed a new boron prepreg tape, which was 1/8 in. wide. Isaac Daniel and his co-workers made their first advanced composite sample from this tape, the size of a postage stamp. The government announced a large program and encouraged the collaboration of universities, research institutes and industry. Isaac’s director at IITRI wrapped the postage stamp sized specimen in his wallet and they both flew to Fort Worth to meet with General Dynamics. At a high level technical meeting they asked Isaac how much experience he had with advanced composites. Before he had a chance to say “not much” his boss kicked him under the table and said that they had expertise in fabricating composites and showed them the specimen. The General Dynamics people were impressed. It was proposed then that the first application be a flat part, namely the horizontal stabilizer of the F-111. Isaac Daniel was appointed manager of the Experimental Stress Analysis Section at IITRI, a position previously held by Durelli and Riley. It was a heady time for research in composites and the funding was lavish. There were no established testing procedures for composites and the common practice was to adopt the same methods used for metals. For example, tensile testing was done by ultrasonically machining a narrowed section in a boron/epoxy coupon. This process, besides being very expensive and time consuming, did not work. It was then that Isaac tried a straight-sided coupon tabbed with glass/epoxy tapered tabs, a method that has become an ASTM standard. In the

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following years his work with composites expanded in all directions including processing, development of test methods, micromechanics, material characterization, special test methods (torsion, impact, creep/relaxation, fracture toughness, hygrothermal effects, high rate testing, multi-axial testing, fatigue), stress and failure analysis (stress concentrations, joints, warpage), damage mechanics and nondestructive evaluation. In the early 70’s he worked with Bob Rowlands and Jim Whiteside of Grumman on the effects of stress concentrations in composite panels. They applied birefringent coatings and moiré techniques to study the deformation and failure around holes. This work was extended later to investigations of size effects and biaxial loading. He built a new biaxial testing system for flat plates. He had a chance to work on speckle interferometry with Mike Hung during his brief sojourn in his group at IITRI. In the 70’s he also worked with Ted Liber on developing an embedded strain gage technique for determination of curing residual stresses, biaxial testing using tubular specimens, high strain rate testing and nondestructive evaluation of composites. He developed and built a fully controlled test system for multiaxial testing of thin wall tubular specimens. He also developed a new method for testing composites at very high strain rates using thin ring specimens subjected to an internal pressure pulse produced by an explosive inside a liquid container. This method did not have the limitations of other methods using test coupons such as the Hopkinson bar method, where stress wave propagation over the length of the specimen produces a nonuniform stress distribution. In the case of the thin ring specimen under internal pressure there are no end effects and the relevant dimension is the thickness of the ring. He also found out that they could determine dynamic compressive properties by loading thin wall cylinders under external pressure without producing buckling, if the pressure were applied at a sufficiently high rate. Isaac Daniel received two best paper awards from the Society for Experimental Stress Analysis (SESA and later SEM). He became Fellow of SESA in 1981 and was invited to be the keynote speaker at the 7th International Conference on Experimental Mechanics in 1982 in Haifa. In the same year he coauthored with Jim Whitney and Byron Pipes a monograph on Experimental Mechanics of Composites. In January 1982 he was invited to join IIT as Professor and Director of the Experimental Stress Analysis Laboratory (later Mechanics of Materials Laboratory). The first grant he received at IIT was from ONR (Yapa Rajapakse) which has continued its support to date. During this period he worked on damage mechanics of composites, including damage characterization, evolution and accumulation, accelerated testing and life prediction; dynamic fracture toughness, and nondestructive evaluation of composites. In 1984 he received the B. J. Lazan Award for outstanding original contributions in Experimental Mechanics. In the same year he spent the summer in Poitiers, France, as a visiting Professor at the invitation of Professor Alexis Lagarde, and gave several seminars on composites. In early 1986 he was invited to give a seminar at Northwestern University. During lunch, Toshio Mura, in his inimitable fashion, said to him: “After you join us in the fall, we will write a proposal together on solitons.” That was the first hint he got of the offer from Northwestern that came later and he couldn’t refuse. He joined Northwestern and he wrote the proposal with Toshio Mura to NSF. It received two “excellent” and one


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“very good” rating, but it was not funded. He continued the work in damage mechanics and damage evolution in composites. This included development of models for stiffness and strength degradation and closed form expressions for constitutive behavior of cracked composite laminates with students Jae-Won Lee and Cho-Liang Tsai, now at Samsung Company in Korea and Yun Lin University in Taiwan, respectively. With Tsai he developed a unique method for determining all three shear moduli of an orthotropic composite. He developed advanced ultrasonic and acoustic emission techniques for flaw and damage detection and characterization with his student ShiChang Wooh, now Professor at MIT. This work included ultrasonic methods for threedimensional imaging of internal damage, detecting porosity and for determination of elastic properties of composites. He worked with Tao-Ming Wang (now at Illinois Tool Works) on control of residual stresses and warpage in circuit boards, a problem of great interest in electronic packaging. One result of this work was a method for measuring the chemical cure shrinkage in composite laminates. A multi-year program, started in the late 80’s, was the investigation of constitutive behavior of metal matrix composites sponsored by NASA-Lewis (Chris Chamis). He worked with his students Dimitri Karalekas and Heoung-Jae Chun, now Professors at the University of Piraeus, Greece and Yonsei University in Korea, respectively. They did a very extensive characterization of elastic and viscoelastic behavior of metal matrix composites at ambient and elevated temperatures. Starting in the late 80’s and continuing into the 90’s he embarked on the study of micromechanics of failure and constitutive behavior of ceramic matrix composites with his students Jae-Won Lee, George Anastasopoulos and Jyi-Jiin Luo. Based on real time observations of microscopic failure modes and acoustic emission output, they proposed analytical constitutive models describing material response up to failure. The investigation was extended to elevated temperatures and determination of residual stresses. He worked with his former student and now research associate Jyi-Jiin Luo on the development of an analytical model and a general theory for characterizing deformation of damaged composites. One of the most satisfying activities for Isaac Daniel in the early 90’s was writing the textbook “Engineering Mechanics of Composite Materials,” with Ori Ishai of the Technion in Israel. The book has been very well received by the community and is widely used as a textbook throughout the world. He is currently working with Ishai on the second edition. In the early 90’s the composites thrust of ONR shifted to the compressive behavior of thick composites. With his student Hao-Ming Hsiao he developed methods for manufacturing thick composites of uniform quality and a new method for compressive testing of such composites using a specially designed fixture (NU fixture). They studied and recorded the failure mechanisms of kink band formation and its propagation. They investigated experimentally and modeled the behavior, both linear and nonlinear, of thick composites with fiber waviness. The above studies were followed by extensive investigations of the effects of high strain rate on the stress-strain behavior and properties of composites under tension, compression and shear. This work was followed by research in the area of composite sandwich structures with the help of Emmanuel Gdoutos, visiting Professor from Democritus University of Thrace, Greece, and former student and now research associate Jandro Abot. They

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developed methods for fabrication and testing of sandwich structures statically and under impact and studied experimentally and analytically the various failure modes and criteria for their occurrence. A major expansion of his activity in composites occurred at the end of 1997 when Professor Daniel became Director of the new Center for Intelligent Processing of Composites. The objective of this center is to develop a comprehensive, commercially viable approach for fabrication of affordable, functional and reliable composite structures. This is an interdisciplinary program conducted by a team including University faculty, industrial firms and the government. Contributions of Daniel’s group in this Center include the design and fabrication of an innovative computer controlled system for resin transfer molding (RTM); new methods for characterizing reinforcement preforms; flow and cure models; and a new noninvasive approach for online quality control of reinforcement quality, a method of great commercial potential. Isaac Daniel received several recognitions and awards in the 90’s. He became Fellow of the American Academy of Mechanics (1994); was invited to be a keynote speaker at the International Conference on Composites Engineering in New Orleans (1995); received the Distinguished Research Award of the American Society for Composites (1996); was elected Chairman of the Theoretical and Applied Mechanics program at Northwestern University (1996); received the William M. Murray Medal of the Society for Experimental Mechanics and gave the associated prestigious lecture to the society (1998); he was appointed the Walter P. Murphy Professor of Civil and Mechanical Engineering at Northwestern University (1998); became Fellow of the American Society of Mechanical Engineers (1999); received the Professional Achievement Award from his alma mater, the Illinois Institute of Technology (1999). Besides all the research and professional activities, Isaac leads a busy and happy family life. His wife Elaine Krule Daniel of Chicago shares many interests with him. She studied at the Sorbonne and the University of Illinois. Elaine is a polyglot with degrees in French and teaching English as a second language and has taught at the University of Illinois in Champaign-Urbana and the University of Louisville in Kentucky. She is also a gourmet cook and a quilter. They have three children, Belinda, 13, Rebecca, 11, and Max 9. Isaac spends a lot of time with his family taking them along on trips to summer conferences in the United States and abroad. He attends and participates in the children’s activities, such as concerts and Boy Scout projects. He is active in community affairs, reads, travels and is interested in photography. He has a sense of humor which Elaine appreciates by laughing at his jokes even when she hears them for the fifth time. He shares a special interest with Elaine in linguistics and their dream is to write a Ladino-English dictionary. Isaac was very close to his father and mother who passed away in 1993 at the ages of 94 and 86. His older brother, Aaron, is an architect living in California. When he lived in Chicago he contributed to the design of the Sears Tower and the Civic Center plaza. His sister, Sarah Spector, has a degree in nutrition and lives in Chicago. His younger brother Sam, works for Motorola on artificial intelligence and lives in Arizona. Isaac Daniel has no immediate plans for retirement. People say that his children keep him young and active. He hopes to continue working and help his children with their education and, maybe, have them in his class.

1. Mechanical Behavior of Materials

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HOPKINSON TECHNIQUES FOR DYNAMIC TRIAXIAL COMPRESSION TESTS

JACOB ROME, JON ISAACS, SIA NEMAT-NASSER Center of Excellence for Advanced Materials Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, CA 92093-0416

Abstract

A modified Hopkinson technique is developed to simultaneously load a sample in both the axial and radial directions. This is accomplished by using a composite incident-bar and a composite transmission-bar system. The incident-bar system consists of a largediameter bar, followed by a small-diameter bar that fits inside a tube whose outside diameter is equal to that of the large bar. The transmission-bar system consists of a tube and a bar that fits inside the tube. The sample is placed inside a Teflon sleeve, and the Teflon sleeve is placed inside an aluminum sleeve. The Teflon sleeve is loaded during the test by the stress pulse traveling in the incident tube, producing a radial stress on the sample. The stress pulse traveling in the incident bar loads the sample that is sandwiched between the small-diameter incident and transmission bars. The radial stress is calculated from the hoop strain measured by a strain gauge placed on the outer surface of the aluminum sleeve. Mortar samples have been tested under a range of confining stresses and strain rates, and some of the results of these experiments are presented. 1.

Introduction

Compressive properties and failure modes of many materials, particularly brittle materials, are dramatically affected by the stress triaxiality. This has been demonstrated through various laboratory experiments, since the early work of Bridgman [1] who demonstrates several failure modes peculiar to high pressures, leading to seemingly paradoxical results. The common feature of these paradoxes is that failure always occurs by the formation of tension cracks in specimens subjected to pure compression. While all these paradoxes have been fully understood and resolved (Scholtz et al., [2]; Nemat-Nasser & Horii [3]; Horii & Nemat-Nasser [4]; Horii and Nemat-Nasser [5]) they do emphasize the importance of controlled triaxial experiments in material characterization. Indeed, in axial compression, brittle materials such as rocks or ceramics fail by axial splitting, faulting, or plastic deformation (barreling), depending 3 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 3–12. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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on the relative magnitude of the confining pressure which may accompany the axial compression (Nemat-Nasser & Horii [3]; Horii & Nemat-Nasser [4]; Horii and NematNasser [5]). Model studies show that similar results emerge at high strain rate (NematNasser and Deng [6]; Nemat-Nasser and Hori [7]). The Split Hopkinson pressure bar (SHPB) was invented by Kolsky [8], following the early work of John Hopkinson [9] and his son, Bertram Hopkinson [10,11], and Davies [12], on elastic stress pulse traveling in a single elastic bar. Several investigators (Harding et al. [13]; Lindholm [14]; Lindholm & Yeakley [15]; Follansbee [16]; Hartman et al. [17]; Nicholas and Bless [18]) have since developed new methods for Hopkinson bar testing, including recovery experiments (Nemat-Nasser et al. [19]). A Hopkinson bar can be modified to subject a sample simultaneously to axial and radial compressions (Nemat-Nasser et al. [20]). In the present paper, this technique is described in detail, and the results are illustrated using mortar samples. 2.

Test Apparatus and Procedure

2.1. CLASSICAL SYSTEM A classical split Hopkinson bar (the Kolsky bar) is shown in Figure 1, in its simplest configuration. It consists of a gas gun with a perforated barrel (not shown), a striker bar, an incident bar, and a transmission bar with common cross-sectional area, and Young's Modulus, The sample of length and cross-sectional area is sandwiched in between the incident and transmission bars. Strain gauges are placed on the incident and transmission bars, equidistant from the sample.

A test begins by firing the gas gun that propels the striker bar through the barrel at a prescribed velocity, impacting with the incident bar and producing in it an incident elastic pulse, which travels through the incident bar at the bar's elastic wave speed The corresponding elastic strain in the bar, is measured as a function of time by the strain gauge. Once the incident pulse reaches the sample, part of it is reflected from the sample and travels back through the incident bar, while the remaining part is transmitted through the sample into the transmission bar, as it loads the sample. The reflected elastic strain in the incident bar, and the transmitted one, are measured by the corresponding strain gauge.

DYNAMIC TRIAXIAL COMPRESSION TESTS

5

In a well-designed test, the magnitude of the reflected pulse is proportional to the strain rate in the sample, The sample stress is proportional to the strain in the transmission bar. The sample strain is obtained by integrating the corresponding strain rate with respect to time. In this manner, the stress-strain response of a sample can be measured. The sample strain rate and sample stress are calculated based upon onedimensional elastic waves in the bar,

2.2. TRIAXIAL SYSTEM In order to test a sample in triaxial compression at a high strain rate, several modifications are made to the Hopkinson bar. The modified Hopkinson-bar system now consists of a gas gun with a perforated barrel, a striker bar, incident bar 1, incident bar 2, incident tube, transmission bar and transmission tube (Figure 2).

The sample, which has the same diameter as the transmission bar and incident bar 2, is placed inside of a Teflon sleeve. The Teflon sleeve has roughly the same cross section as the incident and transmission tubes; there is a slight interference fit (0.03mm) between the sample and the Teflon. The Teflon, with the sample inside of it, is placed inside an aluminum sleeve with an inside diameter, and an outside diameter, that can be adjusted for each test. The sample is placed between incident bar 2 and the transmission bar. The Teflon, slightly longer than the sample, fits around incident bar 2 and the transmission bar, and is sandwiched between the incident and transmission tubes. The aluminum sleeve, slightly longer than the Teflon sleeve, fits around the incident and transmission tubes. We have used Teflon and aluminum sleeves in our tests to show the basic approach, but other material may be used as appropriate.

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In the current study, a 19.1-mm Hopkinson bar was used. The system configuration where sample is to be sandwiched between the incident and transmission bars, is shown in Figure 3. The striker bar and the first incident bar are each 27.1mm in diameter. The second incident bar and the transmission bar have a diameter of 19.1mm. The incident and transmission tubes both have an inside diameter of 19.1mm and an outside diameter of 27.1mm.

2.3. TEST PROCEDURE Before the test, the entire Hopkinson bar apparatus is pressed together to ensure that there are no gaps between the contacting bars, the sample or the Teflon sleeve. This is vital to ensure that the radial load is applied at the same time as the axial load, and to ensure that the data collected during the experiment is an accurate measure of the sample response. Incident bar 1 must be in direct contact with both incident bar 2 and the incident tube. The incident tube must be in a tight contact with the Teflon sleeve, which in turn must be in contact with the transmission tube. In our setup, incident bar 2 is slightly longer than the incident tube. Due to the radial expansion accompanying the sample deformation, it is necessary to use a pre-split Teflon sleeve to facilitate sample recovery without any damage (Figure 4). Pushing the Teflon sleeve through the aluminum sleeve easily separates the sample, the pre-split Teflon sleeve and the aluminum sleeve. Since the fit is very important, the Teflon is sliced with a knife instead of being machined. The residual stresses in the Teflon then produce expansion, leading to a tight fit with the sample and the aluminum sleeve. Experiments that follow this technique will simultaneously load the sample in the axial and radial directions, depending on the mechanical properties of the sleeves and the geometry of the setup. Refer to Figures 5a,b.


7

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J. ROME, et al.

The gas gun propels the striker bar (A) through the barrel at a prescribed velocity. The striker bar (A) impacts the first incident bar (B), generating the incident pulse. The incident pulse travels through the first incident bar (B) and the strain it causes in the bar, is measured at the strain gauge. When the incident pulse reaches the end of the first incident bar (B), it continues into the second incident bar (C) and the incident tube (F). The resulting strain in the second incident bar (C) and the incident tube (F) is still When the incident pulse reaches the end of the second incident bar (C), the sample (D) is loaded in the axial direction. Part of the incident pulse, proportional to the sample strain rate, is reflected back into the second incident bar (C); the reflected pulse (being tensile) does not travel into the first incident bar (B) and therefore can not be measured directly. The rest of the incident pulse travels through the sample (D) and generates the transmission pulse that travels through the transmission bar; the strain it generates, proportional to the sample stress, is measured at the strain gauge. Simultaneous with the incident pulse loading the sample, the incident pulse also reaches the end of the incident tube (F) and loads the Teflon (G) as in much the same way as a normal incident bar loads a sample. As the Teflon (G) is loaded between the incident tube (F) and the transmission tube (I), it is restrained outwardly by the aluminum sleeve (H), and inwardly by the sample (G). As a result, a hydrostatic stress is produced in the Teflon (G), which loads the sample (D) in the radial direction. The hoop strain in the aluminum sleeve (H) is measured, and the pressure in the Teflon (G) (equal to radial confining stress on the sample) is calculated. The radial stress can be controlled independently from the axial stress and strain to a limited extent, by, e.g., altering the thickness of the aluminum sleeve to control if and when the sleeve yields, or using a different material for the outer sleeve. The Teflon sleeve could also be replaced by another appropriate material to produce different loading conditions. The simultaneous loading in the radial and axial directions is assured by the design of the bar. The elastic waves in the incident bar and incident tube are generated at the same time. The bar and the tube are made of the same material and they have nearly the same length. Thus, the elastic wave in the incident bar reaches the sample (loading it axially) at the same time as the stress wave in the incident tube reaches the Teflon (loading the sample in the radial direction). 3. Data Analysis With this design, the reflected pulse can not be measured directly. Instead it is calculated as the difference between the incident and reflected pulses. To do this, the incident and reflected pulses must be time-shifted very precisely to ensure reliable results. The constructed reflected pulse is:

The axial strain rate and axial stress in the sample are then calculated according to equations (1).


9

In our tests, 7075 aluminum (Young's modulus, Poisson’s ratio is used as the sleeve to confine the Teflon, with outside diameters of 28.6mm, 31.8mm and 38.1mm, depending on the test requirement. This material is chosen for its high strength (yield modulus, and low degree of hardening. The relation between the hydrostatic pressure in the Teflon (which is equal to the radial confining stress on the sample) and the (measured on the outside surface of the aluminum sleeve) hoop strain, falls into one of three distinct regions. The first is before the aluminum sleeve has yielded. In this region, the stress is found using basic elastic theory for a thick-walled cylinder. The radial stress at the inner surface (the hydrostatic pressure in the Teflon), is:

The second region is when the aluminum sleeve is elastic-plastic. The third and final region is when the entire material has undergone plastic deformation. When plastic deformation occurs in the metal sleeve, additional information on the elasticplastic properties of the material is required in order to calculate the radial stress on the sample; see, for example, Mendelson [21]. 4.

Results and Discussion

Several samples have been tested using this method. The material being studied is a concrete mortar. This material has been chosen because it is known to have a much different response under triaxial loading than it does under uniaxial loading. Under uniaxial compression tests, the elastic modulus has been measured as 32GPa, and the failure strength is 50MPa. If the aluminum sleeve was suspected to have undergone plastic deformation, it was replaced and a new sleeve was used in its place. New Teflon sleeves were used for each test. The samples were tested at average strain rates of about 200/s, 400/s or 600/s; the strain rate during each test was not constant due to the increasing stress in the sample during the experiment. Figure 6 shows a comparison between the response of an unconfined sample and a confined sample. The confined sample carried a much greater load and underwent significantly more strain after the unconfined sample had failed. Figure 7 compares three samples that had the same nominal strain rate (400/s), but each had a different radial stress. It is shown that the greater the confining stress, the greater the axial stress at a given strain. Figure 8 demonstrates that mortar has rate sensitivity at high strain-rates under triaxial compression. In this case, the radial stress is about the same, but the axial stress is higher for the sample with the higher strain rate. Even in these graphs, the effect of the confinement can be seen. As is seen, at about 2.5% strain, the radial stress for sample tested at 200/s increases; at the same time, the axial stress rises relative to the axial stress in the higher strain rate sample.

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5.

11

Concluding Comments

This new method to test materials under triaxial compression at high strain rates is important for a number of reasons. First, the radial stress is applied at the same time as the sample is loaded axially. Since the radial stress is generated by the incident tube compressing the Teflon sleeve, this method does not wholly depend on the dilatancy of samples to create the radial stress, rather, the magnitude of the radial stress can be controlled by proper design. 6.

Acknowledgement

This work has been supported by an ARO MURI DAAH04-96-1-0376 grant to the University of California, San Diego. 7. References 1. 2.

3. 4. 5.

Bridgman, P.W. (1931) The Physics of High Pressure, Bell, London. Scholtz, C.H., G. Boitnott, and S. Nemat-Nasser The Bridgman ring paradox revisited, PAGEOPH, 124, (1986), 587-599. Nemat-Nasser, S. and H. Horii Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst, J. Geophys. Res., 87, (1982), 6805-6821. Horii, H. and S. Nemat-Nasser Compression-induced microcrack growth in brittle solids: Axial splitting and shear failure, J. Geophys. Res., 90, (1985), B4, 3105-3125. Horii, H. and S. Nemat-Nasser Brittle failure in compression: splitting, faulting, and brittle-ductile transition, Phil. Trans. Roy. Soc. Lond., 319, (1986), 1549, 337-374.

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6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19.

20. 21.

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Nemat-Nasser, S. and H. Deng Strain-rate effect on brittle failure in compression, Acta Melallurgica et Materialia, 42, (1994), 3, 1013-1024. Nemat-Nasser, S. and M. Hori (1999) Micromechanics: Overall properties of heterogeneous materials, Elsevier Science Publishers, 2nd edition. Kolsky, H. An investigation of the mechanical properties of materials at very high rates of loading, Proc. R. Soc. Lond. B, 62, (1949), 676-700. Hopkinson, J. Original Papers by J. Hopkinson, 2, B. Hopkinson Ed., Cambridge at University Press, (1901), 316-324. Hopkinson, B., The effects of momentary stresses in metals, Proc. R. Soc. Lond. B, 74, (1905), 498. Hopkinson, B., A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets, Phil. Trans. R. Soc. Lond. A, 213, (1914), 437-456. Davies, R.M., A critical study of the Hopkinson pressure bar, Phil. Trans. R. Soc. Lond. A, 240, (1948), 375-457. Harding, J., E.O. Wood, and J.D. Campbell, Tensile testing of materials at impact rates of strain, J. Mech. Eng. Sci., 2, (1960), 88-96. Lindholm, U.S., Some experiments with the split Hopkinson pressure bar, J. Mech. Phys. Solids, 12, (1964), 317-335. Lindholm, U.S. and L.M. Yeakley, High strain-rate testing: tension and compression, Exp. Mech., 8, (1968), 1-9. Follansbee, P.S., Metals Handbook, 8, American Society of Metals, (1985), 198-203. Hartman, K.H., H.D. Kunze and L.W. Meyer, Metallurgical effects in impact loaded materials, Shock Waves and High-Strain-Rate Phenomena in Metals, Editors: M.A. Meyers and L.E. Murr) (1986), p. 325, New York: Plenum. Nicholas, T. and S.J. Bless, Metals Handbook, 8, American Society of Metals, (1985), 208-214. Nemat-Nasser, S. J.B. Isaacs, and J.E. Starrett, Hopkinson techniques for dynamic recovery experiments, Proc. Royal Soc. London, 435, (1991), 371-391. Nemat-Nasser, S., J. Isaacs and J. Rome, Triaxial Hopkinson techniques, ASM Mechanical Testing and Evaluation Handbook, 8, (2000), 516-518. Mendelson, A., Plasticity: Theory and Application, Chp. 8, A. Mendelson Ed., Macmillan Company, New York, (1968), 156-163.

ASYMPTOTIC SCALING OF GRADIENT THEORY OF MICRO-SCALE PLASTICITY OF METALS

Departments of Civil Engineering and Materials Science Northwestern University Evanston, IL 60208

Abstract The paper deals with the question of asymptotic behavior of Gao, Huang, Hutchinson and Nix’s dislocation-based theory of strain-gradient plasticity on the micrometer scale, which is currently of keen interest for micro-mechanics. Certain peculiar features of the nano-scale extension of the existing theory are identified and possible remedies pointed out. 1. Introduction The recent interest in small-scale extensions of continuum models capturing the transition to atomic lattice models for metallic crystals and polycrystals brings about interesting problems of scaling. In these problems, one can benefit from a formal analogy with the previous studies of scaling in quasibrittle heterogeneous materials such as concrete (e.g., and Planas [1], and Chen [2], [3-5], and Novák [6]). Exploiting such analogy, the small-scale asymptotic scaling properties of the recent theory of Gao, Huang, Nix and Hutchinson [7,8], which is based on dislocations that are geometrically necessary for producing a curvature or twist of the atomic lattice (Fig. 1 top) are examined in the lecture briefly summarized in this paper. A full presentation of the present scaling analysis is given in [9]. 2.

Constitutive Model with Strain Gradients

The constitutive relation proposed by Gao et al. [7,8] and Huang et al. [10] reads:

where 13 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 13–18. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

14

SCALING OF MICRO-SCALE GRADIENT PLASTICITY

Here K = elastic bulk modulus; strains;

15

deviatoric strains, 2nd

and 3rd order tensors of components

displacement curvature (or twist), reflecting the effect of geometrically

necessary

dislocations

(Fig.

1

top)

[the

strain

gradient

volumetric (hydrostatic) part of order stresses work-conjugate to

third-

While Gao et al. (7,8) characterize the plastic

constitutive properties by the semi-empirical formula interesting to consider a slightly more general formula

with positive exponents p and q;

is

yield stress,

it is

stress and strain intensities;

effective strain gradient proportional to the density of geometrically stored dislocations (i.e., to lattice curvature or twist);

classical plastic hardening

function, which reflects the effect of statistically stored dislocations and is an increasing function of monotonically decreasing slope, size of the so-called ‘mesoscale cell’ or the spacing of geometrically necessary dislocations (which is the material length characterizing the transition from standard to gradient plasticity and is interpreted by Gao et al. [7,8] as the minimum volume on which the macroscopic deformation contributions of the geometrically necessary dislocations may be smoothed by a continuum); magnitude of Burgers vector of edge or screw dislocation (e.g., 0.255 nm for copper); positive dimensionless material characteristics expressed in terms of Taylor factor, Nye factor and the ratio of elastic shear modulus to the dislocation reference stress. The principle of virtual work yields the following field equations of equilibrium [7,8]:

The following dimensionless variables (labeled by an overbar) may be introduced:

While the derivatives with respect to are denoted by subscript preceded by a comma, the derivatives with respect to dimensionless coordinates are denoted as

16

To avoid struggling with the formulation of the boundary conditions, one may consider that the applied surface tractions and applied couple stresses vanish at all parts of the boundary where the displacements are not fixed as 0, and that all the loading characterized by nominal stress is applied as body forces whose distributions are assumed to be geometrically similar; is considered as a parameter of these forces having the dimension of stress. As for the loading by applied surface tractions and applied couple stresses, one may consider them replaced by equivalent body forces (proportional to ) that act within a surface layer of a very small thickness proportional to D. 3. Small-Size Asymptotic Scaling

Now the problem of scaling and size effect. For one must recover the standard theory of plasticity, and this is indeed the case. But it is also of interest to check the opposite asymptotic behavior for It can be shown that for the asymptotic form of the field equations is:

where

From this it follows that the asymptotic scaling law is:

where constant, and the exponent According to Gao et al.’s theory, and so (Fig. 1 middle) This asymptotic size effect is curiously strong. It is much stronger than the size effect of linear elastic fracture mechanics (LEFM) for similar sharp cracks on the macroscale, which is It is also possible to determine the asymptotic load-deflection curve for the special case in which the displacement distribution (or relative displacement profile) remains constant during the loading process. This is for example typical of the micro-bending and micro-torsion tests. It can be generally shown that

SCALING OF MICRO-SCALE GRADIENT PLASTICITY

17

In particular, for Gao et al.’s theory, the exponent is 3/2 (see Fig. 2 bottom). Now it is hard to escape noticing that for metals such a behavior is queer. The tangential stiffness of the structure for an infinitely small load is infinitely small, which is physically hard to accept, and then it increases with increasing deflection. It must be emphasized, however, that the small-size asymptotic behavior is approached only for much smaller sizes than those of the available experiments, and at dimensions much smaller than the theory was intended for. Nevertheless, a gradual transition to such odd behavior is doubtless the reason that the comparisons with test data reveal this theory to be too soft for small loads and too stiff for high loads. 4. Closing Comments

Thus it seems that some more fundamental modification of Gao et al.’s (1999) theory might be desirable in order to eliminate the strangely strong small-size asymptotic size effect and the queer small-size load-deflections curve. Some ways to do that are being studied at Northwestern University by Z. Guo. The lecture will also present the scaling properties of the original theory of Fleck and Hutchinson [11,12] and of a new modified theory of Gao and Huang [13]. As found by and Guo, the small-size asymptotic scalings of these theories are as follows:

and

respectively, where is the hardening exponent of the power-law stress-strain curve on the large scale. The size effect in the former is still curiously strong. The size effect in the latter, which is the limiting case of the scaling according to [7,8,13], is of the same type as in the case of similar cracks in linear elastic fracture mechanics. 5.

Acknowledgement

Partial support under U.S. National Science Foundation Grant CMS-9732791 to Northwestern University is gratefully acknowledged. 6.

References

1.

Z. P. and J. Planas, Fracture and Scaling of Concrete and Other Quasibrittle Materials, CRC Press, Bocca Raton, Florida, and London, 1998. Z. P. and E.-P. Chen, “Scaling of structural failure,” Applied Mechanics Reviews, ASME 50 (1997) (No. 10), 593-627. Z. P. Scaling of Structural Strength. Hermes Scientific Publications, Oxford and Paris 2002.

2. 3.

18

4.

5. 6.

7.

8. 9. 10. 11. 12. 13.

Z. P. “Scaling of quasibrittle fracture: Asymptotic analysis,” Int. J. of Fracture, 83 (1997) (No. 1), 19-40. Z. P. “Size effect on structural strength: a review,” Archives of Applied Mechanics (Ingeniur-Archiv, Springer Verlag) 69 (1999), 703-725. Z. P. and D. Novák, D., “Probablilistic nonlocal theory for quasibrittle fracture initiation and size effect. I. Theory. II. Application,” J. of Engrg. Mech., ASCE 126 (2000) (No. 2), 166174, 175-185. H. Gao, Y. Huang, W. D. Nix and J. W. Hutchinson, “Mechanism-based strain gradient plasticity –I. Theory,” J. of the Mechanics and Physics of solids 47 (1999), 1239-1263. H. Gao, Y. Huang and W. D. Nix, “Modeling plasticity at the micrometer scale,” Naturwissenschaften , 86 (1999), 507-515. Z. P. “Scaling of dislocation-based strain-gradient plasticity,” J. of the Mechanics and Physics of Solids – in press. Y. Huang, H. Gao, W. D. Nix and J. W. Hutchinson, “Mechanism-based strain gradient plasticityII. Analysis,” J. of the Mechanics and Physics of Solids, 47, (2000), 99-128. N. A. Fleck and J. W. Hutchinson, “A phenomenological theory for strain gradient effects in plasticity,” J. of Mechanics and Physics of Solids, 41 (1993), 1825-1857. N. A. Fleck and J. W. Hutchinson, “Strain gradient plasticity,” In Advances in Applied Mechanics, vol. 33, J. W. Hutchinson and T. Y. Wu, eds., Academic Press, New York, pp. 295361, 1997. H. Gao and Y. Huang, “Taylor-based nonlocal theory of plasticity,” Int. J. of Solids and Structures, in press.

THE ROLE OF PRESSURE IN THE BEHAVIOR OF TIME-DEPENDENT MATERIALS TED PRODAN, IGOR EMRI Center for Experimental Mechanics University of Ljubljana Ljubljana, Slovenia 1000

Abstract We improved and upgraded the apparatus that Fillers, Moonan, and Tschoegl used several years ago to investigate the influence of pressure and temperature on the mechanical properties of time dependent polymeric materials. The new apparatus can measure the volume and the shear relaxation moduli of solid polymer specimens, subjected to a combination of temperatures from –60°C to +110°C, and pressures from atmospheric to 360 MPa. The paper demonstrates the capabilities of the new apparatus from the scientific as well as engineering stand point. Shear relaxation measurements on poly(vinyl acetate) (PVAc) and styrene-butadiene rubber (SBR) are also presented. For PVAc we present also the bulk creep compliance, coefficient of thermal expansion and the bulk modulus.

1. Introduction Although high isotropic pressure is commonly employed in many demanding structural applications and during several manufacturing processes, there is little information available about its effect on the mechanical properties of polymeric materials. Few instruments exist that can measure the effect of pressure on polymers. This paper describes an apparatus, built at the Center of Experimental Mechanics (CEM) of the University of Ljubljana, for researching the effect of pressure on the time-dependent properties of solid polymers. An extended measurement description is published in another paper [4]. For liquid polymers, Bair built a rheometer that can determine rheological properties of liquid polymer lubricants under high hydrostatic pressure [1]. A detailed account of the history and development of various attempts to describe mathematically the time-dependence of the bulk properties is presented elsewhere [7]. The CEM apparatus is an improved and upgraded version of the apparatus developed originally by Fillers, Moonan, and Tschoegl. The first version [3] measured stress relaxation in uniaxial tension. This was later modified extensively [6] to allow the determination of stress relaxation in shear. The new version, described here, improves the measurement technique in many details and greatly extends the usefulness of the original device. The apparatus now also performs various types of volumetric measurements. One of the more noteworthy features of the new CEM apparatus is that it can perform measurements of shear relaxation either isothermally or isobarically on the same specimen. Thus, it allows the determination of both the temperature shift factors, and the pressure shift factors, required to predict the effect of the 19 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 19–30. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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T. PRODAN, I. EMRI

temperature, or of the pressure, on the mechanical properties of polymeric materials. This is becoming more and more important because of the rather high pressures utilized in the state-of-the-art molding operations. In addition, measurement of the time-dependent properties, not only isothermally but isobarically as well, lead to the removal of the ambiguity in the way the temperature shift factors, are currently determined. The background to this is presented in Section 6 of this paper. This paper describes the CEM apparatus, along with results on two polymeric materials: poly(vinyl acetate) (PVAc) and styrene-butadiene rubber (SBR). The temperature and pressure shift factors were determined for both polymers.

2. The CEM pressure apparatus The assembly is shown schematically in Fig. 1. The pressure is generated by the pressurizing system using silicone fluid. The pressure vessel is contained within the thermal bath, where another silicone oil circulates from the stirred circulator. The apparatus utilizes two separate measuring devices that can be inserted into the pressure vessel: a relaxometer, shown in Fig. 2a, and a dilatometer, shown in Fig. 2b. Signals from the measuring inserts pass through the carrier amplifier prior to being collected in digital format by the data acquisition system (DAQ). The magnet and motor charger supplies current to the electric magnet, which activates the measurements. The same charger also supplies current to the electric motor of the relaxometer, shown in Fig. 2a, which pre-loads the spring that then applies the desired deformation to the specimen. Specimens can be subjected to pressures of up to 360 MPa with a precision of ±0.1 MPa, and to temperatures from –60°C to +110°C with a precision of ±0.01°C. The pressure limit is currently being extended to 850 MPa.

2.1 THE RELAXOMETER The relaxometer insert, shown in Fig. 2a, measures the shear relaxation modulus by applying a constant torsional strain to a cylindrical specimen, and by monitoring the induced moment as a function of time. The two main parts of the insert are the loading device, and the load cell. The loading device, applies the torsional strain by twisting the specimen for a few degrees in less than 0.01 seconds. To effect the deformation of the specimen, the electric motor is used to first pre-load an

APPARATUS FOR PRESSURE AND TEMPERATURE

21

Archimedes spring build into the triggering mechanism. Once twisted, the spring is kept in its pre-loaded position by a rack-and-pawl mechanism. Activation of the electric magnet, mounted outside the pressure vessel (see Fig. 1), pulls the pawl out of the rack, and the energy of the spring deforms the specimen to a pre-determined angle. The angle can range from 1° to 15°. The induced moment is then measured by the load cell, which is attached to the slider mechanism to compensate for possible changes in the length of the specimen resulting from various temperature and pressure conditions. After the shear relaxation measurement is complete, the electric motor brings the specimen to its original undeformed state, while maintaining the pressure vessel fully pressurized. The specimen diameter can range from 2.5 mm to 12 mm, while its length may vary from 50 mm to 58 mm. The relaxometer can measure shear moduli ranging from 0.01 MPa to 10 000 MPa, with an accuracy within ±5% of the modulus value.

2.2 THE DILATOMETER The dilatometer insert is used to measure bulk properties such as the bulk creep compliance, the equilibrium bulk modulus, and the coefficient of thermal expansion. A schematic of the insert is shown in Fig. 2b. The measurements are performed by monitoring the volume change of the specimen, which results from the imposed changes in pressure and temperature, by measuring the change in length of the specimen with the aid of a built-in Linear Variable Differential Transformer (LVDT). The volume estimate can be considered accurate if the change in volume is small (up to a few percent) and the material is isotropic. Specimens for the dilatometer can be up to 16 mm in diameter and 50 mm in length. Measurements are accurate within 0.1% of the specimen’s specific volume, which corresponds to resolutions better than 0.0001 Length dilatometry has several advantages over the mercury confinement volume dilatometry. One of the most important is its high accuracy, which, combined with easily automated measurement procedure, allows tracking of any possible transient volume changes. However, there is also an important limitation that one needs to consider when working at temperatures above Above specimens creep downward due to their own weight. Given the arrangement of the LVDT rod (see Fig. 2b), there will be an additional creep downward caused by the

22

T. PRODAN, I. EMRI

weight of the rod. If the combined creep downward is significant, the measurement will over time considerably underestimate the true volume of the specimen.1

3. Materials and specimen dimensions We used two materials: poly(viny1 acetate) (PVAc) and styrene-butadiene rubber (SBR). Pertinent data are assembled in Table 1. The specimen used for shear relaxation measurements had a diameter of 4.3 mm and a length of 58 mm. The specimen used for volume measurements had a diameter of 12.1 mm and a length of 52.2 mm. Both specimens were cast from the same material batch and were both thermally annealed prior to testing.

4.

Experimental methods

We determined the shear relaxation modulus, G(t), by measuring the decaying moment M(t), after applying a sudden angular displacement, to the cylindrical specimen with diameter at time The modulus is calculated from

On the poly(vinyl acetate) (PVAc) sample we additionally measured the timedependent bulk compliance, B(t), the equilibrium bulk modulus as function of temperature, K(T), and its pressure dependence at the reference temperature using the Murnaghan equation,

where the asterisk indicates that the modulus is referred to zero pressure. The measurements started at the lowest temperature and were continued in increasing temperature order. The specimen was held at each temperature for 2.5 hours before G(t) was measured. The bulk creep compliance, B(t), is the time-dependent relative cubical contraction, v(t), measured in the dilatometer in response to an instantly applied hydrostatic pressure, P, while the temperature is maintained constant. It is obtained from

where

and V(t) is the volume at time t, while

is the original undeformed volume.


5.

23

Experimental results

We first present results on the shear relaxation moduli measured on both materials at different temperatures (i.e., isothermally) and different pressures (i.e., isobarically). The left half of each figure shows either the isothermal or the isobaric measurement segments, while the right half of each figure the mastercurves obtained by shifting the segments into superposition at the chosen reference temperature or pressure. The shift factors, and are shown in the inserts on the right side of each figure.

5.1 DETERMINATION OF THE SHEAR RELAXATION MODULI, G(t) The two materials were subjected to shear relaxation measurements in the CEM relaxometer.

5.1.1. PVAc We begin by presenting the results obtained on PVAc. The left plot in Fig. 3 shows G(t) for PVAc at various temperatures and constant atmospheric pressure. 40°C was chosen as the reference temperature. This temperature is roughly 10°C above the glass transition temperature of PVAc, No vertical correction factors were applied.

A second series of measurements of G(t) were performed at various pressures while maintaining the specimen at the reference temperature. The measurements shown in Fig. 4 started at atmospheric pressure and continued in ascending pressure order, as indicated in the plot on the left side. The specimen was held at each pressure for 2.5 hours before G(t) was measured. The reference pressure was 0.1 MPa.

5.1.2.

SBR

Figures 5 and 6 show the same information for the SBR rubber, In Fig. 5 the reference pressure was 200 MPa. At atmospheric pressure the material was in the

24

T. PRODAN, I. EMRI

rubbery region even at the lowest temperature used. The specimen was first pressurized at room temperature. Then it was cooled down to the lowest experimental temperature where it was equilibrated over night. Afterwards, the measurements were performed at the temperatures in the order indicated in the figure. Between each measurement the specimen was equilibrated for 2.5 hours.

The isobaric experiments shown in Fig. 6 were performed at the reference temperature, The specimen was first cooled to the reference temperature at atmospheric pressure, and then subjected to the pressures in increasing order as indicated in the figure. The glassy region could not be reached because of our 360 MPa pressure limit.


25

5.2. DETERMINATION OF THE BULK COMPLIANCE, B(t) The bulk creep compliance, B(t), is determined by monitoring the volume change of the specimen over time, after a pressure drop is applied, while the temperature is maintained constant. The PVAc sample was first heated to the reference temperature pressurized to the (absolute) pressure ( P = 10.1 MPa ), and then equilibrated in this state overnight. A pressure step function of was then applied by rapidly releasing the pressure to atmospheric, and the change in the volume of the specimen was monitored as function of time. The measurement is shown in Fig. 7.

5.3. DETERMINATION OF THE EQUILIBRIUM BULK MODULUS, K(P) The volume change as function of the pressure was measured in the dilatometer at the reference temperature for pressures ranging from atmospheric (i.e., 0.1 MPa) to 100.1 MPa. The specimen was first heated to the reference temperature at atmospheric pressure, which was then gradually increased in steps of

26

T. PRODAN, I. EMRI

MPa. At each pressure the sample was equilibrated for 2.5 hours before taking the measurements. The results are shown in Fig. 8.

We fitted a quadratic equation to these data in order to calculate the equilibrium bulk modulus as a function of pressure, using the equation,

The result is presented in Fig. 9.

Fitting the Murnaghan equation, Eq. (2), to this data we obtained the parameters MPa and which define the pressure dependence of the equilibrium bulk modulus of PVAc at zero pressure and at the reference temperature


27

5.4. MEASUREMENTS OF THE THERMAL EXPANSION COEFFICIENT OF PVAc The volumetric thermal expansion coefficient was measured in the dilatometer at the two different pressures, 0.1 MPa and 50 MPa. The specimen was first pressurized to the experimental pressure, and then heated to the experimental temperatures shown in the figure. After being exposed to these conditions for 2.5 hours, the measurements were taken. The results are shown in Fig. 10. The thermal expansion coefficient was then calculated from

6.

Background to the WLF and FMT equations

We present here the theoretical background that is required to understand the advantage, or more appropriately, the need to carry out time-dependent measurement on a linear viscoelastic material, both isothermally and isobarically, if its mechanical behavior is to be fully characterized. The temperature dependence of the mechanical properties of a polymeric material is commonly described by the well-known WLF (Williams, Landel, Ferry) equation

where an are material constants. This equation is derived from the Doolittle equation, which may be written in the form [2]

where is the fractional free volume, is the fractional free volume in the reference state, and B is a proportionality factor. The free volume is a function of temperature. Letting

28

T. PRODAN, I. EMRl

where is the expansion coefficient of the fractional free volume and chosen reference temperature, leads to the equation,

is the

Introducing the constants

and

leads to Eq. (6). The parameters and are obtained experimentally by shifting isothermal segments of any linear viscoelastic response curve into superposition with the reference segment at Thus we have only the two experimentally determined parameters, and to represent the three parameters of the Doolittle equation, B, and Thus, the WLF equation is underdetermined unless either B or is known. To deal with this predicament, it is customary to set the bothersome proportionality factor B to unity. One then obtains

and

However, as had been shown already by Moonan and Tschoegl (1985), and is confirmed in this paper, B is generally not equal to unity. These authors have also shown that by carrying out isobaric as well as isothermal measurements as function of time the ambiguity can be removed. Measurements of isobaric as well as isothermal time dependence lead to the FMT (Fillers, Moonan, Tschoegl) equation

where

and


29

The first superscripts on the c’s refer to the reference temperature, The second superscripts refer to the reference pressure, and are the Murnaghan parameters in the equilibrium (rubbery) state. The subscript identifies the same parameters for the occupied volume. An asterisk indicates that the bulk modulus data are referred to zero pressure. and are determined in the CEM apparatus using the dilatometer and therefore the constant is known. For the determination of the remaining c’s see [5]. The FMT parameters and are obtained by solving the Eqs. (14-16) and (18-19). At the reference pressure vanishes and we regain the WLF equation. However, since we now have information on B, and may be found from the equations

and

7.

Discussion

The primary purpose of this paper is to demonstrate the capabilities of the CEM apparatus. As could be gathered from Section 5, in its present form, the apparatus is able to determine G(T, P, t), B(T, P, t), K(P,T), and . It is a noteworthy feature that measurements can be made under isothermal as well as isobaric conditions. Such simultaneous measurements prove to be indispensable for the unambiguous determination of the semi-molecular parameters as well as of and These parameters are needed as input into computer codes for numerical analysis of technological processes and the stress analysis of structural elements. As discussed in Section 6, isothermal measurements often lead to incorrect values for these parameters. For the four materials we investigated, this is shown in Table 2. For each material there are two columns. The first, denoted as T, indicates isothermal measurements only, while the second, denoted as T&P, indicates measurements performed isothermally as well as isobarically. Table 2 clearly shows that the assumption that leads to the wrong estimation of and For PVAc the values of the two parameters calculated on the assumption that are underestimated, while for SBR they are overestimated. In both cases the difference is definitely not acceptable for any engineering application.

30

8.

T. PRODAN, I. EMRI

Conclusions

Finally, we conclude that the CEM apparatus is a highly useful tool for acquiring the necessary materials input data for the numerical optimization of processing technologies, as well as for the long-term prediction of the geometric stability and durability of structural elements composed of polymers and composites.

9. Acknowledgement We wish to point out that our endeavors are a continuation of the work initiated by Professor N. W. Tschoegl of the California Institute of Technology. We are grateful for his abiding interest and guidance.

10. References 1. 2. 3. 4. 5.

6. 7.

Bair S. (1996) Normal stress difference in liquid lubricants shared under high pressure, Rheol. Acta, 35, 13-23. Ferry, J.D. (1983) Viscoelastic Properties of Polymers, 3rd ed., John Wiley and Sons, 285-286. Fillers, R.W. and Tschoegl, N.W. (1977) The Effect of Pressure on the Mechanical Properties of Polymers, Trans. Soc. Rheol., 21, 51-100. Kralj, A., Prodan, T., Emri, I (2001) An Apparatus for Measuring the Effect of Pressure on the Time-Dependent Properties of Polymers, J. of Rheology, 45/4, 929-943. Moonan, W.K. and Tschoegl, N.W. (1983) The Effect of Pressure on the Mechanical Properties of Polymers 2. Expansivity and Compressibility Measurements, Macromolecules, 16, 55-59. Moonan, W.K., and Tschoegl, N.W. (1985) The Effect of Pressure on the Mechanical Properties of Polymers. 4. Measurements in Torsion, Polymer Sci., Phys. Ed., 23, 623-651. Tschoegl, N.W., Knauss, W.G., Emri, I. (2002) 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-Dep. Mater, 5/1 (in print).

HIGH STRAIN RATE TESTING OF SANDWICH CORE MATERIALS M. VURAL Graduate Aeronautical Laboratories, California Institute of Technology Pasadena, CA 91125 Department of Aeronautical Engineering Istanbul Technical University Maslak, Istanbul, 80626, TURKEY G. RAVICHANDRAN Graduate Aeronautical Laboratories, California Institute of Technology Pasadena, CA 91125

Abstract

Mechanical response of a cellular sandwich core material, balsa wood, is investigated over its entire density spectrum from 55 to Specimens were compression loaded along the grain direction in both quasi-static and dynamic strain rates from to Results show that while the initial failure stress is very sensitive to the rate of loading, plateau (crushing) stress remains unaffected by the strain rate. Kinematics of deformation and micro inertial effects are suggested and discussed to explain this different behavior. Specific energy dissipation capacity of balsa wood was measured and determined to be comparable with those of fiber-reinforced polymers. As in quasistatic loading, buckling and kink band formation were identified to be two major failure modes in dynamic loading as well. 1. Introduction

Sandwich structures, which are commonly used to increase structural stiffness, are also used in critical applications as energy dissipators. In these applications, energy is transmitted to the system mostly by the impact of a foreign object and dissipated through the compressive inelastic deformation of the core material used in the sandwich structure. Flow (crushing) stress of the core material determines the load transmitted to the system during dissipation process and the deceleration of impactor as well. Thus it is desired to be kept at an optimum level, which is not too high to damage the container and its contents and not too low to dimmish the amount of energy dissipated. Once the flow stress is established the stroke of deformation (or the maximum strain before stress elevation) determines the amount of energy that can be dissipated safely. Therefore, the constitutive response of an ideal energy dissipator is that of a rigid-perfectly plastic 31 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 31–42. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

32

M. VURAL, G. RAVICHANDRAN

material capable of tolerating as large strains as possible. Within this frame balsa wood, as a naturally occurring porous biocomposite, offers salient mechanical and physical properties. As its cellular/porous microstructure allows the application of a long deformation stroke, fine composite nano-architecture of wood cell material increases its specific strength and stiffness, giving rise to a high specific energy dissipation capacity. Moreover the fact that balsa wood can be found in a wide range of densities from 40 to 380 , depending on the average size and the wall thickness of cells, provides the flexibility in design since the strength is a monotonic function of its density. As shown in Fig.1, the cellular microstructure of balsa (Ochroma) is mainly composed of long prismatic cells (fibers, grains) of nearly honeycomb shape and the blocks of these cells are separated by narrower rays in which the cells are smaller and of a different shape. Balsa wood has three orthogonal axes in the longitudinal (L, along the grain), radial (R, across the grain and along the rays) and tangential (T, across the grain and transverse to rays) directions forming a heterogeneous porous composite. This composite is highly anisotropic with a high ratio of longitudinal to transverse properties, the latter having also difference in itself due to so-called ray cells oriented in radial direction. For applications where balsa wood is considered as a potent material to dissipate energy, its dynamic constitutive response in longitudinal direction as well as corresponding failure mechanisms are of particular interest.

Numerous uniaxial quasi-static tests have been performed to determine the mechanical properties and deformation mechanisms of balsa wood in longitudinal and/or transverse directions by Knoell [1], Soden and McLeish [2], Easterling et al. [3] and Vural and Ravichandran [4]. Knoell [1] investigated the effects of environmental and physical variables (temperature, moisture content and ambient pressure) on the mechanical response of balsa wood. Soden and McLeish [2] carried out extensive tests and mainly concentrated on the variation of tensile strength with fiber alignment. They

TESTING OF SANDWICH CORE MATERIALS

33

also reported compressive strength data. Easterling et al. [3] paid particular attention to the micromechanics of deformation in their experiments, during which they performed in-situ SEM observations and defined the end-cap collapse of grains as the dominant compressive failure mechanism in longitudinal direction. Vural and Ravichandran [4] documented the compressive strength, plateau stress and densification strain of balsa wood in its entire density range, identified the variations in failure mechanisms with density and described simple analytical models to represent experimental strength data. On the other hand, dynamic compression behavior of balsa wood (and in general wood) has received limited attention. Daigle and Lonborg [5] performed dynamic compression tests over balsa wood and several other crushable materials using drop towers and compared their specific energy absorption capacities. However, a systematic investigation on the dynamic response of balsa wood for different strain rates and specimen densities that covers its full range is still lacking. In the present study, the quasi-static and dynamic response of balsa wood is investigated at strain rates from to with a particular emphasis on the energy dissipation characteristics and their dependence on the density and strain rate. The dominant deformation modes at micro-scale during the process of energy dissipation (crushing) and their dependence on density are discussed and documented through the examinations performed on the sectioned surfaces of recovered specimens by scanning electron microscope (SEM). 2. Experimental Procedure 2.1. SPECIMENS

Experiments were performed on balsa wood specimens over a wide range of densities between 55 and _ Cylindrical specimens were machined from big blocks of balsa wood at five different nominal densities provided by BALTEK Corporation (Northvale, NJ) and polished using 320 grit sand papers. Densities were measured in a consistent way, paying considerable attention particularly to the moisture content since it has significant effect on both density and properties [6]. Due to the variation of density within each block of a certain nominal density, specimens were prepared with densities varying continuously between the ranges specified above. In order to find moisture contents of balsa specimens a series of specimens were heat-treated in furnace at 110 °C and intermittent weight measurements were performed until they reached a constant weight. Measured moisture content ranged from 8 to 11 %, which is the typical value for well-seasoned dry woods. 2.2. QUASI-STATIC TESTS

The specimens used in quasi-static tests were typically 19.05 ± 0.12 mm in diameter and 25.4 ± 0.12 mm in length, corresponding to a length to diameter ratio of 1.33. A compression fixture was used in the quasi-static compression tests. It ensured that two loading rods were perfectly aligned with each other so that any unwanted shear forces on the specimen are minimized. The specimen was sandwiched in between the loading rods and the compression was applied by a screw-driven materials testing system

34


(Instron, model 4204). The quasi-static tests were performed at an overhead displacement rate of 2 mm/min, which corresponds to nominal strain rate of The use of bonded resistance strain gages on highly compliant balsa specimens was not considered satisfactory because the reinforcing effect of bonding these gages with adhesive might lead to large errors in measurement of strains. Therefore, deformation data of specimens was obtained from the crosshead displacement transducer, which was calibrated to eliminate the machine and fixture compliance. 2.3. DYNAMIC TESTS For dynamic tests, cylindrical specimens of a smaller dimension, 9.5 ± 0.12 mm in diameter and 5.08 ± 0.12 mm in length, were loaded using a modified Kolsky (split Hopkinson) pressure bar as shown in Fig. 2. Kolsky pressure bar, originally developed by Kolsky [7], has been widely used and modified to measure the dynamic compressive behavior of engineering materials. The present modification through the use of a quartz single crystal, first introduced by Chen et al. [8], enables stress measurements three orders of magnitude more sensitive than in a conventional split Hopkinson pressure bar. Hence, low stress profiles experienced by balsa specimens could be measured with high precision by adopting this modification.

The dimensions of the bars in Kolsky pressure bar setup used in this study are 1219, 584 and 610 mm in length for the incident and transmission bars respectively, with a common diameter of 12.7 mm. The striker bars of 12.7 mm diameter were used with varying lengths to achieve desired pulse duration. All the bars are made of 7075T651 precision ground aluminum alloys. Depending on the striker velocity, either a thin, annealed copper (C11000) disc of 4.8 mm in diameter and 0.81 mm in thickness or a layer of tissue paper was typically used as pulse shaper to control the rise time of the incident pulse and to ensure stress equilibration within the specimen [9]. A high resistance strain gage (Micro-measurements, WK-06-250BF-10C) with excitation voltage of 30 volts was used to measure the surface strain on incident bar. The raw strain gage signal was recorded using a high-speed 12-bit digital oscilloscope


35

(Nicolet 440). A Valpey-Fisher X-cut quartz single crystal of 12.7 ± 0.025 mm in diameter and 0.254 ± 0.025 mm in thickness, coated with a 30 nm thick chrome/gold layer on both of the flat surfaces, was glued to the end of the specimen-contact half of the transmission bar using a conductive epoxy (CW2400, Chemtronics). Other surface remained in contact with the other half of the transmission bar without bonding, allowing the compressive transmitted pulse to pass the quartz without reflection. Before each experiment, a thin layer of conductive grease (CW7100, Chemtronics) was applied at the contact surface to ensure electrical conductivity. Thin, silver coated copper wires were wrapped and glued (using the same conductive epoxy) to the bar surface near the quartz crystal. The wires were connected to a charge amplifier (Kistler 5010 B1) whose output is recorded using the same oscilloscope for data processing. Using the modified Kolsky pressure bar system described above dynamic compression tests were performed at nominal strain rates between and 3.


3.1 MECHANICAL RESPONSE

In all experiments, specimens of balsa wood were loaded in longitudinal (L) direction (i.e., along the grain). Characteristic features of the typical engineering stress-strain curves for balsa wood obtained from these tests in both quasi-static and dynamic range are as shown in Fig. 3. The stress-strain curve is almost linear up to a maximum stress (failure strength) beyond which, as the deformation is increased, the stress level either remains nearly constant or experience a sudden drop depending mainly on the density of specimen. After this sudden drop, if there is, the specimen continues to deform at a lower level of stress (plateau stress). Eventually, at high strains (after the wood cells collapse sufficiently so that opposing cell walls touch (or their broken fragments pack together) and further deformation compresses the cell wall material itself. This gives the final, steeply rising portion of the stress-strain curve called densification regime.

36


Figure 4 illustrates the dependence of compressive failure strength and plateau stress, respectively, on the density and strain rate for balsa wood. Solid lines in Fig. 4a represent the best linear fits to the experimental data. It is obvious that both the density and the strain rate have considerable effects on the failure strength of balsa wood. Higher density means increased ratio of cell wall thickness to cell diameter, and thus less porosity, therefore its effect towards fortifying the structure is simply expected and experimentally observed in both failure strength and plateau stress. However the effect of strain rate seems to be more complicated due to the fact that the plateau stress is practically insensitive to the strain rate whereas the failure strength is significantly affected by the increase in strain rate. In other words, experimental evidence indicates that as the critical stress for failure initiation has a strong dependence on the strain rate driving force for progressive deformation remains unaffected by the dynamics of deformation. This different response to increasing strain rate can simply be explained in terms of the differences in the kinematics of deformation, associated inertia and triggering mechanism at micro scale prior to initial failure and during progressive deformation. Initial failure at the peak stress level is preceded by elastic loading. Micromechanics of deformation in this regime is characterized by the uniform axial compression of wood cells in longitudinal direction. Initial failure is mostly triggered by an elastic instability in the form of local buckling or kink band formation depending mainly on the density of balsa wood, as will be discussed in the next section. In dynamic loading, both kind of instability is associated with the lateral inertia in deforming elements, which in turn serve as increasing the critical stress for failure initiation in longitudinal direction. Simple calculations based on the kinematics of buckling/bending geometries show that even though the initial effect of this inertial hardening is very strong it rapidly falls down and becomes negligible during the evolution of deformation in a localized zone. However, progressive deformation phase is also characterized, in addition to the continuation of deformation in initially localized regions, by the creation of new localization zones. Following the initial failure/localization, the state of stress experienced by neighboring cells is perturbed and far from being uniaxial due to the kinematics of deformation at micro scale. This new local situation (state of stress) makes the following progressive deformation (localizations) easier and at a lower nominal stress level by taking and exaggerating, in a sense, the role of local geometric and material imperfections that trigger the initial failure. Within this frame, regarding the formation of new strain localizations, it appears that the softening effect due to the perturbations in stress field created by preceding localization, suppress the effect of inertial hardening. As conclusion the effect of strain rate, and hence micro inertia, turns out to be restricted to the initiation phase of first instability due to the kinematics of failure (buckling/kink banding). During the following progressive deformation phase the effect of inertia is suppressed either by the kinematical considerations, as in the evolution of deformation within localized region, or by the perturbations in stress field as in the creation of new localization sites. Figure 4b also shows an increased scatter in plateau stress when the specimen density exceeds 170-190 interval. This is attributed to the transition in deformation mode from buckling to kink band formation and resulting increased susceptibility of plateau stress to the random nature of increased perturbations in stress field. As the length scale of the deformation geometry increases from a fraction of cell


37

diameter in buckling mode to as much as 10-15 times the cell diameter in kink band formation mode, an increased level of perturbation is induced in the stress field and it generates the increased scatter in plateau stress for denser specimens.

Another aspect of stress-strain response is the significant decrease observed in densification strain with increasing density and strain rate (see Fig. 5), which is defined in this study as the strain at the last local minimum before the stress starts rising steeply. For quasi-static loading rates, densification strain decreases as a function of

38


specimen density and varies between 0.68 and 0.86. As expected, the denser is the balsa wood the lower is the densification strain due to its close correlation with the level of initial porosity. In dynamic loading regime, the same declining linear trend in densification strain is preserved. However, it is translated downwards and the bounds decrease to the range between 0.58 and 0.74, suggesting that cellular packing efficiency at microstructural level degrades by the increasing rate of strain. This could be attributed to the micro-inertial effect associated with the cell wall collapse.

Plateau (crushing) stress and densification strain are two important quantities that directly determine the energy dissipation capacity of the material under consideration. Specific energy dissipation capacity of balsa wood is shown in Fig. 6 for both unit volume and unit mass. Data of Fig. 6 were obtained by integrating the area under stressstrain curves, rather than simply multiplying plateau stress and densification strain, up to the strain schematically denoted by in Fig. 2. Therefore, the data presented represent the maximum amount of energy that can be dissipated safely, i.e., by limiting maximum transmitted force to the one associated with the failure strength of balsa wood. The effect of transition in progressive deformation mode from buckling to kink band formation is apparent in Fig. 6 either in the form of plateau region (volume based data) or in the form of a temporary decrease (mass based data). When considered together with the decrease in densification strain with density (Fig. 5), slight leveling of plateau stress (see Fig. 4) during this transition results in the obvious leveling of the energy dissipated per unit volume (Fig. 6). The effect of increasing density in this


39

transitive plateau region can be seen as a transient drop in the energy dissipated per unit mass. As a whole the energy dissipation capacity of balsa wood lies between 30-90 kj/kg, which is comparable or better than the most fiber reinforced composite tubes. Mamalis et al. [10] report in their review for the crashworthy capability of composite material structures that specific energy dissipation/absorption capacities of fiber reinforced hallow tubes made of Carbon/Epoxy, Aramid/Epoxy, Glass/Epoxy and Carbon/PEEK lie between 50-99, 9-60, 30-53, 127-180 kj/kg respectively, variations arising mostly from the fiber orientation angle and stacking sequence. Actually balsa wood owes its remarkable energy dissipation capacity to its naturally tailored microand nano-structural features such as the tubular honeycomb geometry of its cells and the fine nano-architecture and composite nature of its cell walls.

Figure 6 also shows the effect of elevated strain rate on the energy dissipation capacity. Even though there appears a slight increase in dynamic data for low densities it is highly scattered and intermixed with the quasi-static data for higher densities. Once again this phenomenon is attributed to the transition in deformation mode and, thus, to

40


the increased effect of random stress perturbations on the plateau stress level for high densities. For low-density balsa the cells deform by buckling. Since the length scale of local deformation is limited to the length of buckles, which is much smaller than that in kink banding mode, stress perturbations are also small and therefore plateau stress benefits from micro inertial effects. However, with the transition in deformation mode, the length scale involved becomes higher and, therefore, induces larger stress perturbations which, in turn, both suppress the inertial effects and result in a wider scatter. 3.2. FAILURE MODES In order to reveal microstructural failure modes some of the balsa specimens are first compressed to predetermined strains and then carefully sectioned to perform SEM examination on the failure surfaces. The SEM observations on recovered specimens from both quasi-static and dynamic tests clearly show that the failure mode undergoes transition from elastic/plastic buckling of cell walls to kink band formation as the density of the specimen tested increases. The micrographs of two specimens showing the typical dynamic deformation mode in low- and high-density balsa wood are presented in Figure 7. The rate of loading does not change the deformation mode. Similar micrographs for quasi-statically loaded specimens can be found in Ref. [4]. The major mechanism during both initial failure and progressive deformation is buckling for low-density balsa wood. When the specimen density goes beyond 170-190 range, thickness to diameter ratio of tubular cells increases and the critical stress for buckling becomes too high so that another mechanism (kink band formation) starts operating. Initiation of this new deformation mode is very sensitive to and triggered by initial fiber (cell, grain) misalignments which are naturally present in the microstructure. Therefore, the appearance of kink band formation in denser specimens is attributed to the increase in fiber misalignments caused by the thicker ray cells penetrating the entire microstructure in radial direction. Ref. [4] reports the experimental evidence of larger fiber misalignments in denser balsa wood as well as analytic models that explain the transition in failure modes. 4. Conclusions The mechanical behavior of balsa wood over its entire range of densities has been investigated under quasi-static and dynamic loading conditions. In both cases, the compressive failure strength and plateau stress of balsa wood increase as a function of its density while the densification strain decreases. Even though the application of high loading rates elevates the initial failure stress as much as 1.5 to 2 times, plateau stress remains unaffected by the strain rate. This different response is attributed to and explained by the kinematics of deformation and associated micro inertial effect. Specific energy dissipation capacity (per unit mass) of balsa wood, as a naturally occurring material, was found to be comparable with advanced polymeric composites. The post-test SEM examinations on dynamically loaded specimens reveal that failure mode undergoes transition from elastic/plastic buckling to kink band formation with increasing density, as also observed in quasi-statically loaded specimens.


5.

41

Acknowledgements

This research was supported by the Office of Naval Research (Dr. Y. D. S. Rajapakse, Scientific Officer) and is gratefully acknowledged. MV gratefully acknowledges the support provided by (The Scientific and Technical Research Council of Turkey) through the NATO Advanced Science Fellowship.

42


6. References 1. 2.

3. 4. 5. 6.

7. 8. 9. 10.

Knoell, A.C.: Environmental and physical effects of the response of balsa wood as an energy dissipator, JPL Technical Report No. 32-944, California Institute of Technology, Pasadena, CA, 1966. Soden, P.D., McLeish, R.D.: Variables affecting the strength of balsa wood, J. Strain. Anal. 11(4), (1976), 225-234. Easterling, K.E., Harrysson, R., Gibson, L.J., Ashby, M.F.: On the mechanics of balsa and other woods, Proc. R. Soc. Lond. A 383, (1982), 31-41. Vural, M. and Ravichandran, G.: Microstructural aspects and modeling of failure in naturally occurring porous composites, Mech. Mater., (2001), submitted for publication. Daigle, D.L., and Lonborg, J.O.: Evaluation of certain crushable materials, JPL Technical Report No. 32-120, California Institute of Technology, Pasadena, CA, 1961. Dinwoodie, J.M.: Timber – a review of the structure-mechanical property relationship, Journal of Microscopy 4(1), (1975), 3-32. Kolsky, H.: An investigation of mechanical properties of materials at very high rates of loading, Proc. Phys. Soc. Lond. Ser. B 62, (1949), 676-700. Chen, W., Lu, F., and Zhou, B.: A quartz-crystal-embedded split Hopkinson pressure bar for soft materials, Experimental Mechanics 40(1), (2000), 1-6. Ravichandran, G., and Subhash, G.: Critical appraisal of limiting strain rates for compression testing of ceramics in split Hopkinson pressure bar, J. of the American Ceramic Society 77(1), (1994), 263-267. Mamalis, A.G., Robinson, M., Manolakos, D.E., Demosthenous, G.A, Ioannidis, M.B. and Carruthers, J.: Review: Crashworthy capability of composite material structures, Composite Structures 37, (1997), 109-134.

DEVELOPMENT OF A SHEAR TEST FOR LOW MODULUS FOAM MATERIALS AJIT K. ROY Air Force Research Laboratory ARFL/MLBC, Wright-Patterson AFB OH 4543 3-7 750 JOHN D. CAMPING University of Dayton Research Institute Dayton, OH 45469-0168 Abstract The shear properties of carbon (graphitic) foam, being brittle and highly porous, are not reliably measured with most of the standard test methods, such as, single rail, double rail, Iosipescu shear, etc. A new testing device is developed to accurately measure the shear stiffness and strength of carbon foam or other porous materials. Specimens of cylindrical cross-section are used to reduce high stress concentration that normally occurs in the vicinity of the grip section. Since strain gages could not be installed on the specimen surface (due to porosity), the shear strain is determined from the specimen end rotation. A high accuracy in the rotational measurement is achieved by using a stepper motor with multiple gear reduction. In view of testing low modulus material, the load cell of fixture was mounted on an axial roller to relieve axial constraint while twisting the specimens. The accuracy of the measurement and calibration of the test fixture was demonstrated by measuring shear modulus of PVC plastic.

1. Introduction The intrinsic superior mechanical and thermal properties of carbon material (carbon fiber, in particular) than metals make carbon (graphitic) foam an attractive ultralightweight material for space and thermal management applications. Its process tailorability makes it work as an insulator and a radiator as well. Its macroscopic near isotropic property makes it an attractive core material for sandwich structures over traditional honeycomb. The material also a good candidate for net shape fabrication of structural components providing three-dimensional reinforcements through its ligaments. The micrograph of a graphitic (open-cell) foam microstructure is shown in Figure 1. Based on the minimum surface energy during the foaming process (i.e., the bubble nucleation process), the microstructure of an open-cell foam usually possesses a tetrahedral structure of the foam ligaments oriented approximately 109° with each other. Due to the tetrahedral cell microstructure, the macroscopic properties, such as foam modulus and strength, are critically influenced by the deformation characteristics of the cell ligaments. Thus, to develop a basic understanding of the performance of open-cell foam materials, as a first step, there 43 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 43–54. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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A. K. ROY, J. D. CAMPING

is need to measure its bulk (macroscopic) properties and then correlate the bulk properties with the foam ligament microstructure.

There is an abundance of literature on modeling foam properties, but very limited on measuring foam properties in the contrary. Among the modeling efforts, the most widely used reference is that of Gibson and Ashby [1]. Christensen [2, 3] developed a few closed form relations for bulk properties of foam assuming membrane behavior of foam ligaments. Warren and Kraynik [5] incorporated foam ligament bending in their model. Zhu, et at [6, 7] analyzed the elastic properties and compression characteristics of open cell foam with tetrakaidecahedral cell geometry. Although some effort has been undertaken to measure foam properties in the past, such as the work of Kau et al. [8, 9] and Bilkhu [10], a comprehensive effort of correlating foam ligament structure with foam bulk properties appears to be lacking to assess or validate the existing foam models. Christensen [4] also reiterates the need for comprehensive experimental efforts for measuring foam properties in his recent review paper. The foam material considered here contains porosity of around 80 percent or even higher, with pore size of in diameter (Figure 1). In general, foam being a highly porous and low modulus (about one order of magnitude lower than that of standard PVC plastic) material, traditional strain measuring devices (such as, surface mount strain gages) fail to work reliably in measuring its strains. Surface mount strain gages are not compliant enough to measure the true strain of foam. Strain gages tend to add stiffness to this type of material; hence its apparent measured strain is lower than the strain in the material. Clip extensometers, another strain measuring device used in many engineering materials, do not work with foam due to open material porosity and lack of fixed extensometer clipping point on the material surface. Further, measuring shear strain even gets more difficult in porous materials like foam. The objective of this study is to develop a test method to

SHEAR TEST FOR FOAM MATERIALS

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measure the shear modulus and strength of foam. In developing the test method a careful attention is given on the low load requirement and the shear strain measurement accuracy, for measuring shear modulus and strength of foam (porous) materials.

2. Test method considerations for shear properties of carbon foam There are several standard shear tests, such as Single Rail, Double Rail (ASTM Standard D 4255), and Iosipescu, (ASTM Standard D 5379) which are used in measuring shear properties of engineering materials [11]. The Single and Double Rail shear tests are not suitable for measuring low strength, brittle and porous materials, such as foam, because the adhesive that is used to bond foam to the rails tend to ingress through the foam pores and induce an artifact in the test data. The present ASTM standard for measuring shear properties of sandwich core materials (ASTM C 273-94), using the single rail configuration suffers from the above deficiency [11]. In addition, in order to eliminate the effect of stress concentration due to bonding of the rails with the specimen, specimen dimension (width and length) in these two test configurations need to be sufficiently large. Such large specimen configurations, in general, are difficult to meet in testing new materials like carbon foam, when these materials in its developmental stage are produced in small quantities.

It is known that Iosipescu shear test configuration requires a relatively small specimen compared to that of single or double rail shear. Thus, Iosipescu shear test was initially performed to measure the shear strength, although the test configuration is not suitable for measuring foam shear modulus. The standard Iosipescu shear test specimens were machined with v-notches at the specimen midlength (see ASTM Standard D 5379, [11]). In order to prevent a premature failure under the loading pins the loading faces of the specimens were reinforced by bonding tab materials, as shown in Figure 2. Even with reinforced loading faces specimens mostly failed near the face-tab ends and under the loading pins, as shown in Figure 2. This undesirable failure mode may be attributed to high porosity and the brittle nature of carbon foam. Thus, Iosipescu shear test appeared not to be suitable for highly porous and brittle carbon foam materials.

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3.


Miniature torsion fixture design and calibration

The difficulty in generating a desirable failure mode in Iosipescu shear specimens may be attributed to excessive stress concentrations of the foam ligaments in the vicinity of loading rollers. Thus, a cylindrical specimen configuration (for example, circular rods) subjected to an axial torsional load is considered to be a preferred test configuration for this material to measure its shear properties. The schematic configuration of the specimen is shown in Figure 3, as well as, the foam specimen with slotted end tab is shown in Figure 4.

The cylindrical specimen of applying torque at the specimen ends, in contrast to Iosipescu, Double rail and Single rail shear tests, significantly reduces the stress concentration at the loading locations, i.e., the specimen ends. A special miniature torsion fixture was designed to measure the shear stiffness and strength of the material, as shown in Figure 5 [12]. Due to low bulk modulus of carbon foam (generally, one order of magnitude lower than that of PVC plastic), a careful attention was given to the low load requirement in designing the test fixture. Since strain gages could not be installed on the specimen surface, shear strain was determined from specimen end rotation with a special design to obtain an angular rotational resolution of 0.005 degrees. Further, to relieve the axial constraint while twisting (applying torque) the specimen, the specimen-gripping end attached to the load cell is mounted on a linear roller. In addition to testing foam, this miniature torsion tester is also designed to allow the convenient and accurate torsional testing


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of small samples of various low stiffness materials. The fixture consists of several components described separately as follows.

General Configuration: The test machine consists of a torsion drive section and a reaction section that are both mounted on an aluminum base. The torsion drive section uses a computer-controlled stepper motor to apply a torsional displacement to the material under test. The reaction section contains a torque load cell that measures the applied torque (see Figures 6(a), 6(b)).

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Support Base: The base is made of a 76.2x44.5x6.35 mm (3" x 1.75" x 0.25") aluminum channel with 12.7 mm (0.5") thick aluminum end plates. The overall size of the base is 127 x 343 x 76.2 mm (5" x 13.5" x 3"). The main consideration in constructing the base is that it be able to resist the torque produced by the machine without significant deflection. The top surface of the base is machined flat to provide a stable platform for the drive and reaction sections. Drive Section: The drive section is constructed on a 89 x 165 x 12.7 mm (3.5" x 6.5" x 0.5") aluminum plate, which is bolted to one end of the base. The drive itself consists of a Frame 23-stepper motor with a 30-oz capacity geared to the output shaft through two sets of gear reduction. The first gear reduction is a 3:1 timing belt drive that drives a 60:1 worm gear reducer to the output shaft for a total reduction of 180:1. All elements of the torsion drive are mounted in precision ball bearings to reduce friction in the system as much as possible. The worm drive consists of a hardened and ground steel worm driving a brass worm gear. The output shaft is a 6.35-mm (0.25") diameter steel shaft aligned parallel to the long axis of the base. All of the mechanical parts of the drive section are protected by an extruded aluminum cover. Angular Measurements: Provisions have been made in the drive section for either a potentiometer or a shaft encoder to be added to the drive unit to measure the angular deflection applied to the specimen. The current version accomplishes this measurement in the control software by simply counting the number of steps applied to the motor. The capacity of the motor coupled with the high torque multiplication due to the gear reduction ensures that the motor will never "stall" so the software calculation of the applied angle will always be accurate to within one step. The stepper motor is a standard 200-step-per-revolution motor so the final resolution of the drive is: Steps/Degree (or 0.01 Degree/Step) Since the control software operates the motor in "half-step" mode, this resolution doubles to 200 Steps/Degree or 0.005 Degree/Step step Radian/Step). This resolution was deemed satisfactory for the preliminary testing. Reaction Section: The reaction section consists of a reaction base, which contains two commercial ball slide assemblies aligned, parallel to the test axis of the machine (the long axis of the base) and the mounting plate for the reaction torque cell. The purpose of the ball slides is to allow the torque cell the freedom to move along the torque axis of the machine while resisting the applied torque. This ensures that no axial loads are applied to the specimen. The total range of motion of the torque cell is approximately 1 inch (25.4 mm). This allows the operator sufficient room to insert each specimen into the gripping fixtures of the machine. The location of the reaction section on the base is also adjustable over about a 4inch (101-mm) range. Once the reaction section has been located in a suitable position for the specimen being tested, it can be locked in place using two locking screws. This allows specimens to be tested in a range from less than 0.5 inch (12.7 mm) to about 3.5 inch (76 mm) in length. Torque Measurement: The torque cell is a commercial unit with a 100-lb capacity (Transducer Techniques, Inc., TRT-100). The signal from the torque cell is amplified by a commercial strain gage signal conditioning system and fed to a 16-bit resolution Data Translation data acquisition card in an IBM PC-compatible computer. The torque cell is calibrated before each set of tests using a 10.00-inch (254-mm) lever arm and a set of standard laboratory gram weights.


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Specimen Attachment: The carbon foam specimens for this program were limited in size to approximately 0.3-inch (7.6 mm) diameter by 2 to 3 inch (50 to 75 mm) long due to the small size of the foam preforms. The foam specimens were fitted with slotted end ring or solid tabs (Figure 4 shown with slotted ring tab) for easy gripping to the torsion tester. Due to this limitation, the torsion tester was fitted with grooved collar adapters to grip the specimens through the slotted end tabs. These adapters can be easily modified or replaced to accommodate specimens of different diameters. Control Software: The control software was written in Microsoft Quick Basic, Version 4.5. The program allows the operator to enter specific information about the specimen and designate a file name for the data. The output data file contains the torque and angular displacement values in engineering units for each step of the motor (every 0.005°). This data is in ASCII format and may be imported into a spreadsheet program, such as Excel, for further analysis or plotted using any of the commercially available plotting programs. Calibration of the Torsion Fixture: The tester was calibrated in two stages. First, the response of the load cell was calibrated to known applied torque and then the whole torsion fixture by measuring shear modulus of PVC plastic. The PVC plastic was chosen as the material for calibrating the fixture because of its modulus being closest to that of foam that we could obtain. To assess the response of the load cell, the real time response (dynamic response) of the load cell due to applied quasi-static torque was recorded and is shown in Figure 7. The quasi-static torque to the load cell was generated by hanging a known mass at the end of a torsion arm attached to the fixture perpendicular to the torsional axis. A known amount of torque was applied to the tester for a few minutes and then changed to another value several times to assess the stability of the load cell response. The real time response of the load cell shown in Figure 7 indicated the reliability and stability of the load cell response

The objective of developing this test fixture was to measure shear properties of brittle porous (carbon foam) materials. Since strain gages could not be installed on porous specimen surface, the emphasis for this test method was to determine the shear strain in the specimen from the specimen end rotation rotation between

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the specimen gage length. To calibrate the torsional response of the fixture, a solid PVC tubular specimen (Figure 8) with slotted ring aluminum tabs was tested in torsion to calibrate the torsional response of the fixture. The modulus of aluminum tabs is generally over 10 times of that of PVC and over 100 times of that of the foam considered here for testing. Thus the portion of the PVC or foam specimen that was bonded into the aluminum ring tabs, assuming a perfect bond between the specimen surface and the tabs, was considered to

behave rigid compared to the rest of the PVC or foam specimen. Thus, the specimen length excluding the end tab portion is considered the gage length (G.L.) of the specimen. The specimen was also instrumented with a strain gage to compare the strain gage strain data with that of specimen end rotation. The torque versus the specimen end rotation of the specimen is shown in Figure 9. The expression for calculating shear modulus from the torque (T) versus the Specimen End Rotation plot is given below.

where, G is the shear modulus of the material, T is the applied torque, L is the specimen gage length, is the specimen end rotation, J is the torsional rigidity of the specimen. For circular cross-section the torsional rigidity (J) is where and are the outer and inner diameter of the specimen, respectively. In Equation (1), is the slope of the torque (T) versus Specimen End Rotation curve, as shown in Figure 9. The shear strain on the specimen surface of circular cross section can be calculated from the Specimen End Rotation by using the following relation

The values of the specimen surface shear strain associated with the specimen end rotation of the PVC specimen tested is shown at the top abscissa in Figure 9. The initial shear modulus of the PVC material obtained from the data shown in Figure 9 was found to be 1.112 GPa (0.161 Msi). In order to assess the accuracy of the torsional measurement, a flat specimen of dimension 101 x 12.7 x


51

1.016 mm (4”x0.5”x0.040”) made out of the same PVC material (Figure 10), was tested in tension to measure its Young's modulus and Poisson’s ratio. In view of low modulus of PVC, two extensometers were used to measure its strain in longitudinal and transverse directions. In our experience, adhesion of strain gages to the specimen surface tends to locally stiffen PVC by as much as by 19 percent; thus strain gages were not used measuring strain in PVC. The tensile stress-strain curves of the PVC flat specimen are shown in Figure 11. Using the stress-strain data in Figure 11, the measured Young's modulus (E) and Poisson’s ratio (v) of the PVC specimen were found to be 2.987 GPa (0.433 Msi) and 0.3683, respectively. Assuming material isotropy, the initial shear modulus of PVC was independently calculated from its Young's modulus (E) and Poisson’s ratio (v) using the following expression

The initial shear modulus of the specimen calculated from the shear test (Equation (1) and Figure 14) was within 1.8 percent of the shear modulus calculated from the tensile test (using Eq. (3) and Figure 11) that illustrated the accuracy of the test method.

4. Measured Shear Properties of Foam Materials The shear modulus and strength of graphitized foam of two different densities were measured using the miniature torsion fixture described above. The measured shear stress-strain curves of the two materials are shown in Figures 12 and 13, respectively. The densities of the two carbon foams were 0.16 and 0.56 g/cc, respectively. The shear modulus and strength of the foam, as expected increased with its density. All the specimens failed in the gage section in shear, as revealed from the failure surface shown in Figure 14. The test data in these to figures

52


indicate that both the ring and solid tabs are equally efficient for gripping the foam specimens to the test fixture.

The failure surface of the specimens in Figure 14 revealed that shear failure occurred at approximately 45° with respect to the specimen longitudinal axis, which is the usual failure mode observed in specimens subjected to pure shear. It is known from the principle of strain transformation that the shear strain when transformed to a 45° rotation, becomes orthogonal tensile and compressive strains (see Figure 15), yielding Thus a comparison was made between the tensile failure strain (measured from the tensile tests) and the tensile strain calculated from transformed shear failure strain (measured from the shear test). The failure shear strain of the foam of density 0.16 g/cc (data shown in Figure 12) was compared with its tensile failure strain measured by Roy, et al. [13]. The average tensile failure strain of the same foam reported in Figure


53

7(b) in [13], was (with standard deviation of . The failure shear strain of the 0.16 g/cc foam found in this study to be , which yielded a tensile strain of value transformed at an angle of 45° with respect to the specimen longitudinal axis. The value of this tensile strain, obtained after transforming the shear failure strain, was within the scatter of the failure tensile strain data obtained in reference [13], which inferred that the shear failure was in fact a tensile failure at an angle of 45° with respect to the shear plane.


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5. Conclusions The shear properties of carbon (graphitic) foam, being brittle and highly porous, are not reliably measured with most of the standard test methods, such as, single rail, double rail, Iosipescu shear, etc. A new testing fixture is developed to accurately measure the shear stiffness and strength of carbon foam or other porous materials. A digital step motor was used in the fixture to exert torsion to specimens of cylindrical cross section. Since strain gages could not be installed on the specimen surface (due to porosity), the shear strain is determined from the specimen end rotation. To ensure a high accuracy in the specimen end rotation, the test fixture was designed to a rotational resolution of 0.005 Degree/Step (or Radian/Step). In view of testing low modulus material, the load cell of fixture was mounted on an axial roller to relieve axial constraint while twisting the specimens. The accuracy of the measurement with the test fixture was demonstrated by measuring shear modulus of PVC plastic.

6. 1.

References

Gibson, L. J. and Ashby, M. F., 1988, Cellular Solids – Structures and Properties, Pergamon Press, Chapters 5 and 6. 2. Christensen, R. M., 1986, Mechanics of Low Density Materials, J. Mech. Phys. Solids, Vol. 34, No. 6, pp. 563-578. 3. Christensen, R. M., 1996, On the Relationship of Minimal Conditions to Low Density Material Microstructures, J. Mech. Phys. Solids, Vol. 44, No. 12, pp. 2113-2123. 4. Christensen, R. M., 2000, Mechanics of Cellular and Other Low-density Materials, Intl J. Solids and Structures, Vol. 37, pp. 93-104. 5. Warren, W. E. and Kraynik, A. M., 1988, The Linear Elastic Properties of Open-Cell Foams, J. Applied Mechanics, Vol. 55, pp. 341-346. 6. Zhu, H. X., Knott, J. F., and Mills, N. J., 1997, Analysis of the Elastic Properties of Open-cell Foams with Tetrakaidecahedral Cells, J. Mech, Phys. Solids, Vol. 45, No. 3, pp. 319–343. 7. Zhu, H. X., Mills, N. J., and Knott, J. F , 1997, Analysis of the High Strain Compression of Open-cell Foam, J. Mech. Phys. Solids, Vol. 45, No. 11/12, pp. 1875-1904. 8. Kau, C., Huber, L., Hiltner, A., and Baer, E., 1992, Hard Elastic Behavior in Polyurethane Foams, J. Applied Polymer Science, Vol. 44, pp. 2081-2093. 9. Kau, C., Huber, L., Hiltner, A., and Baer, E., 1992, Hard Elastic Behavior in Polyurethane Foams, J. Applied Polymer Science, Vol. 44, pp. 2069-2079. 10. Bilkhu, S. S., Founas, M., and Nusholtz, G. S., 1994, Determination of Cellular Foam Material Properties for use in Finite Elements Analysis, Experimental Techniques, pp. 23-25. 11. Annual Book of ASTM Standards, Vol. 15.03, Standards C 273-94, D 4255, D 5379, ASTM, 1916 Race Street, Philadelphia, PA 19103, USA. 12. Roy, A. K. and Camping J. D., A Novel Miniature Test Fixture to Measure Shear Stiffness and Strength of Foam (Highly Porous Materials), Invention Disclosure, Submitted, August 1999, U. S. Air Force. 13. Roy, A. K., Pullman, D., and Kearns, K. M., Experimental Methods for Measuring Tensile and Shear Stiffness and Strength of Graphitic Foam, Proc. Intl. SAMPE Conference and Exhibition, Anaheim, CA, 31 May – 4 June, 1998

INDENTATION OF A PVC CELLULAR FOAM

E. E. GDOUTOS and J. L. ABOT Department of Civil Engineering Northwestern University Evanston, IL 60208 Abstract

An investigation of the indentation response of a PVC closed-cell cellular foam (Divinycell H80) by a cylindrical indenter was undertaken. The material has a density of 80 and shows a linear elastic almost perfectly plastic behavior in simple compression up to a critical strain at which densification takes place. The plastic Poisson’s ratio of the material approaches zero. The load-indentation depth relation was monitored for a progressively increasing applied load. It consists of an initial linear part followed by a softening behavior. A simple model based on a rigid plastic behavior of the foam in compression and a maximum stress failure criterion was used to predict the load-indentation depth curve. Results for the indentation hardness indicate that it takes values close to one for small indentation depths, while it increases as the indentation depth increases. 1. Introduction

The compressive stress-strain behavior of many cellular foams consists of an initial relatively small and stiff elastic regime, an extended stress plateau regime and a final regime in which densification of the material takes place [1]. In the stress plateau regime the cells of the foam collapse, while the average stress remains almost constant during the instability spread through the material. Axial compression produces little lateral spreading resulting to almost zero Poisson’s ratio. When all of the cells collapse the material is densified and its stiffness increases. As a consequence of such behavior foams change their volume during plastic compression. This is contrary to dense solids which are incompressible during plastic deformation. Because of this, the indentation behavior of foams is quite different than that of dense solids. The first study on the plastic behavior and indentation hardness of cellular materials was conducted by Shaw and Sata [2]. They observed that during the plastic deformation Luder’s-like bands and upper and lower yield points appear, as in mild steel. Furthermore, they found that indentation hardness was close to the yield stress in compression and they attributed this behavior to the very low Poisson’s ratio. Patel and Finnie [3] presented a model for the prediction of properties of foams assuming a unit cell in the form of a pentagonal dodecahedron. An analysis of the indentation of compressible foams by cylindrical and spherical indenters was conducted by Wilsea et al.[4]. They presented a simplified model for the prediction of the indentation hardness based upon idealization of the stress-strain behavior. Cousins [5] presented a simplified theory to describe the impact behavior of padding materials with rate dependent deformation when struck by flat or spherical indenters. 55 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 55–64. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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E. E. GDOUTOS, J. L. ABOT

Following the above early investigations, a number of works on the indentation behavior of honeycombs have appeared. Klintworth and Stronge [6] studied the stress and deformation fields in an anisotropic half-space made of a ductile honeycomb loaded by a flat punch. They found that after a small regime of elastic deformations the honeycomb crushes in localized bands. They obtained a lower and upper estimate of the indentation force. Papka and Kyriakides [7,8] studied the in-plane crushing of hexagonal aluminum honeycombs between parallel rigid surfaces. The forcedisplacement behavior is initially stiff and it is then followed by an instability behavior under constant load. Localized crushing occurs and is spread through the material under constant load. When the whole specimen is crushed the load-deformation response stiffens again. A finite element analysis was performed to simulate the crushing process. The crushing response of a polycarbonate circular cell honeycomb subjected to in-plane biaxial loading was studied by Chung and Waas [9]. Fortes et al. [10] based on the micromechanical model of Gibson and Ashby [1] analyzed the contact behavior of a cellular solid for both open cell and closed cell materials. They derived a simple equation for the prediction of the indentation behavior. A thorough investigation of the mechanical behavior of a PVC closed cell foam under uniaxial and multiaxial stress conditions was conducted by Gdoutos et al. [11,12]. Two types of Divinycell, H100 and H250, with densities 100 and 250 , respectively, were investigated. The uniaxial tensile, compressive and shear stressstrain curves along the in-plane and the through-the-thickness directions of both materials were obtained. A series of biaxial tests using strip, ring and tube specimens subjected to uniaxial compressive or tensile loads and combined tension, torsion and internal pressure were conducted. The failure envelopes of the materials were constructed and satisfactorily modeled by the Tsai-Wu failure criterion. An experimental and analytical study of the indentation behavior of composites facesheets supported by a foam foundation and subjected to a concentrated load was conducted by Abot et al. [13]. The load-deflection curve includes a linear range followed by a nonlinear portion. In the linear range the load-indentation behavior was predicted by the Hetenyi model based on Winkler’s solution of an elastic beam resting on an elastic foundation. In the nonlinear range the load-indentation relationship was obtained by fitting experiments. A new method was described for direct measurement of the foundation modulus using cylindrical and rectangular section indenters on foam blocks. In the present work the indentation behavior of Divinycell H80 by a cyclindrical indenter was studied. The load-indentation depth and indentation hardness curves were obtained experimentally and were modeled by a simplified plastic analysis. 2.

Characterization of foam material

2.1 MATERIAL

The material studied in this work was a fully cross-linked PVC closed-cell cellular foam (Divinycell H80) with density of 80 This is a commonly used core material in sandwich construction by marine structures. The material is produced by introducing gas bubbles into the hot PVC polymer and allowing the bubbles to grow and stabilize. The material was produced by the manufacturer in the form of 25.4 mm (1 in.) thick plates with the rise direction along the thickness direction.


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For modeling the indentation response of the material its compressive stress-strain behavior is needed. 2.2 COMPRESSION TEST Divinycell H80 is relatively isotropic. Its mechanical properties along the in-plane and the through-the-thickness directions are almost equal. To determine the in-plane stressstrain behavior of the material in compression, prismatic specimens of dimensions 25.4 x 25.4 x 76.2 (1 x 1 x 3 in.) were tested quasi-statically in an Instron servohydraulic testing system. Both longitudinal and transverse strains were measured with extensometers. The longitudinal strains were monitored on opposite sides of the specimen to insure that there was no bending effect during loading. The tests were terminated after the load dropped and remained almost constant following a peak value. The above specimens during loading in the stress plateau region of the compressive stress-strain behavior failed due to buckling. Thus, they cannot be used to obtain the complete stress-strain curve. For this reason short prismatic specimens of dimensions 25.4 x 25.4 x 25.4 (1 x l x 1 in.) were tested. The longitudinal strains were determined from the stroke of the grips of the testing machine. Figure 1 shows the in-plane longitudinal and transverse stress-strain curves of Divinycell H80. The longitudinal stress-strain curve shows an initial elastic portion followed by a yield plateau during which the applied stress remains nearly constant under increasing strain. The elastic portion of the curve shows an initial linear part followed by a nonlinear portion with a decreasing tangent modulus. At the end of the nonlinear elastic part the stress drops abruptly from an upper to a lower yield point. The complete stress-strain curve of the material after densification takes place is shown in Fig. 2. Note that the yield portion of the curve extends up to a strain of 50 percent. Following the end of the yield region, densification takes place and the stress rises abruptly. From a micromechanical pint of view this corresponds to collapse of the cells of the foam. Opposing cell walls come into contact and the stress-strain curve corresponds to that of the cell material. This behavior is similar to that of aluminum honeycomb in which localized yielding or crushing occurs after peak load. The transverse stress-strain curve (Fig. 1) shows similar characteristics as the longitudinal stress-strain curve up to the stress drop to the lower yield point. After that point, the transverse strain does not increase any more, that is, longitudinal compression does not produce any lateral deformation. In other words, Poisson’s ratio in the yield region is almost zero. This behavior can be explained from the collapse of the foam cells as the foam is compressed. Thus, foams, unlike dense solids, cannot be considered to be incompressible when deformed in the plastic region. This behavior of foams has been noticed by previous investigators [1,2]. The in-plane longitudinal and transverse stress-strain curve of Divinycell H80 are almost the same as the through-the-thickness curves. 3. Indentation tests A series of panel specimens of width 254 mm (10 in.) and thickness 24.1 mm (0.95 in.) were cut from Divinycell H80 plates. The height of the specimens took the values 25.4, 76.2, 152.4, 254 and 381 mm (1,3,6,10 and 15 in.).

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The panels were supported on a rigid base and loaded by a cylindrical roller of diameter 25.4 and 50.8 mm (1 and 2 in.) in an Instron servo-hydraulic testing machine. The indentation of the cylindrical roller into the specimens was measured from the stroke of the testing machine. The load was applied at a rate 0.0085 mm/sec (0.02 in./min). The load and the stroke of the testing machine were monitored during the test. Figure 3 shows a picture of the experimental setup for a panel specimen of width 254 mm (10 in.) and height 12.7 mm (5 in.). The deformation of a series of straight lines parallel to the surface of the panel near the indenter is shown in Fig. 4. Note that two


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inclined cracks on both sides of the indenter starting from the surface of the panel have initiated outside the indentation zone beneath the indenter.

4. Theoretical modeling For the prediction of the indentation behavior of the foam a simplified model developed by Wilsea et al.[4] will be used. The model is based on an idealized stress-strain behavior of the foam under compression, as shown in Fig. 5. The material presents a small linear elastic region which is omitted. When the stress reaches a critical value

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the material yields under constant strain up to a critical value at which densification occurs. For strains higher than the material hardens and the stress increases. Failure of a material under a multiaxial state of stress occurs when the maximum principal stress becomes equal to and is uninfluenced by the other two principal stresses. This is a simplified idealization of the material behavior. Gdoutos et al. [11,12] have shown that the failure behavior of Divinycell is influenced by all three principal stresses. They showed that the failure surfaces of Divinycell under combined normal and shear stresses along the in-plane and through-the-thickness directions obtained from a host of biaxial tests are predicted well by the Tsai-Wu failure criterion. Under such conditions, the plastic zone has the shape of a lune under the indenter. The indentation depth d corresponding to an indentation load P is obtained from the following equations

where 2a is the width of indent, R is the radius of the indenterand L is the indenter length. The angle defines the plastic lune under the indenter. For the predicaiotn of the load versus indentation depth curve based on Eqs. (1) to (3) the following procedure is followed. For a given value of indentation depth d the value 2a of the width of indent is calculated from Eq. (1) and then the value of angle from Eq. (2) with a material constant determined from the uniaxial compression stress-strain curve of the material. It was obtained Having the values of a and the load P is calculated from Eq. 3 in which is the yield stress of the material in uniaxial compression. Following the above procedure the load versus indentation depth curve can be predicted when the values and defined from the stress-strain curve of the material in compression, the radius, R, and the length, L, of the indenter are known. 5.


Figure 6 shows the load-indentation depth curves for five Divinycell H80 panels of width 25.4 mm (10 in), thickness 25.4 mm (1 in.) and heights 76.1, 152.4, 254 and 381 mm (1,3,6,10 and 15 in.) for indentation depth up to 20 mm (0.79 in.). Note that for height higher than 76.1 mm (3 in.) all curves coincide. This means that the panel height does not affect the indentation behavior. Thus, for panel heights higher than 76.1 mm (3 in.) the indentation curve depends on the foam and the diameter of the indenter. It can be considered as a material property for a given indenter diameter. The initial part of the indentation curves for an indentation length up to 5 mm (2 in.) are shown in Fig. 7. Finally, Fig. 8 shows the indentation curves for a panel of height 150 mm (5.9 in.) indented by a cylindrical indenter of diameter 25.4 and 50.8 mm (1and 2 in.).


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In Fig. 8 the predicted indentation curves based on the simplified rigid-perfectly plastic model proposed by Wilsea et al. [4] are shown. Note that for the case of indenter of diameter 25.4 mm (1 in.) the experimental results coincide with the theoretical predictions for an indentation depth up to 10 mm (0.4 in.), while for an indenter of diameter 50 mm (2 in.) up to 5 mm (0.2 in.). After these indentation depths there is a small deviation of the experimental results from the theoretical predictions.

A characteristic feature of the indentation behavior is the indentation hardness H defined by

H is equal to the indentation pressure assumed uniformly distributed along the width of indent. The relationship between the indentation hardness H and the width 2a of the indenter can be predicted from Eq.s (2) to (4). For a given value of a the angle is calculated from Eq. (2), then the load P from Eq. (3) and finally the indentation hardness from Eq. (4).

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Figure 9 shows the indentation hardness H normalized to the yield stress plotted versus the width of indent 2a divided by the diameter d of the indenter In the same figure the theoretical curves based n the model of Wilsea et al. [4] for different values of are shown. Note that the indentation hardness H for small indentation widths is about equal to the yield stress. H increases with 2a/D up to a value of for . These results are in agreement with those of references 2 and 4.


6.

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Conclusions

The indentation behavior of a PVC closed-cell cellular foam was investigated. Panel specimens of various different dimensions were indented by a cylindrical indenter of two different diameters. The load versus indentation depth response was monitored during the test. From the results of this study the following conclusions may be drawn: 1.

The indentation behavior of the foam material studied in this work consists of a nonlinear load versus indentation depth curve with decreasing stiffness as the load is increased.

2.

The load versus indentation depth response becomes independent of the height of the panel for the two indenter diameters used for heights higher than 75 mm (3 in.). Under such conditions the indentation curve is a material property and depends only on the diameter of the indenter.

3.

The indentation hardness is almost equal to the yield stress of the foam material for small indentation depths. As the indentation depth increases the hardness increases up to a limiting value of for indentation depth versus indenter diameter ratio equal to 1.5. This behavior is contrary to incompressible materials, like metals, where the indentation hardness is equal to three tunes the yield stress. This is because the foams are compressible materials with Poisson’s ratio almost equal to zero in the plastic region. This behavior has been reported in the literature.

4.

The indentation behavior can well be predicted by a simple model based on a rigid plastic behavior of the foam in compression and a maximum stress failure criterion.

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7. Acknowledgements The authors wish to thank Professor I. M. Daniel for his advice, guidance and support during the course of this work. This research was sponsored by the Office of Naval Research (ONR). We are grateful to Dr. Y. D. S. Rajapakse of ONR for his encouragement and cooperation and to Mrs. Yolande Mallian for typing the manuscript. 7. References 1. 2. 3. 4. 5. 6.

7. 8. 9.

10. 11. 12.

13.

L. J. Gibson and M. F. Ashby, Cellular Solids, Cambridge University Press, Cambridge, 1997. M. C. Shaw and T. Sata, The plastic behavior of cellular materials, Int. J. Mechanical Sciences, 8 (1966) 469-478. M. R. Patel and I. Finnie, Structural features and mechanical properties of rigid cellular plastics, Journal of Materials, JMLSA, 5 (1970) 909-932. M. Wilsea, K. L. Johnson and M. F. Ashby, Indentation of foamed plastics, Int. J. Mechanical Sciences, 17 (1975) 457-460. R. R. Cousins, A theory for the impact behavior of rate-dependent padding materials, J. Applied Polymer Science, 20 (1976) 2893-2903. J. W. Klintworth and W. J. Stronge, Plane punch indentation of a ductile honeycomb, Int. J. Mechanical Sciences, 31 (1989) 359-378. S. D. Papka and S. Kyriakides, In-plane compressive response and crushing of honeycombs, J. Mechanics and Physics of Solids, 42 (1994) 1499-1532. S, D. Papka and S. Kyriakides, Experiments and Full-scale numerical simulations of in-plane crushing of a honeycomb, Ada Metallurgies, 46 (1998) 2765-2776. J. Chung and A M. Waas, In-plane biaxial crush response of polycarbonate honeycombs, J. Engienering Mechanics-ASCE, 127 (2001) 180-193. M. A. Fortes, R. Colaço and M. Fátima Vaz, The contact mechanics of cellular solids, Wear, 230 (1999) 1-10. E. E. Gdoutos, I. M. Daniel and K.-A. Wang, Failure of cellular foams under multiaxial loading, Composites: Part A 33 (2002) 163-176.. E. E. Gdoutos, I. M. Daniel and K.-A. Wang, Multiaxial characterization and modeling of a PVC cellular foam, J. Thermoplastic Composite Materials, to appear. J. L. Abot, I. M. Daniel and E. E. Gdoutos, Contact law for composite sandwich beams, J. Sandwich Structures and Materials, to appear.

NANOMANIPULATION AND CHARACTERIZATION OF INDIVIDUAL CARBON NANOTUBES RODNEY S. RUOFF Department of Mechanical Engineering Northwestern University 2145 Sheridan Road, Evanston, IL 60208 MIN-FENG YU Advanced Technologies Group Zyvex Corporation 1321 North Plano Road, Richardson, TX 75081 HENRY ROHRS Mass Sensors, Inc. 1350 Baur Blvd. St. Louis, MO 63132-1904 Abstract We describe the implementation of new tools for the measurement of the mechanics of nanostructures. These tools have been used to measured the stiffness, fracture strength, and various tribological properties, of carbon nanotubes.

1. Introduction The rapid advancement in miniaturization in the last decade has seen the discovery and generation of numerous types of nanomaterials whose size is in the nanometer scale. In particular, carbon nanotubes (CNT) have received significant attention. Multi-walled carbon nanotubes (MWCNT) were discovered in 1991[1] and single wall carbon nanotubes (SWCNT) in 1993 [2,3], and since then over 4000 scientific publications on CNTs have appeared. The electrical and mechanical properties of CNTs have been the subject of numerous theoretical/modeling and experimental studies. CNTs can be considered to be a seamless cylinder (SWCNT), or a set of nested cylinders (MWCNT), of graphene sheet(s), and can have open or capped ends. CNTs made to date typically have a diameter of a few tens of nanometers (MWCNT), and about 1 nm (SWCNT), and lengths of a few microns. Various novel discoveries related to the natural properties of a graphene sheet, of quantum confinement in small dimensions, and the detailed geometry including in particular the chirality of the shell structure of CNTs have been made, and potential applications are likely in materials reinforcement, field emission panel display, electronic nanowires and devices, chemical sensors, gas storage, as tips for scanning microscopy, for use in batteries, and so on. It has been theoretically predicted that CNTs should possess mechanical properties far superior to commercially available carbon fibers, due to their expected structural perfection. High Young’s modulus (~ 1 TPa) similar to the inplane modulus of high quality graphite and high tensile strength (~ 50 to 100 GPa, 65 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 65–74. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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thus much greater than any other known material) are predicted, and for the modulus experimentally verified, for CNT [4-7]. For comparison, the highest strength carbon fiber in industry has a strength of ~ 7 GPa [8]. The small dimensions of CNTs imposes a significant challenge for experimental study of their mechanical properties. The general requirements for such study include: (i) the challenge of CNT placement in an appropriate configuration for testing, such as of pick up followed by release and placement; (ii) in certain cases the fabrication of appropriate clamps and boundary conditions for loading; (iii) successful application of the desired loading; (iv) characterizing and measuring the mechanical deformation at the nanometer or perhaps atomic length scale. Various types of high-resolution microscopes allow the characterization of nanomaterials, and recent developments in the new area of “nanomanipulation,” based on inserting or adapting new tools to such high-resolution microscopes, have enhanced our ability to mechanically test such nanoscale objects. In this paper, we briefly summarize our effort in the development of new tools for nanoscale characterization and the study of various mechanical properties of CNTs. We close by speculating about future directions worth pursuing for mechanics studies on individual nanostructures.

2.

A brief review of related instruments for nanoscale material characterization

Scanning probe microscopy and electron microscopy have been the most widely used methods for resolving and characterizing nanoscale objects. We and others have primarily used SPM and EM instruments to study nanotube mechanics. For this reason, we give a brief review of the methods of operation of these types of microscopes. Electron microscopes (EM) use high-energy electron beams (several keV up to several hundred keV) as a source for scattering and diffraction from a sample, which results in high resolving power down to sub-nanometer resolution because of the extremely short wavelength (a fraction of a nanometer) of high kinetic energy electrons. In TEM, an accelerated electron beam from a thermal or a field emitter is used to interrogate samples. The beam transmits through the sample and passes several stages of electromagnetic lenses, and projects the image of the studied sample region to a phosphor screen or other image recording media. A thin (normally several tens of nanometer or less in thickness) and small (no more than sample is a typical requirement since only a small sample chamber is available and a dedicated holder is typically used in these expensive instruments (which are thus typically time-shared among many users) for sample transfer. TEM is limited by such factors as the spread in the electron beam energy and the quality of the ion optics. It normally has a resolution on the order of 0.2 nm. A good textbook on TEM is reference [9]. Fig. 1C and D shows typical high-resolution images of CNTs [10]. Fringes corresponding to the projection of the shell(s) in each CNT are easily resolved. In SEM, a focused electron beam (nanometers in spot size) is used to raster across the sample surface and the amplified image of the sample surface is formed by recording the secondary electron signal or the back scattering signal generated from the sample. There is no strict limit on sample size in principle, and normally a large sample chamber is available such that samples can be surveyed over large

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areas. SEM is limited by the scattering volume of the electrons interacting with sample material, and high end instruments commercially available are capable of achieving a resolution of a few nanometers. Typical SEM images of a SWCNT sample, and of a purified MWCNT sample, are shown in Fig. 1E and F, respectively. Individual MWCNTs can be seen sticking out from the bundle, while only bundles of SWCNTs having diameters ~10 nm can be seen in the SWCNT sample. Scanning probe microscopes (SPM) use extremely sharp probes (that can have 10 nm or smaller radius of curvature at the tip) controlled by sensitive sensing and actuation feedback electronics for obtaining nanoscale and even atomic scale information. The Atomic force microscope (AFM) has been particularly useful for the mechanical studies of CNTs. Since its invention in 1986 [11], it has been rapidly accepted as a useful tool for many applications related to surface characterization. High-resolution (nanometer to atomic resolution) mapping of surface topology on either conductive or nonconductive material can be achieved with an AFM. The principle of operation of the microscope is relatively simple. A probe with a sharp tip having a force-sensitive cantilever is used as a sensor to scan, in close proximity to, the sample surface. The probe is driven by a piezoelectric tube capable of nanometer resolution translation in the x, y and z directions, and the tip normally has a radius of curvature of ~10 nm. The force interactions between tip and sample results in deflection of the cantilever. The deflection of the cantilever, during scanning of the sample surface in the x and y directions, is constantly monitored by a simple optical method, or other approaches. A feedback electronic circuit that monitors the deflection signal and controls the piezoelectric tube maintains a constant force between the tip and the sample surface; a surface profile of the sample can thus be obtained. Fig. 1A shows a typical image acquired by AFM of MWCNTs on a substrate. For in-depth introduction of AFM see references [11-13].

The detection mechanism of the scanning tunneling microscope (STM) is based on the tunneling current (on the order of nA and less) between a sharp conductive tip and a sample when positioned in close proximity and under a constant bias. A non-insulating sample is thus required for STM imaging. At appropriate imaging

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conditions on many crystalline samples atomic resolution can be obtained, due to the strong dependence of the tunneling current on the distance between the conductive tip and the sample surface (there is an exponential dependence of the current on this distance). Fig. 1B shows an atomically resolved image of a SWCNT by Odom et al. [14]. The lattice structure and thus the chirality of the SWCNT can be determined from such images, as well as the electronic structure from additional scanning tunneling spectroscopy measurements. Similar atomically resolved STM images of SWCNTs were also obtained at the same time by Wildoer et al. [15]. A detailed explanation on the principle of STM can be found in ref. [12].

3. New tools for nanoscale mechanical measurement The high resolution and the large sample chamber space have made SEM a good candidate for inclusion of three dimensional nanomanipulation capabilities. A multiple degree-of-freedom manipulation device inside SEM was developed by Yu et al. to allow handling, characterizing, and even building with CNTs and other nanomaterials [16]. The device possesses the capability and precision to probe a collection of CNTs, and then to isolate an individual CNT and extract it from the ensemble for study. The manipulator can be used as a testing stage once a CNT has been isolated from the group, to allow measurement of the mechanics and transport properties while the CNT is freely suspended in vacuum and thus free from contact with a surface. The device fits well inside the vacuum chamber of an SEM, and can be operated without disturbing the electron microscope's imaging quality or interfering with the microscope's components. The design of the manipulator is depicted in Fig. 2 Only the major components are shown. The manipulator was designed with small size, low-cost, wide translation range, reasonable precision, and rapid assembly in mind. To avoid interference with the SEM electron optics, the x-y and z-theta motions are grouped into two low-profile, opposing stage sub-modules anchored symmetrically on the SEM platform around the axis of the electron beam column. The SEM stage manipulator occupies roughly

As observed with the SEM, the linear stage has a "dynamic" step resolution of 25 nm. The rotational actuator gives angular step sizes of 1.2 mm. 7. Conclusion

An analysis combining the fracture mechanics and plastic collapse is presented to predict the failure mode of resistance spot weld. The analytical results are validated through comparison with available test data. Spot weld under either tensile (cross tension samples) or shear (lap-shear samples) load is considered. Excellent prediction is obtained. In addition, through comparison it is concluded that the prediction may be applied to many different grades of steels ranging from low carbon mild steel to high strength steel as well as under either quasi-static or impact loading conditions. When compared with industry standards for sizing weld nugget, the result obtained in this paper provides a simple, more accurate and better prediction for the transition of failure mode from pullout to interfacial. With further validation, e.g. more test data in thin ( t < 1.0 mm) gage steels, the results, e.g. equations (8) or (10), could be recommended for sizing the spot weld in steel sheet for auto industry. 8. Acknowledgement

The author acknowledges the partial support of this work by NSF through grant CMS0116238 and the encouragement from the Program Director, Dr. Kenneth P. Chong. Valuable discussion with Dr. P. C. Wang of General Motor Corporation is specially noted. 9. References 1.

2. 3. 4. 5.

6.

Section 5.7, Weld Button Criteria, Recommends Practices for Test Methods for Evaluating the Resistance Spot Welding Behavior of Automotive Sheet Steel Materials, ANSI/AWS/SAE/D8.997, An American National Standard (1997). Ewing, K. W., Cheresh, M., Thompson, R., and Kukuchek, P., “Static and Impact Strengths of Spot-Welded HSLA and Low Carbon Steel Joints,” SAE Paper 820281 (1982). Radaj, D., “Stress Singularity, Notch Stress and Structural Stress at Spot-Welded Joints,” Engineering Fracture Mechanics, 34(2), (1989), 495-506. Zhang, S., “Approximate Stress Intensity Factors and Notch Stresses for Common Spot-Welded Specimens,” Welding Journal, (1999), 173-179. Zuniga, S. and Sheppard, S.D., “Resistance Spot Weld Failure Loads and Modes in Overload Conditions,” Fatigue and Fracture Mechanics: Volume, ASTM STP 1296, R.S. Piascik, J.C. Newman, and N.E. Dowling, Eds., American Society for Testing and Materials, (1997) 469-489. Lin, S.H., Pan, J., Wu, S., Tyan, T., and Wung, P., “Spot Weld Failure Loads under Combined Mode Loading Conditions,” SAE 2001-01-0428 (2001).

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Smith, R.A., “Sizing of Spot Welds by Elastic/Plastic Analysis”, Fracture, Radon, J.C., London, (1980) 49-56. 8. Lee, Y., Wehner, T., Lu, M., Morrissett, T., Pakalnins, E., “Ultimate Strength of Resistance Spot Welds Subjected to Combined Tension and Shear,” Journal of Testing and Evaluation, 26(3), (1998), 213-219. 9. VandenBossche, D.J. “Ultimate Strength and Failure Mode of Spot Welds in High Strength Steels,” SAE 770214 (1977) 10. Tada, H., Paris, P. and Irwin, G. The Stress Analysis of Cracks Handbook, 25.5, Del Research Corp., St. Louis, Missouri. (1973) 11. Chao, Y.J., “Failure of Spot Weld under Tensile, Shear and Mixed Tensile and Shear Loads” manuscript in preparation. (2001) 12. Rivett, R.M. Welding Institute Research Bulletin, 20, (1979), 235-239.

INVESTIGATING THE EFFECTS OF SPECIMEN THICKNESS AND PRESSURE ON THE CRACK GROWTH BEHAVIOR OF A PARTICULATE COMPOSITE MATERIAL

C. T. LIU AFRL/PRSM 10 E. Saturn Blvd. Edwards AFB CA 93524-7680

Abstract

In this study, the effects of specimen thickness and confined pressure on the crack growth behavior in a particulate composite material, containing hard particles embedded in a rubber matrix, were investigated. The experimental data were analyzed and the results are discussed. 1.

Introduction

An important engineering problem in structural design is evaluating structural integrity and reliability. It is well known that structural strength may be degraded during its design life due to mechanical or chemical aging, or a combination of these two aging mechanisms. Depending on the structural design, material type, service loading, and environmental condition, the cause and degree of strength degradation due to the different aging mechanisms differs. One of the common causes of strength degradation is the result of crack development in the structure. When cracks occur, the effects of crack sizes and the rate of growth on the fracture resistance of the material need to be investigated. In recent years, a considerable amount of work has been done studying crack growth behavior in particulate composite materials under different loading conditions at ambient pressure [1-4]. This work was based on linear fracture mechanics. The principles of classical fracture mechanics are well established for single-phase materials. However, experimental evidence indicates that linear fracture mechanics theories have been applied to particulate composite materials with varying degrees of success. In this study, pre-cracked specimens were used to study crack growth behavior in a particulate composite material, containing hard particles embedded in a rubbery matrix, under a constant strain rate condition at ambient and 8697 kPa confined pressure. The effects of specimen thickness and pressure on crack growth behavior was investigated and the results are discussed. 257 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 257–266. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The Experiments

In this study, single-edge notched tensile specimens made from polybutadiene rubber embedded with hard particles were used in crack propagation tests. The specimens were 1.0 in. wide, 3.0 in. high and the thicknesses of the specimens were 0.2in., 0.5 in., 1.0 in., and 1.5 in. The geometry of the specimen is shown in Fig. 1.

Prior to testing, a 0.76 cm. crack was cut at the edge of the specimen with a razor blade. The pre-cracked specimens were tested at an 8 cm/cm/min strain rate under ambient and 8697 kPa confined pressures. During the tests, a video camera was used to monitor crack growth. The raw data obtained from the tests were the crack length a; the time, t; and the load, p, corresponding to the measured crack length. These raw data were used to calculate the crack growth rate, da/dt, and the Mode I stress intensity factor . The recorded experimental data, a, t, and p, were used to calculate the Mode I stress intensity factor, and the crack growth rate, da/dt. In calculating the stress intensity factor, for a given set of values of a and p, a nonlinear regression equation, which relates the normalized stress intensity factor, to the crack length, a, was used. The values of for different crack lengths were determined from the ABAQUS computer program. In calculating the crack growth rate, da/dt, the polynomial method was used. In the polynomial method, a high order polynomial function was selected to fit the crack length versus time data, and the crack growth rate was computed by taking the derivative of crack length with respect to time at a given time. To avoid the timeconsuming process of data reduction, a computer program was written to calculate and da/dt.

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It is well known that, on the microscopic scale, a highly filled polymeric material can be considered an inhomogeneous material. When these materials are stretched, the different sizes and distribution of filled particles, the different crosslink density of polymeric chains, and the variation in bond strength between the particles and the binder can produce highly nonhomogeneous local stress and strength fields. Depending on the magnitude of the local stress and the local strength, damage can be developed in the material, especially near the crack tip region. The damage developed in the material may be in the form of microvoids or microcracks in the binder or dewetting between the binder and the filler particles. Damage growth in the material may occur as material tearing or as successive nucleation and coalescence of the microcracks. These damage processes are time dependent and are the main factor responsible for the time sensitivity of strength degradation as well as the fracture behavior of the material. A typical photograph showing the crack surface during the earlier stage of crack growth under ambient pressure is shown in Fig. 2. Experimental results indicate that crack tip blunting takes place both before and after crack growth. The material at the tip of the crack suffers very large elongation and is nearly straight. The highly strained or damaged zones extend ahead of the crack tip, appearing as an equilateral triangle with the crack tip as its base. This damage zone is known as the failure process zone, which is a key parameter in viscoelastic fracture mechanics [5-6]. When the local strain reaches a critical value, small voids are generated in the failure process zone. Due to the random nature of the microstructure, the first void is not restricted to the surface where the maximum normal strain occurs. Since the tendency of the filler particle to separate from the binder under a triaxial loading condition is high, it is expected that voids, or a damage zone, will also be generated in the specimen’s interior. Consequently, there are a large number of strands, essentially made of binder material, which separate the voids that form inside the failure process zone.

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As the applied strain increases with time, material fracture occurs at the blunted end of the crack tip. This will always be the location of the maximum local strain. The failure of the material between the void and the crack tip causes the crack to grow into the failure process zone. This kind of crack growth mechanism continues until the main crack tip reaches the front of the failure process zone. When this occurs, the crack tip resharpens temporarily. The above paragraphs discusses the damage and crack growth mechanisms when the specimen is strained under ambient pressure. The formation of voids in the material, especially in the highly damaged zone near the crack tip, is a typical damage mechanism observed at ambient pressure. In order to obtain a fundamental understanding of the effect of confined pressure on the crack growth behavior, the effects of pressure on damage initiation and evolution processes need to be investigated. The results of a preliminary study are shown in the following paragraph. The effects of confined pressure and applied strain on volume dilatation, which is due to the formation of voids in the material, in smooth specimens, are shown in Fig. 3. It is seen that the volume dilatation decreases as the confined pressure is increased. At 8697 kPa confined pressure, the volume dilatation approaches to zero. This phenomenon is mainly due to the suppression of the formation of voids and increase the debond stress at the interface between the filler particles and the matrix. Under this condition, it is conjectured that, instead of the development of voids, micro cracks are developed in the matrix material. The number of micro cracks increases with increasing applied strain, and eventually, a macro crack is formed as a result of the coalescence of the micro cracks. The propagation of the macro crack leads to the fracture of the specimen. Experimental data revealed that for specimens without pre-crack, micro cracks started to develop approximately at 30% applied strain and the number of the micro cracks increased significantly at 40% applied strain (Fig.4) and specimens fractured at 50% applied strain as a result of the fast propagation of a long macro crack.


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The above paragraph describes the damage mechanisms in smooth specimen under confined pressure. In the following paragraphs, the crack growth behavior in pre-crack specimen under ambient and 8697 kPa confined pressure is discussed. Under ambient pressure, experimental data reveal that the crack start to propagate when the applied stress reaches a critical value, which is in the neighborhood of the maximum stress, as shown in Fig. 5. Once the crack starts to grow, the high crack growth rate results in a fast fracture of the specimen. In other words, unstable crack growth occurs as soon as the crack starts to grow. This type of crack growth behavior is similar to the brittle fracture observed in metals. Therefore, in this study, there is no crack growth analysis conducted under ambient pressure, and only the critical Mode I stress intensity factor, for the onset of crack growth is calculated. The results of analysis are shown in Fig. 6. From Fig. 6, it is seen that the variations of among different specimen thicknesses are within experimental scatter. Therefore, as a first approximation and for engineering applications, it can be assumed that for the onset of crack growth is independent of specimen thickness. A similar conclusion can be draw for under 8697 kPa confined pressure as shown in Fig. 7. It is known that one of the most important practical aspects of linear elastic fracture mechanics for single-phase materials is the concept of the plain strain value of the critical stress intensity factor, This is because this value is perceived to be the minimum critical value and that, for a given material and test environment, it exhibits a constant value to within a certain experimental scatter. This value is usually obtained from a tensile test where the specimen size is adjusted to produce sufficient thickness for the through-the-thickness cracked specimen to induce a brittle fracture. Under this condition, the crack front will bow in the direction of the crack growth, creating a thumbnail shape. This suggests that there is a plane strain constraint in the center portion of the specimen, which diminishes near the side boundary. However, this may not be true for highly filled polymeric materials. Experimental data obtained from crack

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propagation tests on highly filled polymeric material specimen revealed that the crack front exhibited no “thumbnailing” both before and after growth. In other words, the crack front was ideally straight, but with local irregularities. The straight crack front observed in solid propellants is believed to be due to the development of a highly damaged zone at the crack tip. The development of the highly damaged zone together with the straight crack front suggests that, within the highly damaged zone, the transverse constraint is very small and that the plane strain fracture toughness does not exist for these materials. The independent of to the variation of specimen's thickness indicates that the highly filled particulate composite material behaves differently from metals, and concepts that appeared clear in one instance are only found to contradict physical reality in another.


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In order to obtain a fundamental understanding of the effect of damage at a crack tip on the fracture toughness, and also to determine damage initiation and evolution processes near the crack tip, three-dimensional numerical modeling analyses were conducted, based on a micro-macromechanical approach [7]. A detailed description of the numerical model and micro-macromechanical analysis are shown in Reference 7. Therefore, only the relevant results are presented in the following paragraph. The initiation and evolution of damage at the crack tip near the center and near the surface of the specimen were determined and the results are shown in Fig. 8. It is noted that damage initiated earlier near the center than near the surface of the specimen. Since the damage state is closely related to the stress state, the difference in the damage initiation processes is due to the differences between the stress states near the center and near the surface of the specimens. It is known that near the center of the specimen, the stress state is close to the plane strain stress state as a result of relatively high constraint developed near the center of the specimen. However, near the surface of the specimen, the stress state is close to plane stress conditions. It is known that under a triaxial loading condition, it is relatively easy to develop microcracks in the binder and/or debonds at particle-binder interface. Therefore, it is expected that damage will initiate earlier near the center of the specimen. When the material is damaged, the stiffness, the magnitude of the stress, and the constraint in the damaged region will be reduced. Consequently, redistribution of the stresses occurs and the material adjacent to the damaged material will be subjected to a higher stress that, in turn, will induce damage to the material. These stress redistribution and damage evolution processes continue, and eventually all of the material in the immediate neighborhood of the crack is damaged, i.e., the thickness of the damaged material at the crack tip is equal to the thickness of the specimen. Under this condition, the constraints, both in-plane and out of plane, become insignificant. The uniform distribution of the damage along the crack front will result in a uniform distribution of stress. Since the crack growth behavior is controlled by the

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local stress at the crack tip, the uniform distribution of stress along the crack front will also result in a relatively straight crack front, as observed experimentally.

Referring back to Fig. 8, it is seen that in the early stage of damage evolution, the damage rate, expressed as the damage parameter per the applied strain, or the slope of the damage parameter versus the applied strain curve, is higher near the center than that near the surface of the specimen. However, the damage rate near the center of the specimen starts to decrease when the applied strain reaches 3% and the opposite is true near the surface of the specimen. Finally, damage saturation occurs near the center and the surface of the specimen at 6% applied strain level, resulting in a uniform distribution of damage at the crack tip and a straight crack front. In order to investigate the effect of damage at the crack tip on the distribution of along the crack front a three-dimensional linear elastic finite element analysis was conducted by Liu [8]. The results indicate that if there is no damage at the crack tip, the value of at the center of the specimen is about 12% higher than that near the surface of the specimen. However, when through-thickness damage occurs at the crack tip, the distribution of along the crack front is uniform. Since crack growth rate is control by it is expected that a constant along the crack front will lead to a straight crack front as mentioned before. A typical plot of the stress–strain curve of pre-crack specimen under 8697 kPa confined pressure is shown in Fig. 9. It is noted that, unlike ambient pressure test, the crack starts to grow in the linear region of the stress-strain curve and a considerable amount of stable crack growth occurs after the crack propagates. Under this condition, we calculated the crack growth rate, da/dt, and the Mode I stress intensity factor, and determined the relationship between da/dt and The results are discussed in the following paragraphs.


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The determination of the crack growth rate requires an analysis of discrete data relating the instantaneous time, t, to the corresponding crack length, a. Due to the nonhomogeneous nature of the particulate composite material, the measured data shows a considerable scatter. Therefore, it is anticipated that a smooth and steadily increasing relationship between the crack growth rate and time is difficult to obtain, and the different methods of da/dt calculation may result in different solutions [ 9]. Therefore, when selecting a method to calculate the crack growth rate from the raw experimental data, the accuracy and the scatter that introduced into the calculated crack growth rate by the selected data reduction method should be considered. As pointed by Liu (9), the secant method introduces a pronounced fluctuation of da/dt whereas the polynomial method results in a continuous smooth crack growth velocity curve. It is important and interesting to note that the fluctuation of da/dt, calculated by the secant method, is consistent with experimental observation. Based on experimental evidence, in general, the crack does not grow in a continuous and smooth manner. During the crack growth process, crack growth rate both accelerates and decelerates. Therefore, the secant method appears to provide the best estimate of both the actual crack growth process and the actual crack growth rate. However, it was found that the crack growth rate calculated by the secant method oscillates approximately around the smooth crack growth rate curve, obtained by the polynomial method. Therefore, if one is interested in knowing the average crack growth behavior, the fluctuation in crack growth rate seems unimportant and the polynomial method can be use to calculate the crack growth rate. It is based on this argument the polynomial method was used to calculate the crack growth rate in this study. A typical plot of crack growth rate da/dt versus the stress intensity factor for initial crack lengths of 0.1 in and 0.3 in. is shown in Fig. 10. From Fig. 10, a power law relationship exists between log (da/dt) and log which can be mathematically expressed as:

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Conclusions

In this study, the effects of specimen thickness and confined pressure on damage mechanisms and crack growth behavior in a particulate composite material were investigated. Experimental results indicate that, the critical Mode I stress intensity factor, for the onset of crack growth is insensitive to the specimen's thickness. Therefore, on the first approximation, it can be assumed that is independent of specimen’s thickness and plane strain fracture toughness does not exist for this material. Experimental results also indicate that brittle fracture occurs under ambient pressure, whereas a considerable amount of stable crack growth occurs under 8697 kPa confined pressure. Under the stable growth condition, a power law relationship exists between the crack growth rate and the Mode I stress intensity factor. 5.

References

1.

Beckwith, S.W. and Wang, D.T (1978)., Crack Propagation in Double-Base Propellants,” AIAA Paper 78-170 Liu, C.T. (1990) Crack Growth Behavior in a Composite Propellant with Strain Gradients - Part II, Journal of Spacecraft and Rockets, 27, pp. 647-659. Liu, C. T. (1990) Crack Propagation in a Composite Solid Propellant, Proceedings of the Society of Experimental Mechanics, Spring Conference, pp. 614-620. Liu, C.T. and Smith, C.W.,(1996) Temperature and Rate Effects on Stable Crack Growth in a Particulate Composite Material, Experimental Mechanics, 36 (3), pp. 290-295. Schapery, R.A. (1973) On a Theory of Crack Growth in Viscoelastic Media, Report MM 2765-731, Texas A&M University. Knauss, W.G. (1970) Delayed Failure – The Griffith Problem for Linearly Viscoelastic Materials, International Journal of Fracture Mechanics, 6, pp.7-20. Liu, C.T. and Kwon, Y.W. (1999) Numerical Modeling of Damage Initiation and Evolution Processes in a Particulate Composite Material, International Conf. on Computational Engineering and Science. Liu, C. T. (1990) Three-Dimensional Finite Element Analysis of a Damaged Fracture Specimen, Paper No. AIAA 90-2088, AIAA/SAE/ASME/ASEE Joint Propulsion Conf. Liu, C. T. (1990) Critical Analysis of Crack Growth Data, Journal of Propulsion and Power, 6 (5), pp. 519-524.

2. 3. 4. 5. 6. 7. 8. 9.

DYNAMIC FRACTURE EXPERIMENTS USING POINT IMPACT

D. RITTEL Faculty of Mechanical Engineering Technion, Israel Institute of Technology 32000 Haifa, Israel

Abstract This paper summarizes experimental results on dynamic crack initiation, using onepoint impact technique. Two specific issues will be addressed here. The first concerns the determination of the mode I dynamic fracture toughness of small cracked beam specimens. Next, some basic results related to the testing of standard Charpy specimens at high impact velocities will be presented. All these tests have in common the fact that the impact is applied in one point to an unsupported specimen, so that fracture is triggered by inertial effects only.

1.

Introduction

The concept of dynamic fracture of unsupported specimens is not a new concept. In fact it has been noted, e.g. by Kalthoff et al. (1977, 1983), that an impacted specimen which is originally supported (such as three-point bend) may fracture at times where the contact with the supports is lost. This observation has been applied to the development of impact tests of unsupported specimens. Similar observations have been made by Rittel and Maigre (1996), who studied the dynamic fracture of Compact Compression Specimens using a (split Hopkinson) bar (Kolsky, 1963). In these experiments, it was noted that fracture proceeded before the stress wave induced on the impacted incident side had exited the specimen on the transmitter side. Such an observation led to the conclusion that the transmitter bar was not needed in that case. In general, as long as fracture originates before the stress wave reaches the supports, one point impact experiments can be performed. This also implies that the investigated material is sufficiently “brittle” to fracture at the first passage of the stress wave. Such experiments have been also performed by Giovanola (1986), and a number of analytical solutions have been proposed for cracked beams, e.g. by Kishimoto et al. (1990), and by Rokach (1994). In the present paper, we will briefly describe the application of the one point impact technique to mode I and Charpy fracture studies. 267 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 267–274. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Experimental Results

2.1 DYNAMIC MODE I CRACK INITIATION IN SHORT BEAM SPECIMENS

Dynamic fracture toughness testing is not a standardized procedure, by contrast with static fracture toughness. Yet, the typical specimens are generally of a finite size, reaching sometimes several tens on centimeters. However, the final dimensions of manufactured products may be such that the specimens that can be machined are of limited size. Weisbrod and Rittel (2000) investigated the problem of testing short beam specimens machined in the radial direction of 25.4 mm diameter extruded bars. The central assumption was that linear elastic fracture mechanics concepts can be used to determine the dynamic stress intensity factor. It was therefore suggested that the linearity of the problem could be used advantageously, once the response of the cracked structure to a unit impulse is determined (dynamic weight function). The stress intensity factor K(t) can be calculated by a time convolution integral of the applied load with the dynamic weight function,

In the present case, the finite geometry of the specimen precludes the use of analytical solution, and the dynamic weight function was calculated using the finite element method. The dynamic fracture toughness is the value of the stress intensity factor at the onset of fracture, as indicated by a single wire fracture gage. A great advantage of the one point impact technique is that the boundary conditions are extremely simple and they reduce to the applied load. This is a hybrid experimental-numerical approach in which measured loads are blended with the calculated response to yield the stress intensity factor. Several experiments were performed using an instrumented bar as shown in Figure 1. The specimen was instrumented using a single wire fracture gage on one side and a small strain gage on the other side, both positioned in the immediate vicinity of the crack-tip. The validity of the approach was assessed by comparing the measured crack-tip strains with those obtained through the hybrid experimental-numerical procedure outlined in eq. (1), using the relationship between strain and stress intensity factor. Fracture was assessed in a straightforward manner from the single wire fracture gage on the one hand, and interpreted as causing a significant change of slope in the strain gage signal. An excellent correlation was observed between these two way of assessing the onset of crack-propagation. Figure 2 shows the dynamic stress intensity factors (SIF) as mentioned above. One can note the marked similarity between SIF’s, determined directly and

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indirectly. This observation validates the overall approach, including the assumptions related to the applicability of linear elastic fracture mechanics.

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Consequently, it can be concluded that the above-mentioned approach is suitable for the testing of any geometry of specimens, provided that linear elastic fracture mechanics (lefm) applies. This includes the present case of the small specimens presented here.

2.2 ONE-POINT IMPACT OF CHARPY SPECIMENS One-point impact can also be applied to standard or precracked Charpy specimens, provided the material is sufficiently “brittle” to fracture. In this paper, we will address an experimental issue related to the test itself, that is the kinematics of the specimen. Specifically, the problem dealt here is to determine how long does the specimen remain in contact with the incident bar. Moreover, a related issue is that of the correlation of the specimen kinematics with the strain gage readings of the incident bar.


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To address these issues, a series of experiments were carried out with Charpy steel specimens tested at room temperature. The selected steel (A508) was tough enough not to fracture at room temperature. In this case, The motion of the specimen was recorded using a high speed Imacon 790 camera. Special care was paid to the synchronization of the strain gage signals and the firing of the camera. Figure 3 shows a typical record of these experiments. The specimen is noted to remain in contact with the incident bar during the first 4 frames, and careful examination of the frame number 5 shows a very faint gap between the specimen and the bar that grows in the subsequent frames.

In parallel, the incident and reflected signals, recorded by the strain gage on the bars are shown in Figure 4. It can be noted that the two signals reach a similar level at the same time, marked as time 2. Keeping in mind that the applied force, F(t), is proportional to the sum of the incident and reflected signals, the equality of these just indicates that the applied load This is consistent with a free end

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condition for the bar, that corresponds to the specimen “take-off”. One can now correlate the load drop to zero to the instant at which the specimen separates from the incident bar. Several tests showed that the characteristic “take-off” time is of the order of which is much larger than the typical needed for a roundtrip of the stress wave, based on one dimensional wave propagation in a steel specimen. At this stage, low temperature fracture experiments can be carried out, during which the fracture time is recorded, using fracture gages. It is interesting to note whether fracture occurred while the specimen was still in contact with the bar or during its flight. It must be emphasize that the presence or lack of contact between the specimen and the bar at fracture does not affect the validity of the results whatsoever. However, if a measure of the fracture energy is to be assessed, the contact case is preferable since kinetic energy may be neglected at this stage (Rittel et al., 2002).


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Discussion

We have presented various aspects of one-point impact loading experiments. While such experiments have been carried out in the past, the present work emphasizes the advantages of the method which is relatively simple to operate and model numerically as it overcomes all the contact related issues of other methods, such as Charpy testing. Concerning fracture mechanics, a very simple way to determine the dynamic fracture toughness has been shown, that also applied to the case of relatively small specimens. In this method, one takes advantage of the linearity of the problem to determine the stress intensity factor as a function of time. Yet, issues such as loss of contact between the specimen and the bar and accuracy of fracture time determination are important factors in such experiments. Consequently, additional experiments have been carried out to assess the duration of the contact phase between the specimen and the bar. It was observed that for a standard Charpy specimen, this time is of the order of Of course, additional work is needed to obtain a more general relationship between the specimen dimensions and contact time. 4.

Conclusion

One-point impact testing is a convenient technique for dynamic fracture studies of relatively brittle materials. For mode I dynamic fracture toughness characterization, the technique provides valuable results on the basis of simple lefm concepts. The determination of the stress intensity factor of the selected geometry is based on the prior calculation of the structural response to unit impulse. The fracture toughness is determined as the value of the stress intensity factor at the onset of crack propagation. Next, one point impact can be applied to testing of standard notched Charpy specimens, by extending the range of impact velocities. While restricted to the lower shelf domain of steels, this technique can provide reliable values of the fracture energy of brittle materials, which is known to be a delicate issue. 5.

Acknowledgement

This research was supported by the fund for promotion of research at the Technion (030-094). Mr. G. Weisbrod and Prof. A. Pineau are gratefully acknowledged for their contribution on dynamic fracture and Charpy testing respectively.

6.

References

1.

Giovanola, J.H., (1986), “Investigation and application of the one-point bend impact test”, Fracture Mechanics: Seventeenth Volume, ASTM-STP 905, (Edited by Underwood, J.H., Chait, R. et al.), 307-328.

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4.

5. 6.

7. 8. 9.

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Kalthoff, J.F , Winkler, S. and Beinert J, (1977), “ The influence of dynamic effects in impact testing, International Journal of Fracture, Vol. 13, 528-531. Kalthoff, J.F., Winkler, S., Böhme, W., Shockey, D.A., (1983), in Proceedings, International Conference on the Dynamical Properties and Fracture Dynamics of Engineering Materials, Brno, Czechoslovakia. 12. Kishimoto, K., Fujino, Y., Aoki, S. and Sakata, M. , (1990), ” A simple formula for the dynamic stress intensity factor of an impacted freely supported bend specimen ”, JSME Int. J., Series I, Vol. 33, No.l., 51-56. Kolsky, H., (1963), Stress Waves in Solids, Dover Publications Inc., New York, NY. Rittel, D. and Maigre, H., (1996), "An investigation of dynamic crack initiation in PMMA", Mechanics of Materials, Vol. 28, 229-239. D. Rittel, A. Pineau, J. Clisson and L. Rota, “On testing of Charpy specimens using the one point bend impact technique”, (2002), in press, Experimental Mechanics. Rokach, I..V., “Comparison of simplified methods of dynamic stress intensity factor evaluation", (1994), Mechanika Teoretyczna I Stosowana, Vol. 1, No. 32, 203-212. Weisbrod, G. and Rittel, D., (2000), “A method for dynamic fracture toughness determination using short beams”, Intl. Journal of Fracture, Vol. 104, No. 1, 89-103.

EXPERIMENTAL AND NUMERICAL INVESTIGATION OF SHEARDOMINATED INTERSONIC CRACK GROWTH AND FRICTION IN UNIDIRECTIONAL COMPOSITES

A. J. ROSAKIS, C. YU, M. ORTIZ Graduate Aeronautical Laboratories, California Institute of Technology Pasadena, CA 91125 D. COKER Division of Engineering, Brown University Providence, RI 02912 A. PANDOLFI Dipartimento di Ingegneria Strutturale, Politecnico di Milano 20133 Milano, Italy

Abstract

Dynamic crack growth in unidirectional graphite/epoxy composite materials subjected to in-plane impact loading is investigated experimentally and numerically. The experiments are conducted using CGS (Coherent Gradient Sensing) Interferometry in conjunction with high-speed photography to visualize the crack growth events. Cracks are found to propagate at subsonic speeds in the Mode-I case, whereas in both mixed mode and Mode-II the crack tip speed clearly exceeds the shear wave speed of the laminate. For these intersonically growing shear (Mode-II) cracks a shock wave emanating from the crack tip is observed. This provides direct evidence that the cracks propagate faster than the shear wave speed of the composite. The crack tip speed is initally observed to jump to a level close to the axial longitudinal wave speed along the fibers (7500 m/s) and then to stabilize to a lower level of approximately 6500 m/s. This speed corresponds to the speed at which the energy release rate required for shear crack growth is non-zero as determined from asymptotic analysis. The CGS interferograms also reveal the existence of large-scale frictional contact of the crack faces behind the moving shear cracks. In addition high speed thermographic measurements are conducted that show concentrated hot spots behind the crack tip indicating crack face frictional contact. These experiments are modeled by a detailed dynamic finite element calculation involving cohesive elements, adaptive remeshing using subdivision and edge collapse, composite elements, and penalty contact. The numerical calculations are calibrated on the basis of fundamental material properties measured in the laboratory. The computational results are found to be in excellent agreement with the optical 275 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 275–288. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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experimental measurements (crack speed record and near tip deformation field structure). For shear crack growth, the numerics also confirm the optical observation of large-scale crack face contact. 1.

Introduction

Dynamic crack growth along weak planes is a significant mode of failure in composites and other layered materials. In the past years bimaterial fracture specimens, fabricated by the bonding of a stiff material to a compliant material (featuring a mismatch in wave speeds), have been used to demonstrate the importance of highly transient and dynamic crack growth in heterogeneous solids. It was observed that interfacial crack tip speeds rapidly approached and exceeded the shear wave speed of the more compliant material [1-5]. Thus they reached intersonic speeds with respect to the more compliant material in which intersonic crack tip speed is defined as the speed in the open interval between the shear wave speed and the longitudinal wave speed. For crack tip speeds above the shear wave speed a ray emanating from the crack tip representing a line of strong discontinuity (shear shock wave) was observed experimentally and also predicted theoretically [2]. These high crack tip speeds were obtained under loading conditions that promoted locally shear dominated deformations at the crack tip which were further enhanced by the stress wave mismatch due to the bimaterial nature of the solid [6]. In homogeneous materials and in the absence of a weak prescribed crack growth path, Mode-II crack propagation is not possible because the crack naturally kinks and propagates in a direction that causes a local Mode-I stress state around the crack tip. This is not necessarily true in solids that may be homogeneous regarding their properties but may involve preferable crack paths in the form of weak planes of lower toughness. Unidirectional composites may indeed fit this description if viewed from the "macroscopic" point of view of a homogenized anisotropic theory. Although from the microscopic viewpoint, unidirectional composite materials are locally inhomogeneous and are related to bimaterials, in the sense that they also involve a preferred crack growth direction, and are also composed of a stiffer material (the fiber) bonded together with a more compliant material (the matrix), from a macroscopic point of view such solids can be viewed as homogeneous anisotropic materials as far as their elastic properties are concerned. However, they are still inhomogeneous regarding their fracture toughness properties. Indeed, from both the macroscopic or microscopic view points the common characteristic between unidirectional composites and bimaterials which also seem to be the most relevant is the existence of a weak straight line crack path which may accommodate growing cracks of both modes and not the existence of the two phases. Because of the above observations and because of our previous discovery of intersonic crack growth in bimaterials, unidirectional composites seem to be natural candidates for studying maximum allowable crack tip speeds of cracks of both modes in a system that is macroscopically at least homogeneous. While the theory excludes intersonic growth of mode-I cracks, it has not excluded the possibility of intersonic mode-II dominated crack growth in isotropic or anisotropic homogeneous elastic solids in the case of a self similar crack growing in the direction of the plane of the crack [7]. Several researchers [7-11] calculated the critical speed at which intersonic crack growth is stable in isotropic materials to be For stable intersonic

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crack growth this is the only speed at which the energy release rate is finite. Analysis of the intersonic crack growth in anisotropic materials was developed by Piva and Hasan [12], Huang et al. [8] and Broberg [13]. They predicted that, just as in isotropic materials, intersonic shear crack growth is possible at a specific critical speed, above the shear wave speed. However, in the case of anisotropic materials, this speed is the product of a complex function of the material properties and the shear wave speed. To our knowledge, very few experimental studies of dynamic crack propagation in fiberreinforced composites have been performed. Liu et al. [14], investigated quasi-static fracture of polymer composites using the optical technique of CGS. Khanna and Shukla [15] extended the theoretical results of Piva and Viola [16] to in-plane stress and used it to analyze the results of strain gages and to thus determine mode-I stress intensity factors for cracks propagating in unidirectional glass-epoxy composite laminates at constant speed. Lambros and Rosakis [17], using CGS shearing interferometry technique in conjunction with high speed photography, looked at the initiation and growth in thick unidirectional graphite-epoxy composite plates under symmetric 1-pt impact loading. They observed crack tip speeds for mode-I loading which approached 900 m/s or where is the speed of the Rayleigh waves travelling in the direction of the fibers. In another investigation, Lambros and Rosakis [18] also studied the dynamic delamination propagation in real time of a cross-plied laminate subjected to out-of-plane impact. In this paper we summarize some recently obtained experimental results of dynamic crack propagation in unidirectional graphite-epoxy composite plates under mode-I and mode-II loading. Many of the experimental details have already appeared in an extensive article by the first two authors (Coker and Rosakis [21]) and will not be discussed again here. The optical technique of CGS was used in conjunction with high-speed camera to obtain interferograms during crack growth. Crack tip speeds are then computed from the interferograms and the limiting crack tip speeds are determined for both the mode-I and the mode-II cases. These experiments are modeled by finite element analysis involving cohesive elements and penalty contact. In addition, high speed thermographic measurements are conducted which show regions of contact behind the crack tip. The numerics demonstrate good agreement with the experiments for both crack tip speeds and predicted contact behavior at the crack faces.

2.

Material Characterisitcs

A cross-section of the composite material is shown in Fig. 1. The unidirectional graphite/epoxy composite plates were manufactured from 48 layers of graphite fiber and epoxy matrix pre-pregs laid up in the thickness direction to form a 6.3-mm thick plate. The fiber volume fraction in the prepreg is 65% while the fiber diameter is 7.3 The surface on one side of the composite plate was made optically flat by adding a thin layer of epoxy on one surface and then curing the composite specimen upon an optically flat glass plate. This surface was then made specularly reflective by coating with a thin layer of aluminum of 1-2 thickness in a vacuum deposition chamber. The specimen geometry is shown in Fig. 2. The orientation of the axes with respect to the composite plate is shown in Fig. 1. The axis is defined to lie along the fibers, the axes is perpendicular to the plane of the composite surface while the axis is perpendicular to

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the plane. The orthotropic elastic constants, the compliance matrix and stiffness matrix values for graphite/epoxy unidirectional composite material are given in Table 1.

In Table 2 denotes the dilatational wave speed parallel to the fibers while denotes the dilatational wave speed perpendicular to the fibers. The density is 1478 The shear wave speed as well as the longitudinal wave speeds parallel and perpendicular to the fibers are shown in Table 2. The Rayleigh wave speed parallel to the fibers is 1548 m/s.


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Experimental Technique

Schematic of the experimental set-up and the optical technique of coherent gradient sensing, CGS, is shown in Fig. 3. The specimen was subjected to impact loading through a projectile fired from a gas gun. The dynamic stress field produced by the impact loading wave leads to a changing out-of-plane deformation on the surface of the composite plate. The optical technique of reflection CGS, in conjunction with highspeed photography, was used to record in real time the slopes of the out-of-plane deformation field. CGS is a full-field, lateral-shearing interferometer. The details of CGS can be found in several articles [19, 20] and its application to crack initiation in composite materials has been demonstrated before [14, 17]. CGS, when used in reflective mode, measures the in-plane gradients of out-of-plane displacement.

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Symmetric Mode-I crack tip deformations were attained by impacting the specimen symmetrically along the notch line by a steel projectile as shown in Fig. 3 with projectile speeds varying from 10 m/s to 57 m/s. A typical sequence of experimental CGS interferograms for mode-I crack initiation and propagation are shown in Fig. 4 for the highest impact speed of 57 m/s. The first frame shows the CGS interference fringe pattern around the notch tip just before crack initiation. The fringe pattern around the crack tip during propagation at times 1.8 and 3.0 is shown in Figs. 4(b) and (c). Using the above sequence of pictures the crack tip history was recorded as a function of time for Mode-I symmetric loading. The crack tip speed history was then obtained by using 3-point fit to the crack tip history. The crack started growing subsonically at 1350 m/s and accelerated to 1550 m/s which is the Rayleigh wave speed of the composite in the direction of the fibers. The crack tip speeds never exceeded the Rayleigh wave speed. Asymmetric 1-pt bend impact experiments were conducted at different impact speeds by impacting the specimen below the notch. Projectile speed varied from 21 m/s to 30 m/s. For an impact speed of 21 m/s a sequence of three frames taken with the high speed camera at an interframe time of 1.39 is shown in Fig. 5. The notch tip as well as the stress concentration at the notch tip is clearly visible in the first frame. The impact wave has propagated from one end of the plate to the other, was reflected as a tensile wave below the notch and loaded the notch tip in a predominantly shear mode. Notch-tip stress concentration is seen to increase as the interferogram fringes increase in size and number. The nature of the fringe pattern shows that the loading is primarily mode-II [14] featuring one fringe lobe in the back and two fringe lobes in the front. These pictures show the fringe patterns around the notch tip just before and after crack


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initiation (last photograph). However, in the last frame, a significant change takes place in which the fringe loops are squeezed backwards indicating very high initial crack tip accelerations. Indeed, the average crack tip speed between these two frames is 2100 m/s, which is above the shear wave speed 1800 m/s for this material. This dramatic change in shape is not observed in the mode-I crack growth experiments. The notch tip is loaded by the arrival of the loading wave initially at 11 after impact and is fully loaded by the reflected wave at 15 Fig. 6 shows a sequence of three CGS interferograms featuring shear dominated dynamic crack growth. The rear loop shape changes from rounded to a triangular wedge bounded by a line of high intensity fringes emerging from the crack tip at an angle. This line is caused by a steep change in the stress gradients in a localized area, which later forms a discontinuity in the shear stresses, i. e. a shear shock wave. Although CGS is sensitive to the sum of the normal stresses in homogeneous, isotropic materials, the normal stresses and shear stresses are coupled in stresses in our experiments. This would not hold for isotropic solids. Finally, this line broadens into two parallel lines (a double shock wave) and intercepts the crack surface over a finite area between the crack tip and 4-5 mm behind it. We propose for the reason of the double shock wave structure as the existence of a contact region behind the crack tip or alternatively as the cohesive zone in front of the crack tip. The slope of the shear shock wave changes as you move away from the crack tip which may be due to the unsteady growth of the crack tip. The crack tip location and speed history is shown in Fig. 7(a) and 7(b). The crack-tip speed approaches 5200 m/s, a speed 3 times higher than the shear wave speed and is clearly intersonic. In this experiment the crack tip jumps immediately from rest to 2100 m/s becoming intersonic in the first frame after initiation, thus not passing through the subsonic regime. The crack tip acceleration was also very high and was of the order of From Fig. 7 it can be seen that the crack propagation had not reached steady state within our window of observation. A separate experiment was conducted under similar conditions with the field of view further downstream of the notch in order investigate whether the crack tip speed eventually attains steady state conditions following the initial acceleration stage. In this case the impact velocity was 28 m/s and the interframe time was 0.83 Fig. 8 shows a selected set of CGS fringe patterns. As already noted from the previous case, shear shock waves are also observed. The first recorded crack tip speed is 4000 m/s as the crack appears in our field of view. The crack tip speed then climbs up and oscillates around and an average speed of 7000 m/s. This is the highest crack tip speed ever observed in laboratory setting (for details see reference [21]).

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Discussion

A view of the general trends in crack tip position history characteristics of many different experiments corresponding to very similar impact conditions have been collected and displayed in Fig. 9(a). This figure shows the reproducibility of the observed experimental trends (within the same geometry) and allows us to obtain an average sense of the trends in the variation of the intersonic crack tip speed for both mode-I and mode-II cases. Crack tip speed as a function of crack extension is shown in Fig. 9(b). Mode-I cracks initiate around 1300 m/s and subsequently the crack speed increases to the neighborhood of the Rayleigh wave speed of 1548 m/s. However, mode-II cracks initiate around 2000 m/s (intersonic) and rapidly accelerate above 6000 m/s. These cracks asymptotically approach a steady state speed of 6500 m/s. In order to interpret the observed phenomena theoretical analysis was developed by Huang et al. [8]. They obtained the asymptotic stress and displacement fields near an intersonically propagating crack tip in an orthotropic material under steady-state conditions. In this analysis, the composite was modeled as an elastic, orthotropic, homogeneous solid. For both mode-I and mode-II cases a prescribed straight-line crack path was assumed. Mode-I elastic asymptotic field yielded a negative, unbounded crack tip energy release rate in the entire intersonic crack growth regime. This is physically impossible since a propagating crack-tip cannot radiate out energy. Thus, a mode-I crack tip cannot propagate intersonically. This conclusion is supported by our experiments in which the crack tip speed never exceeded the shear wave speed regardless of how large the impact velocity of the projectile was (see Fig. 9b). For mode-II, the energy release rate supplied by the elastic asymptotic field is finite and positive only at a distinct critical crack tip speed [8]. Since a positive and finite energy supply is required to break the material bonds in front of the crack tip this speed corresponds to the only possible steady state intersonic crack growth speed according to the above steady state theory. Using the material properties given in Table 1, this critical crack tip speed is 6600 m/s.

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In our experiments the mode-II crack tip seem to eventually reach this value sometimes from below and sometimes from above [21].

6.

Finite Element Simulations

The shear dominated crack growth experiments were modeled with dynamic finite element analysis using cohesive elements [22,23]. An automatic self-adaptive procedure was used to explicitly reproduce crack propagation by inserting a new cohesive surface along previously coherent inter-element surfaces when the local traction reaches a limit value. The analysis consisted of cohesive elements and contact elements along the line of the initial crack. The numerical calculations were calibrated on the basis of fundamental material properties measured in the laboratory. The loading history and the mesh geometry using a minimum mesh size of 1mm, 44593 nodes and 24685 elements is shown in Fig. 10. The composite is modeled as a homogeneous orthotropic material with the principle axis of symmetry along the original crack. Assuming a cohesive model for the description of the progressive decohesion between two fracture surfaces, the characteristic strength of the cohesive law, as well as the fracture energy, is locally varied with the angle with respect to the fiber orientation. A bilinear cohesive model (see Fig. 10) relating an effective cohesive traction to an effective displacement is chosen to avoid instabilities. Following Camacho and Ortiz [24] the effective opening displacement is defined as where and are the normal and sliding displacements, respectively. The effective traction is given by where defines the ratio between the shear and normal tractions. If we take the shear and opening critical displacements to be the same, is roughly the ratio


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of to of the material. Upon closure the cohesive surfaces are subjected to a unilateral contact constraint and Coulomb’s friction law. The parameters in the model are obtained from fracture and strength experiments (surface energy N/mm, maximum cohesive stress, Mpa, critical displacement, and the weighting coefficient, The contours of constant shear stress obtained from the numerical simulations are shown in Fig. 11. The shear shock wave on the upper part of the crack line can be clearly distinguished in the numerical simulations. The contours resulting from the reflected wave due to impact can be seen to be effecting the lower part. The feature of a double shock wave is captured very well by the numerical simulations. The crack tip speed history obtained from the numerical runs is shown in Fig. 12 for an impact speed of 30 m/s. The crack tip speed history is obtained by differentiation by using a sectional quadratic three-point fit to the crack length history. The plot shows the experimental points (symbols) and the numerical simulations (lines) for different mesh sizes. There is very good agreement between the experimental and numerical results showing the initiation at intersonic speeds and steady climb to crack tip speed between vc and the longitudinal wave speed. 7.

High Speed Infrared Thermal Measurements

Since the events we are trying to observe occur over a few microseconds, a thermal camera designed to capture images at a rate of with a response time of 500 ns is used [25]. The camera features a square array of 64 detectors, in an 8 x 8 format. Each detector is 100 x 100 in size and is separated by a distance of 30 from its neighbor. The detectors are operated at a temperature of 77 K to maximize the signal to noise ratio. The camera is focused on 1.1 mm by 1.1 mm area ahead of the notch tip. Figs. 13(a) show contours of constant temperature lines. In the first image, the crack tip can be seen to be approaching from the left hand side. In the second image the crack has already left our field of view at after impact. Behind the crack tip we can observe local hot spots forming behind the crack tip due to crack surface sliding. These local hot spots of high temperature frequently jump around. The temperature increase is due to frictional sliding of the crack surfaces and later becomes a band of high temperature region enveloping several layers of fibers. This band grows in width and finally saturates the infrared detectors. The area where frictional sliding is taking place is of the order of However, the temperatures measured are being averaged over a area for each detector so that the recorded temperatures are upper lower bounds for the actual temperatures due to sliding. A sequence of numerically calculated temperature fields are shown in Fig. 13(b) showing localized high temperature fields indicating the existence of frictional sliding behind the crack tip. There is qualitatively very good agreement between the experimental and numerically generated temperature fields.

8.

Acknowledgement

This investigation was supported by the Office of Naval Research (Dr. Y. D. S. Rajapakse, Scientific Officer) through grant #N00014-95-1-0453 to Caltech and is gratefully acknowledged.

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References Lambros, J. and A. J. Rosakis, “Shear dominated transonic interfacial crack-growth in a bimaterial: 1. Experimental observations,” Journal of the Mechanics and Physics of Solids, 43(2), 1995, 169-188. Liu, C., Y. Huang, Rosakis, A. J., “Shear dominated transonic interfacial crack growth in a bimaterial: 2. Asymptotic fields and favorable velocity regimes,” Journal of the Mechanics and Physics of Solids, 43(2), 1995, 189-206. Singh, R. P., J. Lambros, Shukla, A. and Rosakis, A. J., “Investigation of the mechanics of intersonic crack propagation along a bimaterial interface using coherent gradient sensing and photoelasticity,” Proceedings of the Royal Society of London Series A, 453(1967), 1997, 26492667.

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7. 8.

9. 10. 11. 12. 13. 14.

15.

16. 17. 18. 19.

20.

21.

22.

23.

24. 25.

A. J. ROSAKIS, et ai.

Singh, R. P. and Shukla, A., “Subsonic and intersonic crack growth along a bimaterial surface,” Journal of Applied Mechanics 63, 1996, 919-924. Rosakis, A. J., Samudrala, O., Singh, R. P. and Shukla, A., “Intersonic crack propagation in bimaterial systems.” Journal of the Mechanics and Physics of Solids, 46(10), 1998, 1789-1813. Lambros, J. and Rosakis, A. J., “Development of a dynamic decohesion criterion for subsonic fracture of the interface between two dissimilar materials,” Proceedings of the Royal Society of London Series A, 451(1943), 1995, 711-736. Broberg, K. B., “The Near-Tip Field at High Crack Velocities.” International Journal of Fracture, 39, 1989, 1-13. Huang, Y., Wang, W., Liu, C. and Rosakis, A. J., “Analysis of intersonic crack growth in unidirectional fiber-reinforced composites,” to be published in Journal of the Mechanics and Physics of Solids, 1999. Burridge, R., “Admissible speeds for plane-strain self-similar shear cracks with friction but lacking cohesion,” Geophysical Journal of the Royal Astronomical Society, 35, 1973, 439-455. Freund, L. B., “The Mechanics of Dynamic Shear Crack Propagation,” J. of Geophysical Research, 84(B5), 1979, 2199-2209. Georgiadis, H. G., “On the stress singularity in steady-state transonic shear crack propagation,” International Journal of Fracture, 30, 1986, 175-180. Piva, A. and Hasan, W., “Effect of orthotropy on the intersonic shear crack propagation,” Journal of Applied Mechanics, 63,1996, 933-938. Broberg, K. B., “Intersonic crack propagation in an orthotropic material,” International Journal of Fracture, to be published, 1999. Liu, C., Rosakis, A. J., Ellis, R. W. and Stout, M. G., “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” International Journal of Fracture, 90, 1998, 355-382. Khanna, S. K. and Shukla, A., “Development of stress-field equations and determination of Stress intensity factor during dynamic fracture of orthotropic composite materials,” Engineering Fracture Mechanics 47(3), 1994, 345-359. Piva, A. and Viola, E., “Crack-propagation in an orthotropic medium,” Engineering Fracture Mechanics, 29(5), 1988, 535-548. Lambros, J. and Rosakis, A. J., “Dynamic crack initiation and growth in thick unidirectional graphite/epoxy plates,” Composites Science and Technology, 57(1), 1997, 55-65. Lambros, J. and Rosakis, A. J., “An experimental study of dynamic delamination of thick fiber reinforced polymeric matrix composites,” Experimental Mechanics, 37(3), 1997, 360-366. Tippur, H. V., Krishnaswamy, S. and Rosakis, A. J., “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” International Journal of Fracture, 48, 1991, 193-204. Rosakis, A. J., “Two optical techniques sensitive to the gradients of optical path difference: The method of caustics and the coherent gradient sensor (CGS),” Experimental Techniques in Fracture, J. S. Epstein, Ed., New York, VCH, 1993, 327-425. Coker, D. and Rosakis, A. J., “Experimental Observations of Intersonic Crack Growth in Asymmetrically Loaded Unidirectional Composite Plates,” GALCIT SM Report No. 98-16, Pasadena, California Institute of Technology, 1998 and Philosophical Magazine A, 81, 571-595, 2001. Ortiz, M. and Pandolfi, A., “Finite deformation irreversible cohesive elements for threedimensional crack propagation analysis,” International Journal for Numerical Methods in Engineering, 44, 1267-1282, 1999. Yu, C., Pandolfi, A., Ortiz, M., Coker, D. and Rosakis, A. J., “Three-dimensional cohesive modeling of impact damage in composites,” submitted to International Journal of Solids and Structures, 2001. Camacho, G. T. and Ortiz, M., “Computational modeling of impact damage in brittle materials,” Int. J. Solids and Structures, 33 (20-22), 2899-2938, 1996. A. T. Zehnder, P. R. Guduru, A. J. Rosakis and G. Ravichandran, “Million Frames per Second Infrared Imaging System”, Review of Scientific Instruments, 71 (10), 3762-3768, 2000.

4. Optical Methods


MOIRÉ INTERFEROMETRY—PAST, PRESENT AND FUTURE DANIEL POST Virginia Polytechnic Institute and State University Prof. Emeritus, Engineering Science and Mechanics Blacksburg, VA 24061

Abstract Moiré interferometry is introduced as an outgrowth of moiré fringe multiplication. Coherent laser light enabled the practical implementation of virtual reference gratings and high-frequency specimen gratings. Reliable analysis of shear depended upon the ideas in a little-known paper. These three facets are the essence of current techniques. Present-day applications are exemplified by work in electronic packaging and composite materials. Several significant advances are forecast to further broaden the usefulness and appeal of moiré interferometry.

1.

Introduction

This paper is dedicated to Professor Isaac M. Daniel, on the occasion of the Symposium presented in his honor at the 14th U.S. National Congress of Applied Mechanics, June 2002. I feel honored, too, to participate in this symposium together with renowned colleagues, gathered to express our admiration and offer tribute to Isaac. His extensive contributions and his gentle, helpful nature are broadly recognized and appreciated. We have had numerous exchanges through many years, but Isaac and I have only one mutual publication [1], co-authored also by our colleague and friend, Bob Rowlands. That work is 30 years old, but it is a fitting entrance to the current topic—the past, present and future of moiré interferometry. 2.

A Perspective of the Past

Moiré interferometry is a special case of moiré fringe multiplication—it is fringe multiplication by a factor of two. In the past, only low frequency gratings could be applied to specimens. However, fringe multiplication utilizing a higher frequency reference grating increased the sensitivity, since sensitivity corresponds to the frequency of the reference grating. In the aforementioned paper with Isaac, fringe multiplication by a factor of 10 was achieved. The glass/epoxy specimen, with a 500 lines/inch specimen grating, was loaded and the deformed specimen grating was photographed. The image on the photographic plate was a transparent replica of the deformed grating. The optical system for fringe multiplication was similar to that of Fig. 1, where the photographic plate was the specimen and the frequency of the reference grating was 10 291 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 291–302. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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times higher. Note that fringe multiplication by a factor of 6 is shown, whereas in the case of multiplication by 10 the beams of +5 and –5 diffraction orders were isolated. It was exhilarating to see the enhanced sensitivity, with 10 times as many moiré fringes as conventional moiré. Several innovations evolved to link the technique 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 two-beam optical interference even when the beams traveled substantially unequal path lengths. The laser led to a practical technique to produce high-frequency specimen gratings. A two-step process was developed. First, a high-frequency grating was made by a socalled holographic process. A high resolution photographic plate was exposed to two coherent collimated beams, incident at angles and After processing and drying, the emulsion shrank in the exposed zones, while shrinkage was restrained in the unexposed (destructive interference) zones by the silver grains. Thus, the plate had an undulating surface of ridges and furrows that formed the high-frequency phase grating.[2] Actually, two exposures were made, with the plate rotated 90° between exposures to produce a cross-line grating with ridges in orthogonal x and y directions. In the second step, the photographic plate was used as a mold to cast, or replicate, the phase grating on the specimen surface. Silicone rubber was used as the replication material. With this development, it was no longer necessary to use high diffraction orders to achieve high sensitivity. Instead, fringe multiplication by a factor of two could be used, whereby the beams of +1 and –1 diffraction orders were combined to form the moiré pattern.

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In addition, it was realized that the high-frequency reference grating was not required at all. Instead, the two beams indicated in Fig. 1 by rays labeled 0 and 1 could be replaced by two coherent beams generated by any optical system.[2,3] The two beams create a virtual reference grating. One such system is illustrated in Fig. 2(a) for a transparent specimen, where the corresponding beams are labeled 0 and 1. The symbol for an eye represents a camera. A similar scheme was used in reflection for opaque specimens. Figure 2(b) illustrates the arrangement used to analyze off-axis composite specimens of graphite/polyimide.[3].

Another important advance relied on an early paper addressing geometrical moiré, the moiré grid-analyzer method.[4] If instrumentation could be arranged for viewing the U and V fringe patterns without moving the sample or the interferometer, it becomes possible to extract shear strains without ambiguity. This realization led to the 4-beam arrangement illustrated schematically in Fig. 3. Numerous implementations of Fig. 3 have been produced, and several are illustrated in Ref. [5]. Improved techniques for specimen gratings evolved. These include molds that utilize photoresist to optimize their diffraction efficiency, and molds with a transferable film of aluminum to maximize reflectivity. The evolution continued with the development of apparatus and techniques for microscopic moiré interferometry, including a robust algorithm called O/DFM, the optical/digital fringe multiplication method.[5] Displacement fields with contour intervals as small as 17 nm were demonstrated.

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The foregoing is a brief perspective on moiré interferometry in terms of my own experience and that of my students. We knew that moiré interferometry would evolve into a practical technique as early as 1982, when Weissman demonstrated superb fringe patterns at 97.6% of the theoretical upper limit of sensitivity, with a virtual reference grating frequency of 4000 lines/mm (101,600 lines/inch).[6] Closely related work progressed elsewhere in the U.S., Europe and Asia. In his paper “A Historical Review of Moiré Interferometry,” Walker reviewed the chronology of many of the contributions.[7] He recounted the pioneering developments in Japan, which predated accomplishments in the Western world by several years, but was not widely known in the West. He outlined work by Sciamarella and by Post and his students in the U.S., and by several investigators in Europe—especially the outstanding work at Strathclyde University by Walker, McKelvie and their colleagues. Moiré interferometry has matured through worldwide efforts. It has gained extensive application in the micromechanics industry for the analysis of thermal stresses in electronic packages, and it serves in numerous other technologies as an important tool of experimental solid mechanics. It will continue to evolve through worldwide activity.


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3. The Present The current status of moiré interferometry is revealed in the open literature, and also in Ref. 5, which is a comprehensive volume on the subject. It has been used effectively in a highly diverse array of disciplines, especially electronic packaging, but also composite materials, fracture mechanics, biomechanics, metallurgy, ceramics, polymers, concrete, mechanical joints, adhesive joints, and undoubtedly other disciplines too. The examples selected here emphasize accommodation for special problems. In the first and third examples, the frequency of the virtual reference grating was 2400 lines/mm, providing contour intervals of 0.417 µm/fringe order. In the second example, the contour interval was 35 nm/contour. 3.1 REAL-TIME THERMAL DEFORMATION

The design of electronic devices is critically dependent upon knowledge and management of thermal deformations, strains and stresses. Figure 4 illustrates the complexity of such analyses. The device is composed of diverse materials, each with a different coefficient of thermal expansion and difference creep/relaxation behavior. It must resist temperature extremes in operation and in shipment without mechanical or electrical failure. The fringe patterns are contour maps of the U and V displacement fields occurring at different temperatures in a thermal cycle. 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. 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. For these tests, the specimen is cycled in an environmental chamber that provides controlled heating and cryogenic cooling. The specimen is not attached physically to the chamber, but instead it is supported by arms extending from the moiré interferometer through the chamber walls, and thus it is isolated from vibrations of the chamber.[8] 3.2 BITHERMAL DEFORMATION In this application of microscopic moiré interferometry, the specimen was a unidirectional boron/aluminum composite subjected to thermal loading. Bithermal analysis was applied, whereby a cross-line grating was replicated on the specimen at 132°C. The specimen was cooled to room temperature and then observed in a microscopic moiré interferometer. The reference grating frequency was f = 4800 lines/mm (122,000 lines/inch); the fringe multiplication factor (by O/DFM) was the contour interval was 35 nm per contour. [5] Figure 5 shows the combined effects of load-induced displacements plus free thermal expansion When is subtracted off for each material, the substantial range of strains occurring in the aluminum matrix becomes evident.

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3.3. 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. Such a case is the ply-by-ply deformation at a central hole in a multi-ply composite plate.[9] 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 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 it 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 in a moiré interferometer and deformation data were recorded by a moiré interferometer as a series of narrow strips, each approximating a flat surface. Figure 6 shows a mosaic of such strips for a 90° portion of the hole in a cross-ply laminate of IM7/5250-4 L(graphite fibers in a bismalimide-cyanate ester matrix). An enlarged view of the fringe pattern at the = 75° location is shown in Fig. 6(d), together with a graph of the ply-by-ply distribution of shear strains. The patterns reveal the data 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 A primary purpose of the work was to provide experimental data for the evaluation of computational techniques of analysis for composite structures. 4.

The Future

I see moiré interferometry as an extraordinarily useful tool—a tool that will continue to expand in scope and utility as technology and science cope with evermore complex materials and structures. It is ideal to meet the challenges of small size and high-level dependability, among others. We can expect advances in several categories, whereby the technique will further broaden in usefulness and appeal. 4.1. REPLICATION OF DEFORMED GRATINGS

The experiments conducted with Isaac in 1972 utilized photographic replicas of the specimen grating, taken with the specimen at different load levels. Subsequently, replication of deformed phase gratings in plastic or elastomer became practical. Replication was used to measure the deformation of an electronic package at a cyrogenic temperature.[5] In the work described above, replication was used to measure deformations along curved surfaces of composite laminates.[8] In a classical paper by McKelvie and Walker, replication was advocated for remote sites and harsh environments, followed by analysis of the replicas in the laboratory.[9] 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


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desired intervals in the loading process, and subsequently the replicas are analyzed in a moiré interferometer.

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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, which otherwise might have to be suppressed for observations with a moiré interferometer, are inconsequential. The technique can be applied conveniently in the field, far from the laboratory, 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, very high magnifications become convenient. Additionally, the replicas become permanent records of the deformation, so there is easy recourse to checking and extending an analysis after an initial investigation. What developments are required? Although the replication method can be practiced with current technology, it would be advantageous to find materials and techniques for quick and routine replication of deformed gratings. I believe these will be optimized and broadly implemented.


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4.2. DATA REDUCTION FROM SINGLE IMAGES 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 in order to reduce these data to graphs of displacement and strain distributions. In many cases, the measurements are conducted at abnormally low load levels in order to reduce the number of moiré fringes in the field, since the phase-stepping algorithms are more effective with sparse fringe patterns. This is artificial suppression of data, and the practice seems counterproductive. Automated analyses do not recognize extraneous inputs like those from scratches or other imperfections of the specimen grating. Automated analyses do not cope well with rapidly changing displacement fields. Special cautions are required if the region of interest extends across dissimilar materials. Thus, for the majority of applications—those that exhibit numerous fringes— another scheme of data reduction would be beneficial. New schemes of computer aided analyses will be developed that produce displacement and strain graphs from a single pair of moiré fringe patterns, i.e., single images of the U and V fields. They will require user input, firstly to choose regions of interest and to determine whether carrier fringes should be used in these regions; and secondly to deal with possible imperfections in the fringe patterns. The computer techniques will likely quantify the location of fringes and graph displacement fields from these; apparent displacement from carrier fringes would be subtracted off. Strains would be calculated from computer measurements of the x and y components of distance between fringes, either the distance between neighboring fringes or between more distant fringes; the operator would decide. We can foresee the development of a variety of algorithms that reduce tedious aspects of the analyses, but retain the logical step-by-step human input that is present in manual analyses. Fourier methods utilize single fringe patterns and they offer current alternatives to phase-stepping analyses. However, they do not offer the high fidelity and user discretion and control that may be desired. As the technology evolves, Fourier techniques may be employed for a global view of the strain field, followed by a direct computer aided analysis for local regions of special interest. The key will be human understanding and control, replacing relatively blind automation. 4.3. INSTRUMENTATION Less complicated designs for moiré and microscopic moiré interferometer systems are anticipated. Instruments designed exclusively for transparent replicas will be the least complex, but those offering the greatest versatility will be simplified as well. Emphasis will be placed on variable image magnification so that regions of high fringe density can be resolved by using higher magnifications, and global patterns can be recorded with lower magnifications. Strong and uniform light distribution will be emphasized to achieve high quality fringe patterns at all magnifications. 4.4. MICROMECHANICS We can anticipate extensive use of microscopic moiré interferometry for micromechanics studies. Special instrumentation will continue to be developed.


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Techniques to enhance the measurement sensitivity will be advanced for microscopic moiré interferometry. Since the basic sensitivity depends upon the frequency, f, of the virtual reference grating, further increases of f will be introduced by utilizing light of shorter wavelengths. Ultraviolet light or immersion interferometry, or both, will provide the shorter wavelengths. The enhancement of basic sensitivity will help, but in most cases the number of fringes in the field will be too small for an accurate analysis. Additional data will be needed and these data will be obtained by phase-stepping. The uncertainties introduced by random electronic noise in the CCD camera and associated hardware will be minimized by repeating the measurements a number of times and averaging the results. Alternatively, electronic components of highest performance will be selected. The additional effort and expense would be justified for analyses involving few fringes, in contrast to those that exhibit an abundance of fringes. Most problems at the finest level of micromechanics—with the smallest region of interest—will utilize replicas of the deformed grating. Then, transmission type moiré interferometers will allow the highest magnification and spatial resolution. This scheme also provides the greatest stability, and thus, the optimum conditions for phase stepping. The algorithms that transform the fringe data to displacement and strain fields will advance too. They will become more interactive to allow compensation for local defects or special conditions. The traditional quasi-heterodyne method will be optimized for microscopic moiré interferometry. The optical/digital fringe multiplication method will be extended to produce a larger number of fringe contours from a given number of phase steps. 4.5. PROPAGATION INTO COLLEGE/UNIVERSITY CURRICULA The theory and practice of moiré interferometry will be taught at many colleges in engineering, science, and technology curricula. Instrumentation will become available for these educational purposes, and it will be more basic and economical. As the technology propagates, it is likely that new innovative designs will emerge. The more advanced courses will surely encounter computer-aided analysis. These experiences will inevitably lead to experimentation with alternate schemes of analysis and superior programs will evolve. Moiré interferometry will become familiar in engineering and science. It will be known by many as a tool for measurement and exploration. 5.

Concluding Comments

Whereas moiré interferometry and microscopic moiré interferometry are currently used in many fields of engineering and science, we can anticipate that they will be used much more extensively in the same fields, and applications will extend into additional fields. The unique combination of capabilities—whole field measurements, very high sensitivity, very high spatial resolution, excellent signal-to-noise ratio, extensive range, etc.—will be put to use for measurements and explorations, as engineering tools and research tools, in widely diverse applications. I am sure Professor Isaac Daniel would be especially pleased to find that these expectations are realized.

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References

Daniel, I.M., Rowlands, R.E., and Post, D., “Strain Analysis of Composites by Moiré Methods, Experimental Mechanics, 13(6), pp. 246-252 (1973). 2. Post, D., “Optical Interference for Deformation Measurements—Classical, Holographic and Moiré Interferometry,” Mechanics of Nondestructive Testing, W.W. Stinchcomb, Editor, Plenum Press, NY, pp. 1-53 (1980). 3. Post, D., and Baracat, W.A., “High-Sensitivity Moiré Interferometry—A Simplified Approach,” Experimental Mechanics, 21(3), pp. 100-104 (1981). 4. Post, D., “The Moiré Grid-Analyzer Method for Strain Analysis," Experimental Mechanics, 5(11), pp. 368-377(1965). 5. Post, D., Han, B., and Ifju, P. G., High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Springer Verlag, New York, (1994). 6. Weissman, E.M., and Post, D., “Moiré Interferometry Near the Theoretical Limit," Applied Optics, 21(9), pp. 1621-1623 (1982). 7. Walker, C.A., “A Historical Review of Moiré Interferometry,” Experimental Mechanics, 34(4), pp. 281-299 (1994). 8. Cho, S.M., Cho, S.Y., and Han, B., “Real-time Observation of Thermal Deformations in Microelectronics Devices: A Practical Approach Using Moiré Interferometry,” Experimental Techniques, (submitted for publication September 2001). 9. Mollenhauer, D.H., and Reifsnider, K.L., “Interlaminar Deformation along the Cylindrical Surface of a Hole in Laminated Composites—Experimental Analysis by Moiré Interferometry.” J. Composites Technology and Research, 23(3), pp. 177-188 (2001). 10. McKelvie, J., and Walker, C.A., “A Practical Multiplied Moiré-fringe Technique,” Experimental Mechanics, 18(8), pp. 316-320 (1978).

OPTICAL FIBRE BRAGG GRATING SENSORS IN EXPERIMENTAL MECHANICS OF COMPOSITE LAMINATED PLATES J. BOTSIS, F. BOSIA, M. FACCHINI, Th. GMÜR Laboratory of Applied Mechanics and Reliability Analysis (LMAF) Department of Mechanical Engineering CH-1015 Lausanne - Switzerland

Abstract Optical fibre Bragg grating sensors have proved in the past years to be extremely useful tools for internal strain measurements in layered composite materials. Two applications of this type of technology are presented in this paper. Firstly, embedded Bragg grating sensors are used to study the through-the-thickness strain field of laminated plates in various quasi-static loading cases. It is shown how in three-point bending the onset of non-linearity in the strains can be detected for decreasing span/depth ratios in the specimens. The strain response of the embedded sensors is also investigated in the presence of a non-uniform strain field generated by a concentrated load. Finally, the use of fibre Bragg gratings is demonstrated in dynamic conditions, both in impulsive transient tests and using harmonic excitation. 1. Introduction

In the early sixties the scientific progress in the optics domain brought important advances in the conception of laser sources and low-loss fibre light guides. As a direct consequence, the first experiments using optical fibre sensors were carried out in the early seventies [1]. Optical fibres show certain properties that make them unique in applications as sensors for single and multiplexed measurements of temperature, pressure, strain, etc. Compared to other traditional sensors, they are compact, lightweight, and minimally invasive due to their small size is the standard diameter of the fibre), furthermore they are immune to electromagnetic interference, have greater resistance to corrosion, higher temperature capacity (~200 ºC) and longer lifetime [2-4]. Thus, fibre optic sensors in general and optical Fibre Bragg Grating (FBG) sensors in particular seem to be ideal for so-called “smart structures” [5, 6], where they are used for process monitoring, non-destructive analysis and structural inspection. While FBG sensors are used in several engineering applications, they are also an important tool in experimental mechanics. They can be used to perform experiments that are very difficult or impossible with other standard techniques. 303 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 303–314. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Characterization of the deformation, damage and fracture characteristics of composite materials is often not trivial. From the experimental point of view, surface displacement or strain distributions can be easily determined by various full-field noninvasive methods (visual inspection, interferometric techniques, etc.), but the internal strains cannot be measured with sufficient resolution using existing non-destructive techniques (X-rays, ultrasonic probing, acoustic emission, etc.). Fibre optic sensors, instead, can be embedded in multi-layered composite materials during the fabrication process, and can provide accurate non-invasive internal strain measurements at selected locations. Thus, key experiments can be designed, and data from such experiments can be used to develop or validate pertinent analytical and numerical models. For example, FBG sensors can be used for direct measurements of bridging tractions in model systems [7], help investigate fibre-matrix interface [7], and most importantly characterize the deformation characteristics of layered polymer composite plates [8]. This report is centered on a study of the through-the-thickness deformation characteristics of layered composite plates using FBG sensors. Measurements are carried out a) on simply supported plates loaded under quasi-static line or concentrated load, and b) on a clamped circular plate subjected to dynamic loads. The surface displacements are simultaneously obtained by electronic speckle pattern interferometry. An introductory part on the working principles of FBGs is presented in section 3; materials and methods used in this work are outlined in section 4, and the results of the quasi-static and dynamic experiments are given in section 5. Finally, a summary of results is presented.

2. Fibre Bragg grating sensors From the practical point of view, a Bragg grating can be described as a spatial modulation of the refractive index created along the core of an optical fibre. The Bragg grating generation is based on the photosensitivity property of silica fibres doped, for example, with germanium when illuminated with UV light, and is usually obtained by two different solutions based on the two-beam interference technique or the phase mask method [9]. The operating principle of an FBG sensor [1, 2] can be summarized in the property of the grating to reflect from a broadband source only the wavelength that matches the grating pitch (Fig. 1) through the equation:

where is the Bragg wavelength, is the grating pitch and n is the mean refractive index of the core. The response of the sensor depends on its physical elongation (and the corresponding change in the grating pitch), and on the change in the fibre index due to photoelastic effects. Both parameters are influenced by strain and temperature. With respect to other widely used fibre optic sensors, FBG show an intrinsic self-referencing capability: the signal of interest is the Bragg wavelength and this output does not depend directly on parameters as the total light intensity, the losses in the connecting fibres and couplers, or the source power. Sensor systems using FBGs usually work by injecting a broad band source into the fibre and detecting the back-reflected Bragg wavelength. Such systems can work in

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transmission or reflection configurations and use as source and detector respectively a broad band source (e.g. a superluminescent LED) combined with a spectrum analyzer, or a tunable laser combined with a photodetector. While the response of FGBs is well understood when the strain field is homogeneous, its response on a heterogeneous field is not completely resolved. In such cases, special iterative procedures should be used to determine the strain distribution along the fibre [10]. In this work, the assumption is made that the axial strain experienced by the fibre is equal to the longitudinal strain in the host (good adhesion between the fibre and matrix is essential to fulfil this condition), and the ambient temperature is maintained constant during experiments.

3. Materials and methods 3.1. SPECIMEN PREPARATION AND CHARACTERIZATION For quasi-static tests, the specimens are in-house fabricated glass/polypropylene (glass/PP) laminated plates whose planar dimensions are 290x250mm. Two symmetrical cross-ply lay-ups are considered, and corresponding to thicknesses of 5 and 10mm, respectively. Similarly, 400x400mm, 5-mm thick, specimens are used for dynamic tests. The prepregs, supplied by Vetrotex (France), are 0.625mm thick and based on a 4/1 fibre weave (four times as many fibres in the weft direction as in the warp direction), which means that the single plies effectively behave as orthotropic unidirectional layers when loaded in the two principal directions. The laminates are autoclave-produced at 10 bar and 190°C, and the FBG sensors are included between plies prior to consolidation. The embedded fibres are 125 µm diameter, polyimide-coated, standard monomode optical fibres. Bragg gratings are written into the fibres by means of a phase mask technique, using a pulsed excimer laser. They are 3 mm in length and the wavelength is centred at 1530 nm. The stripped

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length of the optical fibre at the location of the grating is subsequently treated with silane to ensure strong adhesion with the PP matrix of the composite. The sensors are placed centrally in the specimens, as shown in Fig. 2. To derive strains at different locations through the thickness, various identical specimens are fabricated, differing only in the depth location of the sensor. Thus, in one specimen the FBG sensor is embedded between the first and second plies, in another between the second and third plies, and so on.

Standard quality checks are carried out to verify that the level of porosity and the thickness uniformity are acceptable in order to ensure reproducibility in the mechanical properties of identical specimens. Single-layer material constants are also determined for the glass/PP laminae and for the considered laminates by means of standard tensile tests and modal identification tests [11] The single-layer values are subsequently used in finite element simulations. Further characterization of the fabricated specimens includes checking the proper alignment of the embedded fibres and estimating the uncertainty on their depth location. Having observed various sections of the laminate through an optical microscope, the latter is estimated to be half a fibre diameter, i.e. 62.5 [8]. Significant errors can be made in strain measurements when the sensor location differs from that expected, especially in strongly non-uniform strain fields, as in the region surrounding the application point of a concentrated load (see section 5.1.2). Therefore, the exact position of the grating along the fibre is also checked once the latter is embedded in the laminate, using the Optical Low-Coherence Reflectometry (OLCR) technique [12]. A commercially available HP 8504B precision reflectometer is used for this measurement.

3.2 EXPERIMENTAL TECHNIQUES Experimental inspection of the specimens is carried out by combining different techniques: embedded FBGs for internal local strain measurements, Electronic Speckle Pattern Interferometry (ESPI) for the determination of distributed surface strain, and a laser vibrometer to monitor local surface response in dynamic tests.


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The measurements with the embedded fibre Bragg gratings are carried out using a tunable laser diode source, and the back-reflected signal is directed via a 2x2 coupler to a photodetector. The acquisition and treatment of the signals are done via in-house programmed software. The axial strain in the fibre, which corresponds to the longitudinal strain of the host material in the fibre direction and at the location of the sensor, is related to the Bragg wavelength shift through the well-known relation

where is a sensitivity factor of the fibre [9]. All sensors used in this study are calibrated previous to embedding, and an average value of is found. The system, although quite slow, allows the determination of the peak wavelength shift with a precision of 10 pm, corresponding to a strain resolution of l0µm and under. Electronic Speckle Pattern Interferometry is a powerful, non-contact tool to measure the distribution of displacements occurring on the surface of a specimen with an accuracy of fractions of micrometers [13, 14]. Two standard set-ups exist: the out-ofplane and the in-plane configurations, sensitive to normal and tangent components to the observed surface, respectively. Both are employed in this study. A common 0.632mm wavelength He-Ne laser is used, together with a standard Sony 768x572 pixel CCD camera. Acquisition and processing of the speckle patterns are carried out with commercially available software, and phase maps are derived by means of a standard phase-shifting procedure. In section 5.1, an in-plane configuration is used to measure surface displacements on the deforming specimens, and therefore to derive surface strains at the location where internal strains are measured via the embedded FBG. Instead, dynamic out-of-plane ESPI measurements are described in section 5.2. In this case, the shape of stationary vibration modes can be retrieved by using a special approach called “time-average” [15, 16]. Finally, a commercially available Polytec laser vibrometer is used to measure the surface velocity of specimens subjected to dynamic loading in section 5.2.

3.3 NUMERICAL SIMULATIONS The experimental results of the quasi-static experiments are compared to numerical predictions, using finite-element-method (FEM) models of the laminates in the two loading configurations considered herein. Simulations are carried out using the I-DEAS and ABAQUS codes. The specimens are modelled using solid-element meshes, experimentally-derived single-layer material constants for each layer, as well as the appropriate ply orientation and thickness (0.625mm). A typical model consists of a 54×50×n-element mesh, where n is the number of plies in the laminate considered. The mesh is sufficiently refined to avoid using excessively thin elements. Additionally, in simulations involving concentrated loads, the mesh is more refined in the region surrounding the load-application point. The simulations are carried out using linear hexahedral I-DEAS elements and a selective integration scheme for the computation of the stiffness matrices [17]. Due to some variation in the specimen thickness and possibly due to warping effects, contact between loading/supporting members and the plates does not run uniformly along the whole plate width. Therefore, these contact

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lengths are measured experimentally by using pressure-sensitive film and boundary conditions are chosen accordingly. These amount to enforcing zero z-displacements along the measured line of contact between the plate and the supports, and, in the case of three-point bending, distributing the total applied load uniformly along the measured line of contact between the plate and the loading cylinder. Having included these data in the FEM models, and thus employed the appropriate boundary conditions, the corresponding strain fields are then calculated for the entire structure. Thus, the strain distribution is evaluated at the centre of the plate, in the xy position corresponding to the location of Bragg grating sensors.

4. Experimental measurements 4.1. QUASI-STATIC BENDING TESTS An issue that is of great interest in composites is the study of the type of stress and strain distributions that occur through the thickness of a laminate subjected to flexural deformation. It is well known that because shear moduli are generally smaller than in isotropic materials, shear effects are in most cases non-negligible, and account for nonlinear distributions [18]. It is also known that this effect is all the more pronounced as the thickness of the laminate increases, though in the literature this has often remained a rather qualitative statement. The combined use of embedded FBG sensors and ESPI can therefore be used to experimentally highlight the type of nonlinearity present, and check the range of pertinent parameters in which this occurs.

4.1.1. Quasi-static three-point bending test The first loading configuration considered is a three-point bending set-up, as shown in Fig. 2a Measurements are carried out in a specifically designed loading frame that allows ESPI measurements on the specimen surface. Loading is applied by means of a displacement-controlled motor, and the force is measured by a load cell. The support span is variable, in order to provide the possibility to perform measurements at various span/thickness, or span/depth (s/d) ratios, a parameter used to classify plates as “thin” or “thick”. In this study, s/d varies between 50 and 10. Due to the difference in behaviour of the laminate in tension and compression, experiments are carried out with the FBG sensors situated in the laminate half subjected to tension only. For this reason, results are presented relative to only half of the laminate. All measurements are carried out in the linear elastic range, over two or three loading/unloading cycles. Typically, FBG and ESPI readings are taken at loading steps of about 100N, and a maximum total load of 1500N is reached. Due to the viscoelastic properties of the PP matrix, readings are taken once the load value has stabilized, after a time lapse at least equal to the relaxation time of the material. For each specimen and support span, a strain-load curve is constructed. The data show good linearity and are approximated by a linear fit. Thus, the slope of the fit, corresponding to the strain per unit load at a specific depth location, is derived. Figures 3a and 3b summarize results for s/d=50 and s/d=10, the two extreme cases considered. The experimental strain/load values as a function of normalized laminate thickness are shown, together with the corresponding FEM-calculated distributions. Vertical error bars on the FBG sensor data are due to the uncertainty on the depth location, while the horizontal ones are due to the uncertainty in the determination of the wavelength of the

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Bragg-peak shift. This uncertainty on the strain is in some cases greater than that due to the resolution of the tunable laser. This is because in woven composites, as in this case, the reinforcing fibres can induce micro-bending in the embedded optical fibre and a relatively large loss of signal, and as a consequence give rise to noisier reflected peaks. As for the error bars on ESPI values, these are mainly due to the uncertainty on the determination of the impinging angles of the illuminating beams. Good agreement between experimental and numerical data can be noted in Fig. 3, as well as a clearly nonlinear behaviour in the thicker plate. The experimental data in the latter case would seem to deviate slightly from the numerical curve and display a more marked nonlinearity. This could be due to the afore-mentioned difficulties in accurately modelling the boundary conditions experienced by the plates, especially in the thicker specimens.

4.1.2 Quasi-static concentrated loads The second experimental configuration considered is shown in Fig. 2b. The specimens are simply supported on two sides and are subjected to an out-of-plane concentrated load applied in the centre of the plate. From the experimental point of view, this loading configuration presents some new characteristics with respect to the previously considered one: firstly, the strain field in the vicinity of the loading point displays an abrupt variation, so that the type of FBG response must be monitored for signal modification due to the non-uniform field; secondly, the application of a point load, rather than a line load, eliminates the non-uniform contact problem encountered previously in the load application. The first remark to be made in these experiments is that the detected Bragg peaks do not undergo any broadening once loads are applied, nor is there any secondary peak formation. This is an indication that the strain field experienced by the sensor is

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sufficiently uniform, or in other terms, that the sensor length (3mm) is sufficiently small for the measurement to be considered point-wise. If this were not the case, a more elaborate analysis of the reflected peaks would be called for [10]. Results are shown in Fig. 4 for s/d ratios of 25 and 10, and once more experimental data are compared to FEM-calculated values. There is some discrepancy between ESPI-measured points and numerical predictions. This is probably because these measurements are affected by a systematic error which is hard to estimate, but which in all cases leads to an overestimate, due to the presence of a large out-of-plane displacement component. The latter influences the measurement because due to the curvature experienced by the plates in bending, the reference axes of the interferometric set-up no longer coincide with the directions which are normal and tangent to the observed surface. This effect is more pronounced in the case of concentrated loads, as compared to three-point bending with a line load. It is apparent how the nonlinearity of the through-the-thickness strain distributions increases with respect to three-point-bending. Furthermore, it can be noticed that even though the laminate is symmetrical, the neutral plane no longer coincides with the midplane, rather it is shifted towards the load-application point. These two facts can be explained by observing that the strain field results from two contributions, one deriving from the overall bending, the other from the local effect of the concentrated load [19]. However, it is hard to separate these two effects.

For this reason, it is of interest to monitor the through-the-thickness strain behaviour in other sections of the laminate, at a distance from the load-application point, where local effects are negligible. This can be done by shifting the specimens from their central position in the loading frame by fixed amounts in the x-direction. The resulting loading configuration remains identical, but for the location of the embedded FBG sensor.


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Thus, the x-dependency of the strain distribution can be determined for each layer and once again compared to numerical predictions. An example of such a derived strain distribution is given in Fig. 5. The measurements are relative to the FBGs embedded between the second and third plies in the laminate (1.25mm from the free surface), for a support span of 100mm (s/d=10). The location x=0 corresponds to the sensor placed centrally, and the specimen is shifted by 2-3 mm between successive measurements. There is good agreement between simulations and experimental values, except for the points in the vicinity of the supports This could be again due to nonuniform contact between the supports and the plates, an effect that is hard to model in FEM simulations. To extract the strain distribution through the thickness at various sections of the laminate, several curves such as that shown in Fig. 5 should be constructed for the sensors embedded at different depth locations. Such results will be reported elsewhere.

4.2. DYNAMIC TESTS Although quasi-static conditions can be observed in some cases in real applications, existing structures are most likely submitted to dynamic solicitations. Because health monitoring and reliability evaluation are some of the most important objectives for the application of sensors in composite materials, the potential of FBG sensors to carry out dynamic measurements is fundamental, and their applicability to this domain needs to be verified. Therefore, emphasis in research activities involving FBG sensors at LMAF, EPFL, is also on the study of the mechanical response of composite plates in dynamic conditions. Thus, a series of preliminary tests has been carried out to examine the reliability, advantages and performance of FBG sensors in dynamic tests. The use of FBG sensors for dynamic phenomena is documented in the literature and systems able to detect strain changes in the kHz range already exist. Most of them (MicronOptics and BlueRoad Technologies among others) are based on the ability to track the wavelength of the Bragg peak. The solution adopted in this study, instead, is

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simply based on the intensity change of a selected back-reflected wavelength. The working principle is shown in Fig. 6: once the response curve of the grating in quasistatic conditions has been obtained, the wavelength corresponding to the maximum slope is determined and the injected light is fixed at this value. This working point guarantees the highest sensitivity and the maximum dynamic range. From the plot it is clear that a small change in the Bragg wavelength corresponds to a shift of the complete curve, which in turn results in a change of the voltage read-out of the photodetector. The latter signal is then acquired with a digital signal analyser, and information on frequency and amplitude can be extracted.

As mentioned previously, the considered specimen is an eight-ply square plate of 400-mm side and 5-mm thickness. The plate is clamped between two steel blocks with a circular 300-mm diameter opening, so that the tested assembly can be assimilated to a 300-mm diameter clamped circular plate. The specimen contains a centrally embedded FBG sensor (Fig. 2c), whose grating wavelength is 1529.65 nm and maximum-slope wavelength is 1529.52 nm. The specimen surface is inspected at the same time by an ESPI system and by an interferometric Polytec vibrometer whose beam impinges on the centre of the plate, and the results are compared to the FBG sensor read-out. Two different solicitations are applied: harmonic excitation and impact loads. Firstly, the rear side of the specimen is exposed to spherical acoustic waves generated by a loudspeaker driven by a sinusoidal signal produced by an electronic function generator. The frequency and the amplitude of the signal are tuned and the signals obtained by the three different techniques are acquired and compared. The ESPI technique allows the acquisition of fringe patterns only in the proximity of stationary vibrations: resulting interferograms corresponding to the first four modes are presented in Fig. 7. The vibrometer is sensitive to the velocity of the targeted central point and the FBGs is sensitive to the longitudinal strain component in the composite at the location of the sensor (see Fig. 2c). The signals can be presented in the time domain (Fig. 8a) and,


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as expected, they are in phase opposition. Frequencies can be easily determined if the signal is represented in the Fourier domain (Fig. 8b). In the proximity of the stationary frequencies, detected via ESPI, the response amplitude of the FBG sensor and of the vibrometer increase, as expected. In a second experiment, the specimen is excited by impact loads at different locations. The signals “read” by the FBG system and by the vibrometer are acquired and then transformed in the frequency domain to obtain the transfer functions. A comparison between the FBG and vibrometer signals demonstrates that peak amplitudes occur at approximately the same frequencies, which also correspond to the stationary modes detected with the ESPI approach.

The adopted read-out method can probably give better resolution than those based on Bragg wavelength tracking, and the frequency range can be considerably higher (limited in fact only by the speed of the signal acquisition system). However, some problems related to amplitude fluctuations, probably due to interferometric reasons, are still under investigation.

5. Summary The use of FBG sensors for deformation measurements and inspection of layered composite materials in quasi-static and dynamic loading conditions has been presented. The through-the-thickness strain distribution of laminated plates subjected to bending

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loads has been derived and the feasibility of similar measurements in examples of dynamic solicitation has been demonstrated. While no attempt is made to describe all the multiple applications of FBG sensors, this article intends to show how the use of this type of sensor could be very important in experimental mechanics. Key experiments can be designed to investigate the mechanical behaviour of materials and structures and verify pertinent models.

6. Acknowledgements The authors wish to thank the Institute of Applied Optics, EPFL, for supplying the Bragg sensors used in this study and for support on related matters. The Laboratory of Composite and Polymer Technology, EPFL, is also gratefully acknowledged for advice and collaboration in the preparation of layered specimens. This work is supported by the Swiss National Science Foundation, Grant no. 21-52486.97.

7. References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19.

Culshaw, B. and Dakin, J. (1988-1997) Optical Fiber Sensors, Vol. 1-4, Artech House, Boston Grattan, K.T.V. and Sun, T. (2000) Fiber Optic Sensor Technology: an overview, Sensors and Actuators A, 82, 40-61. Collected papers of the international conferences on Optical Fiber Sensors (OFS), 1983-1997, CDROM, SPIE/IEEE, Washington Kersey, A.D. (1996) A review of recent developments in fiber optic sensor technology, Optical Fiber Technology 2 (3), 291-317. Udd, E. (1995), Fibre-optic smart structures, Wiley, New York. Measures, R. M. (1992) Smart composite structures with embedded sensors. Composites Engineering 2, 597-618. Studer, M. (2001) Étude des forces pontantes dans un matériau composite à 1’aide de réseaux de Bragg dans des fibres optiques. PhD Thesis n. 2451, EPFL. Bosia, F., Botsis, J., Facchini, M., and Giaccari, P. (2001) Deformation characteristics of composite laminates, part I: speckle interferometry and embedded Bragg grating sensor measurements. To be published in Composites Science and Technology. Hill, K.O. and Meltz, G. (1997) Fiber Bragg grating technology fundamentals and overview Journal of Lightwave Technology 15 (8), 1263-1276. Peters K, Studer M, Botsis J, Iocco A, Limberger H, Salathé R. (2000) Embedded optical fiber Bragg grating sensor in a nonuniform strain field: measurements and simulations, Experimental Mechanics 41 (1), 19-28. Cugnoni, J. (2000) Identification des propriétés constitutives des stratifiés par des méthodes mixtes numériques-expérimentales. Internal Report, LMAF, EPFL. Sorin, W. V. (1996) Fiber optic sensing using low-coherence interferometry, Proc. SPIE 2872, 4047. Jones, R and Wykes, C. (1983) Holographic and speckle interferometry, Cambridge University Press. Wykes, C. (1982) Use of Electronic Speckle Pattern Interferometry (ESPI) in the study of static and dynamic displacements), Optical Engineering 21(3), 400-406. Lu, B. et al. (1989) Time-Average Subtraction Method In Electronic Speckle Pattern Interferometry, Optics Communications, 69 (3), 214-218. Lokberg, O.J. and Hogmoen, K. (1976) Vibration Phase Mapping Using Electronic Speckle Pattern Interferometry, Applied Optics, 15, 2701-2704 Hughes, T. (1987) The Finite element method. Prentice-Hall Int Reddy, J.N. (1997) Mechanics of Laminated Composite Plates: Theory and analysis. CRC Press. Timoshenko, S.P. and Goodier, J.N. (1970) Theory of elasticity, 3rd Ed. McGraw-Hill Int.

OPTOELECTRONIC DISPLACEMENT MEASUREMENT METHOD FOR ROTATING DISKS

C.E. BAKIS, B.J. HALDEMAN, R.P. EMERSON Composites Manufacturing Technology Center The Pennsylvania State University University Park, PA 16802

Abstract

Radial displacements on an axial surface of a rotating disk can be measured with optoelectronic sensing devices consisting of infrared emitters and phototransistors. The aim of this investigation is to improve the stability and robustness of a previously described optoelectronic measurement system. Stability is improved by a computerized algorithm for the control of the emitter intensity based on feedback from a compensation portion of an optical pattern placed on the disk. Robustness is improved by using a laser emitter that allows for a larger standoff distance in comparison with the previously used light emitting diode. Radial displacements obtained with automatic intensity control of the laser emitter agreed with theoretical values in a glass/urethane composite disk to within ±1 µm at rotational speeds up to 3300 rpm. With further development, the proposed displacement method has application in the noncontact measurement of radial and hoop strains through the radial thickness of high-speed axisymmetric rotating machinery such as energy storage flywheels. 1.

Introduction

High-speed flywheels are currently attracting interest in energy storage applications where high rates of charge and discharge, high cycle life, and operation in a wide temperature range are required. Ongoing flywheel research activities are focusing on the improvement of specific energy and power density, the reduction of cost, the evaluation of performance, and the prediction of service life. A technical focus area associated with the last two activities mentioned is the measurement of deformation in flywheel rotors. It is beneficial to measure full-field deformations with high fidelity over extended periods of time so that time-dependent structural analyses can be validated. Various experimental methods have been used to measure radial and circumferential strains in high-speed flywheels. Examples of some of the better-known methods include x-ray diffraction, speckle interferometry, and electrical resistance strain gages. A specialized method of interest, attributed to Simpson and Welch [1], is known as the optoelectronic method since it uses simple electro-optical devices and electronic circuits to measure the real-time radial displacements of an optical pattern 315 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 315–324. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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applied on the axial end of a rotating object. Bakis and Emerson [2] subsequently developed a related optoelectronic method integrated with digital data acquisition and processing and demonstrated several key improvements: (i) an order of magnitude improvement in displacement sensitivity at rotor speeds twice as great as had been previously reported; (ii) the ability to separate rigid body vibration from axisymmetric radial displacement; (iii) a unique optical pattern that enabled compensation for changes of intensity and distance between the sensor and rotor during a test. The optoelectronic displacement method relies on the measurement of changes in angular width (duty cycle) of a reflective optical pattern applied to the axial surface of a rotor. An example of a multi-region reflective pattern is shown in Fig. 1, where the shaded triangular areas are considered reflective. In this illustration, four “lobes” of the pattern are evenly spaced at 90-deg. azimuthal intervals on the rotor. Each lobe contains two annular regions where displacements can be measured. For example, as the rotor spins at a known speed, a stationary sensor records angular duty cycles based on the time intervals spent “on” each individual reflective triangular patch encountered along the dashed path shown within one annulus. Since, in practice, the duty cycle versus radius relationship deviates from ideal, a unique relationship for each displacement patch needs to be determined by a calibration procedure with the rotor spinning at a very low speed – the “undeformed” reference condition. This relationship is stored in a look-up table. The duty cycle at each original radius on the rotor can be shown to remain constant for any rigid-body or flexible-body displacement of the patch. Thus, the radial displacement of the rotor at each nominal patch position is found based on the respective change in measured duty cycle. By measuring such total displacements, at 90-deg. (in this example) azimuthal intervals and fitting a sinusoidal function to the measurements as a function of azimuthal angle rigid body radial vibration, can be separated from axisymmetric radial displacement due to strain, u:

where is the angular phase of the vibration. Measurements can be made with sensors at a number of radii in order to calculate radial and hoop strains at several locations through the radial thickness of the rotor:

The main optical requirement of the pattern is that it has appropriately placed regions of contrasting reflectivity. The best radial displacement sensitivity is obtained when the duty cycle of the pattern changes most rapidly per unit change in radial position on the rotor. Additionally, the pattern could incorporate a means of correcting measurement errors that can arise if the intensity of the emitter changes or if the standoff distance between the sensor and rotor surface changes due to vibration of the rotor or thermal expansion of the test apparatus. Such corrections can be easily done if a compensation patch of fixed angular width as a function of radius is placed nearby each displacement patch. One lobe of such a pattern used in this investigation is shown

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in Fig. 2. A method for algebraically correcting a duty cycle based on the deviation of a compensation patch measurement from a presumed fixed value was described in Ref. [2].

An optoelectronic sensor aimed at the reflective pattern consists of an emitter projecting a small spot on the surface of the rotor and a phototransistor (detector) that provides an “on” or “off” logic signal according to the measured intensity of the reflected spot. An illustration of the sensor used in Ref. [2] is shown in Fig. 3. A 20 MHz digital counter triggered by the logic signal records the elapsed time that the sensor is on a reflective region as the target surface rotates. Using a tachometer, the elapsed times are converted to duty cycles. The collection of data and calculation of displacements is done with a personal computer equipped with a data acquisition card and commercially available software for data acquisition and process control. The objective of the present investigation is to explore improvements in the optoelectronic displacement measurement method reported previously [2] by (i) replacing the previously-used infrared light emitting diode (IR LED) emitter with an IR laser emitter and (ii) developing an algorithm for controlling emitter intensity with a computer. The laser emitter offers the potential advantages of higher overall intensity and less divergence, which lead to certain advantages in implementation and accuracy of the optoelectronic displacement measurement system such as larger standoff distance and potentially less sensitivity to out-of-plane vibration. Computer control of the intensity of the emitter simplifies operation of the displacement measurement system by automatically correcting for duty cycle fluctuations induced by external sources such as ambient temperature fluctuations during a test.

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Experimental Setup

Displacements were measured on a glass-fiber-reinforced polyurethane composite disk (3.8-cm ID, 32-cm OD, 25-mm thickness) attached via a ring-shaped aluminum hub (2.4-cm ID and 3.8-cm OD) to a 6-mm dia. steel shaft. The rotor was spun directly by a DC electric motor mounted via elastomeric vibration isolators to a massive steel base. The spin tests were done in open air at speeds as high as 3.5 krpm. The glass/urethane disk was fabricated by a filament winding process in which the fibers are oriented only circumferentially. Because the polyurethane matrix is very compliant, the hoop-direction modulus of elasticity is much higher than the radialdirection modulus. The material properties of the glass/urethane material are as follows: density, hoop modulus, MPa; radial modulus, kPa; major Poisson’s ratio, 0.36 [3]. For computing theoretical displacements in the rotor, a plane stress, anisotropic, axisymmetric, elasticity model was used [4]. The aluminum hub and glass/urethane disk comprised the rotor model. The reflective pattern was applied to the axial surface of the rotor using a standard black and white photographic process. First, the rotor was painted with sequential layers of reflective silver paint, polyurethane, and then (in a darkroom) a light-sensitive emulsion. The pattern, consisting of four lobes such as that shown in Fig. 2, was transferred to the emulsion using a contact negative that had been created with a photoquality ink-jet printer. Since the largest size negative that could be printed at once was smaller than the rotor, the four lobes were printed separately and then carefully aligned relative to each other and attached to a clear acrylic plate. This process admittedly allows the opportunity for errors in aligning the lobes on the rotor, even though much care was taken to minimize such errors. A final layer of polyurethane was applied to the developed rotor to help protect the pattern from scratches. The new emitter chosen for evaluation in this investigation is a 5-mW infrared (IR) laser micro-module with a wavelength specification of 904±10 nm. The IR light emitting diode (LED) used in previous research [2] was also used extensively in the experiments. The IR LED has power and wavelength specifications of 24 mW and 880±30 nm, respectively. The IR phototransistor used with either emitter has a 70 ns rise and fall time. The setup of the laser sensor, shown in Fig. 4, was similar to that shown for the LED in Fig. 3, except that the standoff between sensor and rotor surface was increased to 76 mm versus 4.5 mm in the case of the LED. The greater intensity of the laser enabled the larger standoff. One of the possible benefits of such an increase in standoff is the reduction in sensitivity of the measurements to minute variations in standoff caused by rotor vibrations or differential axial thermal expansions in the test apparatus, although this benefit was not explicitly investigated in the work reported here. 3.

Intensity Control

Given that inevitable changes in intensity of the IR radiation reaching the detector will cause changes in the measured duty cycles on a given optical patch even though there is no real change in displacement on the rotor, it is important to either account for such changes with additional measurements and data processing as was done in Ref. [2] or


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minimize the magnitude of change to an inconsequential amount, as was done in this investigation. The new intensity compensation method adopted in this investigation automatically adjusts the intensity of the emitter to ensure that a fixed angle is always measured on the compensation patch adjacent to the subject displacement patch. In this manner, intensity fluctuations caused by changes in ambient temperature or out of plane rotor displacement can be eliminated. Calibration of the displacement patch using intensity control proceeds as described earlier. A beneficial side effect of this approach for improving the long-term stability of the measurements is that small defects in the angular width of any compensation patch with respect to radius are corrected as well. In an experiment where the rotor undergoes real changes in radial displacement, the intensity control algorithm ensures that the duty cycle of the compensation patch remains constant, which ensures that the radial displacement versus duty cycle information stored in the lookup table is always correct. A flow chart of the intensity control algorithm for the laser sensor is shown in Fig. 5. The algorithm for the LED sensor is similar except for certain details such as the emitter voltage.

5. Results

Figure 6 shows the results of a low-speed calibration experiment relating relative radial position to duty cycle of the displacement patch while the intensity control algorithm enforces a 4.0000±.0001-deg. angle on the adjacent compensation patch using

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the IR LED. Ideally, the duty cycle versus radius relationship for a displacement patch is a smooth, slightly concave-up curve. The undulations in the calibration curve in Fig. 6 are attributed to pattern misalignment and edge roughness either in the displacement patch or the compensation patch that was used to control the intensity of the LED. As evidenced in Fig. 7, the emitter voltage needed to be changed substantially to maintain a constant compensation patch angle during this experiment. No voltage changes should be needed in the ideal case since the compensation patch supposedly has a constant duty cycle with respect to radius. It is demonstrated by the uniformity of the angular width of the compensation patch in Fig. 6 that the intensity control algorithm can effectively perform its intended function.

Spin tests carried out at a constant rotor speed (500 rpm) have been performed using the IR LED emitter to verify that the correction algorithm operates correctly for changes in standoff and in the ambient temperature. Figure 8 shows that over a very large (±200 µm) change in standoff, the apparent radial position of the IR LED sensor changed by only +1/-2 µm. Such a small apparent change could well be an actual change due to the difficulty in ascertaining with comparable or better accuracy the alignment of the instrumented stage used to move the sensor. Changes in standoff beyond ±200 µm cause much larger deviations in apparent radial position, possibly due


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to the illuminated spot leaving the field of view of the detector. Such a limitation can be envisioned by consideration of the triangulation involved in aiming the detector on the spot illuminated by the emitter (Figs. 3 and 4). Significant changes in standoff move the spot outside the field of view of the detector, although the problem is arguably less severe with the smaller included angle associated with larger standoff distances.

Figure 9 shows the effects of a sudden drastic heating of the IR LED sensor with a heat gun while using the intensity control algorithm to maintain a fixed compensation patch angle of 4.0000±0.0005 deg.. Within about 1 minute, the controlled voltage supplied to the LED returned the compensation patch angle to the specified value. Drastic temperature spikes such as this (estimated to be at least 5°C) are not typical in such short time intervals in flywheel testing. Smaller amplitude, longer-duration temperature changes encountered in this investigation were corrected without any outof-specification deviations in compensation patch angle.

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Calibration data for all four displacement patches in the 130-140-mm annulus on the glass/urethane rotor, recorded with the IR LED, are shown in Fig. 10 along with the theoretical relationship between angle and position found according to the algebraic equations given in Ref. [2]. The intensity of the LED for reading displacements in all four lobes was controlled in this experiment based on feedback from the compensation patch of only Lobe 1 rather than each respective lobe. Additional discrepancies among the curves can be caused by pattern roughness and misalignment of the individual lobes. These discrepancies are not of major significance assuming that the calibration curves are not speed dependent. Additional research is needed to determine if this assumption is in fact true. Regardless, since the look-up table created using data from Fig. 10 is used to interpolate radial locations based on measured duty cycles from a deformed rotor, it should be kept in mind that large discontinuities on a given calibration curve within the radii of interest might lead to a less reliable calculation of radial position. Based on the calibration data of Fig. 10, axisymmetric radial displacements along with rigid body vibrations were computed for various rotor speeds in two separate experiments (Fig. 11). Vibration was separated from axisymmetric displacement using eq. (1). Theoretical axisymmetric displacements were found using the plane stress elasticity equations for a dual material (aluminum and glass/urethane) polar orthotropic disk [4] and the material properties given earlier. There is some discrepancy between the theory and experiments, particularly above 2 krpm where the vibration, indicated by the vertical bars in the graph, increases dramatically as a resonant speed of the rotor was approached at 3500 rpm. Additional balancing of the rotor near the resonance could ameliorate this problem.

Although the results shown above demonstrate the robust characteristics of the IR LED sensor with intensity control, similar robustness was realized with the laser in similar types of tests, with the main exception being the amount of electrical noise in the laser measurements. This noise was easy to eliminate using data sampling and averaging, however.


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Figure 12 shows radial displacement versus speed for the same 130-140-mm annulus of the glass/urethane rotor as in Fig. 11, with the laser replacing the LED in this case. As in the LED experiment, the intensity for measurement of all four lobes was controlled based on the compensation patch of only Lobe 1. The measurements are within ±1 µm of the theoretical displacements for all rotor speeds evaluated. A direct comparison of the two types of emitters on a smaller displacement scale in Fig. 13 reveals that the theoretical curve is closer to the laser results than the LED results.

5.

Conclusions

An algorithm has been developed that controls emitter supply voltage using feedback from a compensation patch. In controlling the measured angular width of the compensation patch to remain constant during a test, consistent displacement measurements can be made on the nearby displacement patch regardless of external effects such as changes in ambient temperature and sensor standoff distance that would otherwise adversely influence the measured displacements. Preliminary results show that an IR laser emitter can provide slightly more accurate results in comparison to the previously used IR LED emitter. When using the laser, measured axisymmetric displacements agree to within ±1 µm of theoretical values in a glass/urethane composite rotor spun as fast as 3300 rpm. Implementation advantages of the laser over the LED accrue mainly to its greater intensity and collimation, which allow a larger standoff between the rotor and sensor. Greater standoff distances should reduce the sensitivity of the sensor to out-of-plane displacements that can occur during a spin test due to temperature changes in the support apparatus or out-of-plane rotor vibration. The use of a laser in addition to an intensity control algorithm shows strong potential in the creation of a more robust and accurate optical strain measurement technique for high-speed flywheels. Further investigation is recommended to verify the

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full potential of this improved optoelectronic strain measurement system. In particular, the intensity control feedback signal should be taken from the compensation patch located closest to each individual displacement patch. Doing this could further reduce the potential for intensity- or speed-dependent changes in the calibration curves of the displacement patches. 6.

Acknowledgements

The authors acknowledge the support of the US Army Research Laboratory, Aberdeen, MD, for portions of this research. Dr. Jerome Tzeng was the project monitor. 7.

References

1.

Simpson, M .L., and Welch, D.E.: “Optoelectronic strain measurement system for rotating disks,” Experimental Mechanics, 27 (1987), 37-43. Bakis, C.E., and Emerson, R.P.: “Optoelectronic radial displacement measurement on rotors,” Proc. Annual Conference on Experimental and Applied Mechanics, Soc. Experimental Mechanics, Bethel, CT, (2001), 411-414. Gabrys, C.W., and Bakis, C.E.: “Design and testing of composite flywheel rotors,” Composite Materials: Testing and Design, 13-th Vol., STP 1242, S. J. Hooper, Ed., American Society for Testing and Materials, Conshohocken, PA (1997), 3-22. Genta, G.: Kinetic Energy Storage: Theory and Practice of Advanced Flywheel Systems, Butterworth & Co. Ltd., London, 1985.

2.

3.

4.

DEFORMATION MEASUREMENT OF SHEET METAL FORMING USING PHOTOGRAMMETRY K. ANDRESEN Dr.-Ing., Acad. Director Center of Mechanics, Technical University D-38102 Braunschweig, Germany

Abstract A measuring machine based on a stereo device is developed for analyzing large sheet metal parts deformed by deep drawing. A cross grating on the surface is subdivided into meshes which are uniquely encoded by circular marks. This provides absolute indices for each grating point when taking segments of the surface by the stereo device. Hence overlapping areas can be used to fit together the segments with its spatial coordinates. By means of a theory of large deformation the principal strain and the principal directions are calculated. 1. Introduction

In industry sheet metal forming by deep drawing is simulated with commercial software packages [l]. Since material parameters, friction coefficients and boundary conditions are to be assumed, often the results are not satisfying. To improve this issue, experimental investigations are necessary parallel to the numerical analysis. Therefore the geometry and the flow of material must be measured on the surface of the object. The resulting surface coordinates then can be compared with numerical results and hence the before mentioned parameters may be adapted. Usually a regular grating (circles or crossing lines) is fixed on the undeformed plane surface and its coordinates are measured in the deformed state mainly in critical areas by optical methods. In this paper the method is extended to measuring the whole surface of the object by fitting small local segments into a global coordinate system. This is performed by a measuring machine which scans the surface in overlapping areas with a stereo device, consisting of a stiff framework with 3 CCD-videocameras. In each segment local grating coordinates are determined [2]. which are related to the position of the device. If the position and its orientation is known exactly, the coordinates of each segment can be transformed into a global 325 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 325–334. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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coordinate system. Two different procedures were applied for this function. If the device is moved only in direction of the 3 spatial axes, the shifting coordinates are given very accurate by calibration. Then they can directly be added to the local coordinates to derive global coordinates. If the device is rotated, no adequate calibration is performed. Then overlapping coordinates are used to calculate a transformation matrix fitting neighboring segments successively into the global system. 2. Fitting together of segments

The surface of the object carrying a fixed grating is scanned by the stereo-cameras in segments of about which provides the required accuracy of about 1/10 mm for the spatial coordinates. The device is positioned nearly perpendicular to each segment by a 5 axes measuring machine in a distance suited to the depth of focus of the cameras. Within the images the coordinates of grating points are calculated by picture processing with an accuracy of about 1/10 pixel. From adjoined points in the images the spatial coordinates are determined by ray intersection with the above mentioned accuracy. The camera device is shifted by stepping motors in the 3 spatial directions (X. Y, Z) and rotated around the Y, Z axes by angles and resp. According to the quality of the stepping motors and due to a calibration process, the shifting position is measured with high accuracy (1/100 mm). Therefore its coordinates can be added directly to the local point coordinates in the segment yielding global coordinates. When the stereo device is rotated its related position and orientation are known only roughly. Therefore overlapping points in neighboring segments are taken to calculate a transformation matrix. which transforms the local coordinates into the global system. The search for overlapping points is supported by a global index for every grating point. This is provided by subdividing the whole surface into meshes of grating points. Each mesh carries a unique pattern of circular points. The L–like patterns in Fig. 1 define local grating directions The top of an L represents the origin and the smaller arm of the L identifies the x direction. An index pair (k, l) of each mesh is encoded binary by the points parallel to the x, y axis. eg. in the upper left mesh. Thus local indices (i, j) of grating points will be transformed to global ones by

These global indices are used in two ways. First adjoined points within the stereo images of one device position are supplied directly. They are used to calculate the 3D coordinates by ray intersection. Furthermore the global indices are needed to search for overlapping points in neighboring segments. If at least 3 points not on a straight line are overlapping in segment 0 and 1 resp., there exist a translation vector and a rotation matrix depending on 3 rotation angles (around X–axis). (around Y–axis), (around Z axis)

DEFORMAT1ON OF SHEET METAL FORMING

which transform the coordinates segment 0.

of segment 1 into

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of the neighboring

The two upper left indices, eg 10, mean a transformation from segment 1 to 0. In Eq. (3) are measured values, while are unknown transformation parameters, which explicitly depend on and the rotation angles Putting these values into a parameter vector the nonlinear Eq. (3) can be written symbolically as

This system will be solved iteratively starting with a suitable initial vector

yielding residuals

not equal to zero.

will be improved by linearization

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which evolves to

or

This describes an overdetermined system of linear equations for the increments written as

Application of a least squares method yields

which is a linear equation for the increments

The improved solution then is

This procedure will be repeated, now being the initial vector, until the increments are smaller than a given limit 3. Global transformation The transformation between neighboring segments with overlapping points can be applied successively to all overlapping segments. Only one reference segment must be fixed. because it defines the global coordinate system. The disadvantage of this technique becomes obvious, when looking at an actual segment far from the reference segment. Then the transformation into the reference system consists of a combination of all transformation parameters of the segments lying between the actual and the reference segment. This gives rise to error propagation from segment to segment with decreasing accuracy to the outer segments. Therefore a more global approach was chosen, which directly determines the transformation parameters from each segment i to reference segment 0. This means for overlapping areas in segment i and j:

where are the unknown global coordinates of These equations are setup simultaneously for all overlapping segments considering that for a reference segment the parameters must be given. Then each segment is transformed with the same accuracy, and a higher global accuracy is reached.

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When using the foregoing linearization and the least squares method, it has to be considered, that the overlapping coordinates now are unknown too and hence are included in the parameter vector p. If the values are put into the first part of p, called and the transformation parameters in the second part and the related right hand sides into then one gets the following structure of the normal equations:

Fortunately the very large matrix A, related to the coordinates form and hence easy to invert. Thus the system is split into

is of diagonal

From the first equation one gets

inserted into the second equation yields a linear equation system

Its order is only six times the number of segments, while the order of the original system in Eq.(14) mainly depends on the number of overlapping points, which generally is much larger than the number of segments. This techniques saves up to 90% of computing time. 4. Measuring machine

For scanning curved surfaces of sheet metal parts the stereo device has to be moved along 3 orthogonal axes (X, Y, Z) and it must be rotated around 2 axes, e.g. Y, Z. For these functions a measuring machine was built with standard commercial components as shown in Fig. 2. The components in the upper part of the image realize the linear X, Y–moving and the rotation around the Y–axis, also using a linear device acting on a lever. The lower part shows two linear devices for the Z axis and a table rotating around the Z axis. The linear devices are controlled by stepping motors providing an accuracy of about 1/100 mm. But by assembling the components the orthogonality of the 3 axes does not reach a comparable accuracy. Therefore the deviation from the orthogonal directions will be determined in a calibration process. For this purpose a reference grating on a high quality transparent film (about accuracy 1/100 mm) was fixed on a plane glass plate. The grating is encoded by unique mesh indices as described in section 2. In a first step the grating is used to calibrate the 3 cameras in the stereo device [3]. This defines a projection center and a rotation matrix R for each camera with reference to the global coordinate system, X, Y in the plane of the film and Z perpendicular to it. This system also may be assumed to be fixed to

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the framework of the stereo device. If the device then is moved to a measuring position, the coordinate system is moved likewise, supplying now local stereo coordinates for each segment on the surface. After calibration of the stereo device, the axes of the measuring machine are calibrated. The device is moved successively from the origin into the X, Y and Z direction. Each time pictures are taken from the reference grating and its 3D coordinates are calculated. Then from different positions of the device the difference between two points on the grating is determined. The same difference is known by the grating pitch and the related indices. Usually there is a small


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deviation between these two difference vectors, which are used to calculate two correcting angles for each axis. In practice a large number of points and their differences are evaluated to reduce possible errors. Statistical investigations for the actual setup show the following results for the X-axis.

The angles describe the deviation of the actual X-axis from the global X-axis by a rotation around the Y,Z-axes. is the actual shift in X-direction. Measuring the distance of a grating point from the origin on the reference grating yields the following results:

E is the unit matrix and is the rotation matrix calculated from the rotation of the X-axis by the angles These results show that especially the components in Y, Z direction are significantly improved. Similar calibration in Y-Z-directions show comparable improvements.

4.1. DISPLACEMENT AND STRAIN For the calculation of large strain a thin surface with no extension perpendicular to it’s local plane is assumed. Hence only strain within this plane is available from the grating. The strain perpendicular to the surface, describing the reduction of thickness, is determined afterwards from the condition of volume constancy, which is valid for plastic deformation of metals. The translation of any point x from the plane undeformed surface into its deformed state is described by 3 displacement functions in directions (x, y, z) resp., written in vector form by

An incremental vector into

in the undeformed plane state is deformed by

where the transformation matrix F is known as deformation gradient

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Since F describes a deformation U as well as a rotation

one has to eliminate the rotation

From the Cauchy-Green tensor G

it can be seen, that the square root of G yields the deformation tensor U. According to Ref. [5] one has

with

and det G the related determinant. The elements of the deformation tensor U

include the large strain components in the tangential plane. The displacement functions will be developed similar to Ref. [4]. The four neighboring meshes around a grating point form a basic element, which will be measured in the undeformed state and the deformed state, resp. Both elements will be moved into a virtual origin, where the axes of the deformed state are rotated as to coincide approximately with the undeformed ones. Then for each point except for the origin a displacement vector

is given. Its three components are used to calculate approximating displacement polynomials by a least squares method, e.g. in x–direction

Similar equations with parameters tions the gradients in the origin

Then the deformation gradient is

hold for are

From these approxima-


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According to Eq.(27) two-dimensional strain tensor

is calculated. The principal strain and the related principal direction are derived from the eigenvalues and the eigenvectors of S. The volume constancy then requires

which is used to calculate 5.

the reduction in thickness.

Results

By use of the measuring machine the surfaces of different, parts of sheet metal were measured mainly for comparison with numerical simulations. A project is going on to analyze the spring back of parts when leaving the mould. As an example in Fig. 3 a connecting piece of a tank filling device is shown after the first state of deep drawing (pitch: 2.5 mm, line width: 0.4 mm, sheet metal thickness: 1 mm). The 3D-coordinates of the object are shown in Fig. 4 together with the global reference marks derived from 30 overlapping segments. The grey values (alternative in colors) represent the strain which decribes the reduction in thickness of the sheet metal. In this example also the limits of the method become obvious. Grating lines in regions of large curvature or in areas of deep grooves cannot be determined, because the distance between the lines in the images becomes too small. This results in low contrast, where the picture processing algorithm fail.

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Acknowledgements The research project was partly granted by EFB/AIF 12275 BG. Thanks to the following coworkers. Mr. Hübner constructed and built the mechanical part of the measuring machine. Mr. Algermissen programmed the calibration of the measuring framework and Mrs. Hentrich developed software packages for 3Dmeasuring. Etching of the objects was performed by Mr. Schatz, Techn. University Dresden.

References 1. 2.

”Pam Stamp” from Esi Group France: http://www.csi.fr/products/ Lci,Z., Andreson, K.: Subpixel grid coordinates using line following filtering. Optik Vol. 100 (1995), 125-128 3. Andresen,K., Hentrich K., Hübner. B.: Camera Orientation and 3D-deformation Measurement by Use of Cross Gratings. Optics and Lasers in Engineering Vol 22 (1995), 215-226 4. Andresen, K., Hentrich, K.: 3D Strain measurement from gratings on both surfaces of sheet metal. Arch.Appl. Mech. Vol. 70 (2000) 443-452 5. Stickforth, .J.: The square root of a three-dimensional positive tensor. Acta Mech. Vol. 67 (1987) 233-235

FRACTURE PROCESSES OF QUASI-BRITTLE MATERIALS STUDIED WITH DIGITAL IMAGE CORRELATION J. S. LAWLER AND S.P. SHAH Center for Advanced Cement Based Materials Northwestern University 2145 N. Sheridan Rd. Evanston, IL 60208

Abstract

An understanding of the fracture process is necessary for design and utilization of quasibrittle materials since cracking characterizes their failure. Subregion Scanning Computer Vision (SSCV) is an effective tool for observing these processes in an experimental setting. SSCV, based on Digital Image Correlation (DIC), is a full-field technique for quantitatively examining the development of cracks, with 0.5 micron resolution. DIC uses correlation matching to measure sub-pixel displacement in a sequence of digital images captured of a specimen undergoing failure. Using this technique, the complete behavior of specimens with multiple cracks at incremented levels of fracture can be investigated. SSCV improves on single-image DIC by dividing the specimen into 56 smaller subregions, each of which is imaged separately and so much improves measurement resolution. In this paper, examples of the results obtained in studies of crack growth in a model concrete under compressive loading and in a fiberreinforced mortar under tensile loading are presented. 1.

Introduction

Failure of quasi-brittle materials is governed by the development of cracks. In the case of concrete, cracking follows a tortuous path dictated by material features such as aggregates and pores. While crack development is a three-dimensional phenomena, much can be learned about the fracture process by focusing on cracking visible on the concrete surface. This surface information can be used to form hypotheses about the behavior of the entire concrete specimen. Direct observation of the fracture process is difficult because of the small scale at which microstructural features interact with the failure process. When cracks first initiate, their openings may be less than a micron. In addition, the development of microcracks occurs in a widely distributed manner so a large field of view must be examined. This requires the use of techniques with a much greater resolution than possible with the human eye. Moiré Interferometry [1-2], various forms of Electronic Speckle Pattern Interferometry [3-4] and Laser Holographic Interferometry [5-7] have 335 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 335–344. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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been used to measure deformation with extremely high (sub-micron) sensitivity using interference patterns produced with laser light. These methods, while sensitive, can be difficult to apply and require a vibration free-environment, which is often not easy to achieve around mechanical testing machines. A more robust method for studying fracture processes is Digital Image Correlation (DIC), which has been successfully applied to detect cracks in concrete [8] and other materials [9]. With DIC, full-field surface displacements can be measured with high accuracy for specimens with multiple cracks at incremented levels of fracture. This technique can be used to monitor the testing of a range of specimen sizes and conditions. Subregion Scanning Computer Vision (SSCV) improves on single-image DIC by computing displacement in a mosaic of images of the field of interest. Since the physical features in these images are captured with higher pixel resolution, the measurement resolution achieved using the SSCV technique is improved significantly. This paper describes in detail the SSCV method and its application. In addition, to demonstrate the type of information that may be obtained, results from two separate investigations of the fracture process of concrete materials using SSCV are presented. In both studies, this technique provided insight into the mechanism by which inclusions affect the development of cracks. Such insight is necessary for the efficient and rational design of these materials. 2.

Subregion-scanning Computer Vision (SSCV)

Digital Image Correlation (DIC) is a Computer Vision technique that relies on the optimization of the cross-correlation coefficient to measure the displacement of a portion of an image with respect to the frame of reference for that image. This technique is flexible in its application and lends itself to testing a range of specimen sizes and conditions, requiring only a digital camera capable of imaging the surface of the specimen undergoing deformation [10-11]. Since the analysis is performed by comparing subsequent images of the deformed specimen to an initial image, crack growth can be monitored at numerous selected stages in the loading history without interfering with the fracture process itself. The process of measuring deformations is based on tracking a portion of an image, known as a subimage, as the pattern of pixels it describes moves in a sequence of images of a concrete surface (Figure 1). The displacement field is calculated for a regularly spaced grid of nodes arbitrarily positioned on the reference image. For each node location, the surrounding subimage, typically 48 pixels on a side, is sampled. This subimage is then compared to various subimages from the deformed image in the neighborhood of the original node by calculation of the correlation coefficient. The correlation coefficient is a mathematically defined measure of similarity between two variables, which in this case are the pixel intensity patterns of a sampled subimage from the reference image and from the deformed image at the location being evaluated. The location of highest correlation in the deformed image is determined to be the new location of the subimage and the deformation that occurred is simply the difference between the original and new locations. To account for variations in overall image brightness, the correlation coefficient is normalized and is calculated by equation (1),

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where N is the number of pixels in the subimage, M is the model, undeformed subimage, I is the subimage taken from the deformed image being searched and the summation is over each member pixel of the image. While the correlation coefficient can only be calculated at discrete pixel coordinates, displacements are measured with sub-pixel accuracy by locating the peak on a surface fit to the correlation scores in the neighborhood that likely contains the best match. An example of a displacement field measured with DIC is shown in Figure 2a in vector form, where the circular dots represent the nodes in the arbitrary grid and the lines depict the direction and magnitude of the displacement occurring at the nodes. Figure 2b shows contours of constant displacement that correspond to the vectors displayed in Figure 2a. Discontinuities in the displacement field, represented by tightly packed contours, reveal the location, shape and size of developing cracks.

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The accuracy for DIC is given in terms of pixel size so the resolution of measurement of specimen deformation depends on the amount of physical space imaged by each pixel in the digital images. This is determined by the resolution of the chargecoupled device (CCD) camera and the size of the field of view being imaged. The resolution of this technique is improved significantly in the work presented here through the use of Subregion Scanning Computer Vision (SSCV) [12] and recently updated search algorithms. In SSCV, the digital camera is mounted on a precisely controlled two-dimensional motion stage permitting a set of 56 images, called subregions, to be captured, which together represent the full specimen surface (Figure 3). Since each image describes a smaller area, increased resolution of measurement is achieved. The deformation is measured at each image location separately and these displacement fields are combined in a mosaic fashion to represent the full specimen area.

For the work presented here, an 8-bit camera with 1035 x 1317 pixel resolution was used in combination with a 105 mm macro lens. The correlation search was performed by an algorithm from the Matrox Imaging Library v6.1, which provides a measurement accuracy of approximately 1/20th of a pixel. This combination enables displacements to be measured to with 0.5 µm accuracy for a 75 x 75 mm specimen. Before testing, a fine speckle-pattern is applied to the surface of the specimen using black spray paint, which produces a many-featured and unique pattern, ensuring that a high correlation is found only at the correct location. Note that during testing, the test must be paused to allow the camera to traverse the full specimen area, a process requiring approximately 2 minutes. Since displacement is measured at sub-pixel resolution and this resolution is better than the repeatability of the stage movement, reassembly of the subregion set is a nontrivial problem. To minimize the amount of rigid body motion in the displacement field, a new method has been developed that measures directly the displacement between adjacent images using overlapped areas approximately 100 pixels wide. This relative motion is then accounted for in the displacement calculations. Because this procedure is additive, error accumulates and becomes visible in the form of vertical


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and/or horizontal concentrations in the contour maps further away from the central subregion that is used as the anchor for the assembly process. This full procedure is detailed elsewhere [13]. Through the use of the updated Subregion Scanning system, a resolution was achieved that was approximately 20 times higher than previously possible with single-image DIC. This technique may be easily applied to simultaneously study crack evolution of a specimen during mechanical testing, provided the area examined is a flat surface. Figure 4 shows the arrangement of the testing system as applied to visualize the fracture process in concrete under compression. The CCD camera on the transitional stage was fixed to table approximately 30 cm in front of the specimen. Two white light sources were placed on the left and right of the specimen to produce sufficient lighting of the specimen surface. A automated calibration operation was performed to determine the resolution of the images and to ensure that the movement of the stage was in a plane parallel to the specimen surface [13].

3.


To illustrate the usefulness of the SSCV technique, experimental results obtained from two previous studies are summarized, along with some conclusions that were made about the relationship between the observed fracture development and mechanical response. The first study examined the effect of aggregate inclusions in twodimensional concrete during compressive failure. The second investigated the role of discontinuous, reinforcing fibers in a mortar matrix under tensile loading. 3.1

MODEL CONCRETE IN COMPRESSION

In the investigation pictured in Figure 4 above, the effects of aggregate spacing and properties on the fracture process in a two-dimensional model concrete were examined. Regularly spaced cylindrical aggregates of either granite or limestone, were imbedded in mortar specimens, 25 x 75 x 75mm, in a predefined arrangement. The aggregate

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cylinders, which had a diameter of 12.7 mm, numbered 13, five and one in the configurations tested (Figure 5). These specimens were then loaded in compression, using a closed-loop control signal based on a combination of both axial and lateral deformation [12].

In Figure 6, contour maps, constructed from displacements obtained using SSCV, that describe the evolution of cracking of a specimen containing five granite aggregates is pictured. Cracking begins around the aggregates as a result of tensile and shear stresses generated by differences in the geometric and material properties of the two phases. These interfacial cracks then link from aggregate to aggregate as the load increases, eventually forming continuous cracks that traverse the entire specimen at peak load.


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The crack patterns for all three aggregate configurations at the peak load are pictured in Figure 7 for specimens containing both granite and limestone aggregate. It is clear that the aggregate material had a strong influence on the way cracks developed. The crack path splits the weaker limestone while the stronger granite forced the cracks to travel around the aggregate particles. The stronger bond formed between the porous limestone and the mortar matrix also reduced the likelihood of this type of interfacial cracking. Note also that for both aggregate types, in the specimens containing one and 13 aggregates, one main crack developed, which determined the strength of that specimen. However, for the specimens containing five aggregates, distributed cracking was visible at this point. This is significant because the development of more distributed cracking consumes a greater amount of energy, since more cracks must be formed. This means that a material in which distributed cracking occurs is likely to exhibit a greater amount of ductility than a similar material in which only one crack develops.

3.2

HYBRID FIBER-REINFORCED MORTAR IN TENSION

In a separate investigation, the role of reinforcing fibers in a mortar matrix during failure was examined with the SSCV technique [13]. In this study, hybrid blends of different size fibers were tested since fiber size was believed to be the main factor in

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determining how the fibers interacted with the fracture process. To investigate this hypothesis, mortar specimens containing various dosages and types of fibers were loaded in uniaxial tension (Figure 8). Specifically, PVA microfibers (22 µm in diameter and 12 mm long) were blended with steel macrofibers (500 µm diameter, and 30 mm long). This combination of reinforcement resulted in greater strength and toughness than was achieved with either fiber type individually.

The development of cracking can be traced in Figure 9, which shows contours of vertical displacement obtained with SSCV for a mortar specimen without any fibers. A plot of stress versus the nominal strain is also included in the figure. While two cracks initially open in this specimen, it is clear that only one develops into the post-peak region. The peak load in the specimen coincides with a crack crossing the full specimen width. Since this specimen does not contain fibers, once this crack occurs there is little left to hold the specimen together and the stress in the specimen drops accordingly. Note that while stress relaxation occurred as the axial displacement was held constant during the image capture process, the measured specimen response quickly returned to the envelope of the stress-strain curve once loading was resumed.


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In Figure 10, the crack pattern at the peak load of mortar specimens containing various fiber types are visible. While one single crack dominates the failure of the unreinforced specimens and those with only macrofibers, those with hybrid fiberreinforcement exhibit multiple wide cracks, also known as macrocracks. This increases the deformation capacity of the material and delays the development of the critical through-specimen crack that determines the strength of the specimen.

In the course of this second study, a means for estimating the crack width based on the measured displacements was devised to provide additional insight into the process of crack formation. Since the tests performed were conducted under uniaxial tension and it was observed that cracks formed largely perpendicular to the loading direction, it was assumed their width could be determined based on the measured vertical displacements. The relative displacement of vertically adjacent nodes was calculated

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and when this displacement was higher than a user-defined threshold, this displacement was defined as a crack width. From these results, color coded images of the crack pattern were produced, in which color represents the crack widths at that location. While these figures can not be reproduced here, the crack width information obtained in this manner was used to successfully estimate the water permeability of the mortar based on the assumption of laminar flow occurring through the cracks [13]. These estimates closely matched direct measurements performed on similar specimens. In this way, data obtained using the SSCV technique was used to quantitatively describe cracking and predict material behavior beyond mechanical response. 4.

Conclusions

Failure of brittle materials is governed by the development of cracking. In most cases, this process occurs at a scale below what is visible to the human eye. SSCV is an effective full-field method for detecting crack development because of its robust application and the high resolution with which it measures surface displacement (0.5 µm). SSCV utilizes a DIC search algorithm for determining displacement but improves on single-image DIC by using a mosaic of 56 images to describe the specimen surface. To show how SSCV may be applied in an experimental setting, studies where this technique was used to examine the fracture process in concrete and mortar during loading were presented. In these studies, the topology of cracking was strongly connected to the specimen behavior, particularly the mechanical performance. 5. 1. 2.

References

Dally, J.W., and Riley, W.F.: Experimental Stress Analysis, McGraw-Hill, Inc., New York, 1991. Shao, Y., Li. Z., and Shah, S.P.: Matrix Cracking and Interface debonding in Fiber-reinforced Cement Matrix Composites, Advanced Cement Based Materials 1 (2) (1993) 55-66. 3. Jia, Z., and Shah, S.P.: Two-dimensional Electronic-speckle-pattern Interferometry and Concretefracture Processes, Experimental Mechanics 34 (3) (1994) 262-270. 4. Creath, K.: Phase Measurement Interferometry Technique, in E. Wolf (ed.), Progress in Optics XXVI, Elsevier Science, New York, 1988, 349-393. 5. Miller, R.A., Shah, S.P., and Bjelkhagen, H.I.: Crack Profiles in Mortar Measured by Holographic Interferometry, Experimental Mechanics 28 (4) (1988) 388-94. 6. Mobasher, B., Castro-Montero, A. and Shah, S.P.: A Study of Fiber-reinforced Cement-based Composites Using Laser Holographic Interferometry, Experimental Mechanics 30 (3) (1990) 28694. 7. Castro-Montero, A., Jia, Z., and Shah, S.P.: Evaluation of Damage in Brazilian Test Using Holographic Interferometry, ACI Materials Journal 92 (3) (1995) 268-75. 8. Choi, S., and Shah, S.P.: Measurement of Deformations on Concrete Subjected to Compression Using Image Correlation, Experimental Mechanics 37 (3) (1997) 307-13. 9. Han, G., Sutton, M.A., and Choa, Y.J.: A Study of Stationary Crack-tip Deformation Fields in Thin Sheets by Computer Vision, Experimental Mechanics 34 (2) (1994) 125-40. 10. Chu, T.C. Ranson, W.F., Sutton, M.A. and Peters, W.H.: Applications of Digital Image Correlation Techniques to Experimental Mechanics, Experimental Mechanics 25 (3) (1985) 23244. 11. Pratt, W.K.: Digital Image Processing, 2nd edition, John Wiley & Sons, Inc., New York, 1991. 12. Choi, S., and Shah, S.P. Propagation of Microcracks in Concrete Studied with Subregion Scanning Computer Vision (SSCV), ACI Materials Journal 96 (2) (1999) 255-60. 13. Lawler, J.S.: Hybrid Fiber-reinforcement in mortar and concrete, Ph.D. Thesis, Northwestern University, Evanston, IL, 2001.

ON THE USE OF DIFFERENT WAVELENGTHS TO DIGITALLY DETERMINE THE ISOCHROMATIC FRINGE ORDER

T. Y. CHEN, Y. C. CHOU, H. L. LEE, and S. H. TSAO Department of Mechanical Engineering National Cheng Kung University Tainan, Taiwan 701, R.O. China

Abstract Four approaches, a novel, two modified and a previous developed approaches, are compared for the determination of the total fringe orders using different wavelengths. The results show that an error of 0.05 fringes can be achieved by all methods. The MIN-MAX method gives the best result for fringe orders greater than 0.6, a better result can be given by the rest methods for fringe orders less than 0.5. The three-wavelength criterion could yield better result than that of the two-wavelength for determination of the total fringe order. Owing to various noises, both criteria cannot assure to obtain correct fringe order. Therefore two-wavelength method would be sufficient to determine the total fringe orders with a correctly determined total fringe order.

1.

Introduction

As an experimental technique for stress and strain analysis, photoelasticity is very useful for problems in which stress or strain information is required for extended regions of the structure or member. The birefringence, or fringe order, in a stressed photoelastic model is controlled by the state of stress at each point in the model. Conventionally the measurement of the fringe order is done manually, which can be time-consuming, and require a well-trained operator [1]. By using the computers and digital image processing techniques, the developments in digital photoelasticity have made photoelastic analysis faster and more reliable for solving engineering problems [2]. Chen reviewed the approaches for digital determination of photoelastic fringe order [3]. Earlier developments in digital analysis of photoelastic fringe orders mainly involve the determination of the location of fringe points and interactively assigning the fringe orders from one image only [4, 5]. Recent studies propose to process intensity data of the images recorded to obtain the whole-field total fringe orders automatically with a PC-based image processing system [6-10]. Recent developments using two-load [11], three-load [12], and the combination of load-stepping to phase-stepping methods [13] for more easily or accurately determination of the total fringe order are also reported. Basically one fringe pattern does not provide sufficient information to determine the total fringe orders. Thus most methods mentioned above acquire more information by using either different wavelengths or different loads to determine the total fringe orders. 345 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 345–352. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The use of different wavelengths of light to determine the isochromatic fringe order has been reported. Umezaki [14] and Plouzennec [15] identified the zero order fringes through an image processing process in which two patterns of different wavelengths were superimposed on each other. The computer was then able to identify the other fringes from the located zero order fringes. However, this method fails when no zero order fringes are present. Chen [7] proposed a scheme that uses two light-field isochromatic fringe patterns with different wavelengths to determine the fringe order automatically. The fringe order is determined regardless of whether zero-order fringes are present or not. Kihara [16] and Buckberry [17] used the image data from three different wavelengths to determine the total fringe orders. Use of two-wavelength or three-wavelength method generally involves two steps. The first step is to determine the fractional fringe orders from two or three sets of images obtained from different wavelengths. Then the second step is to search for the total fringe orders by setting a minimum criterion. Since the accuracy of the intensity data might be affected by various factors, such as, non-uniform distribution of light, material inhomogeneities, spatial quantization, gray-level discretization, and variation of light due to rotation of the optical element, the effectiveness of various approaches for determination of the fringe order are studied and compared with an approach developed previously. The principles of the methods are described. Test results from the experiments are given and discussed.

2. Theoretical Considerations In photoelasticity, with a monochromatic light of wavelength the stress-optic law can be written for a two-dimensional plane-stressed photoelastic body as where and are the principal stresses at point; is the total fringe order, is the relative retardation, and is the relative phase shift; is the material fringe value, and C is the relative stress-optic coefficient, which is wavelength dependent. Using the circular polariscope arrangement as shown in Fig. 1, the intensity of light emerging from the analyzer is given by

where is the maximum dark-field intensity of light emerging from the analyzer. Rotating the analyzer 90°, the intensity of light emerging from the analyzer becomes where is the maximum light-field intensity of light emerging from the analyzer. As the total fringe order of the isochromatic fringe pattern is determined, the principal stress difference, can be calculated from eq (1). Since the light intensity varies according to the or function, the intensity of one wavelength can only provide the fractional fringe order of the fringes. Use of different wavelengths to determine the total fringe orders generally involves two steps. The first step is to determine the fractional fringe orders from a set of images obtained from each wavelength used. Then the second step is to search for the total fringe orders from the determined fractional fringe orders having different wavelengths by setting a minimum criterion. Using a circular polariscope

USE OF WAVELENGTHS TO DETERMINE ISOCHROMATIC ORDER 347 setup, the principles of the approaches used for the two steps are described in the following sections.

2.1 DETERMINATION OF FRACTIONAL FRINGE ORDERS FROM ONE WAVELENGTH Four approaches are considered here to determine the fractional fringe orders. Basically the methods proposed or described below allow a normalized isochromatic fringe pattern to be obtained: (1)Use of a light-field unloaded image and a light-field loaded isochromatic fringe pattern (LF-ADJ). This method, proposed by Chen, generates a normalized light-field intensity using an unloaded and a loaded light-field isochromatic fringe pattern of a photoelastic model. After linearly adjusting the normalized intensity to get rid of the background/stray light effect, the fractional fringe order, n, can be calculated from:

where and are the light-field intensities of the isochromatic fringe patterns for the unloaded and loaded model, respectively. (2) Use of a dark-field unloaded image, a light-field unloaded image, and a light-field loaded isochromatic fringe pattern (LF-DF). This approach resembles to the phase shifting method. In this method, the intensities of light emerging from the analyzer of the dark-field unloaded, light-field unloaded, and light-field loaded photoelastic model can be represented by

where a accounts for the amplitude of light and b denotes the background/stray light intensity. From eqs (5)-(7), the fractional fringe order can be calculated from the following equation: (3) Use of the scanned minimum and maximum light-field intensities of the model under various loads and an light-field isochromatic fringe patterns of the model under a specific load (MIN-MAX). This novel approach is similar to method (2). Instead of using the unloaded dark- and light-field image intensities, they are replaced by the minimum and maximum intensities scanned from

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the loaded light-field fringe patterns under various loads. In this case, the variation of light due to rotation of polarizer is avoided. The fractional fringe order thus can be calculated from the following equation: (4) Use of the dark-field and light-field intensities of the polariscope without the model, and a light-field isochromatic fringe patterns of the loaded model (LF-DF-NM). This method may be considered as an alternative of method (2) or method (1), and may be used for analyzing three-dimensional photoelastic slice, in which the unloaded model image is not obtainable. If the maximum fringe order in the field of interest does not exceed 1/2, the fringe order can be determined without ambiguity using the above methods. If the maximum fringe order in an isochromatic fringe pattern greater than 1/2, the total fringe order, N, should be one in a set of multiple-valued fringe orders given by:

or 2.2

DETERMINATION OF TOTAL FRINGE ORDERS USING DIFFERENT WAVELENGTHS

For two different wavelengths and orders are obtained from eq (10) as

used, two sets of multiple-valued fringe

or where the subscript i (= 1, 2) corresponds to the wavelength Since the state of stress at the same point on a model is the same for different wavelengths, the following relationship can be obtained from the stress-optic law as where and are the material fringe values with respect to wavelengths and Therefore, the total fringe order of and at any point must satisfy eq (12), and can be determined by finding a pair that minimizes the error function, D, as follows: Similarly, if three wavelengths are used, the total fringe order of a point can be determined by using the error function, E, defined as:

2.3

and

RANGE OF MEASUREMENT

Since the light intensity function has a period of the difference of the two fringe orders between the two images is limited to 1/2. Hence the range of measurement for using two wavelengths can be derived as

For three wavelengths used, the range of the measurement becomes the lowest common multiple of which is much larger than that for using two wavelengths.

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3. System Configurations and Experiments The system used, as shown schematically in Fig. 1, consists of a diffused-light circular polariscope for producing isochromatic fringe patterns using a white light, three narrow-band filters with a band width of 10 nm for obtaining monochromatic fringe patterns, a Pulnix TM745 CCD camera, a Matrox frame grabber with four frame buffers, a PentiumIII PC, and other peripheral devices. Each frame buffer has 256K bytes of memory that is organized as 512x512 pixels with 256 gray levels. The specimen used is a circular disk (made of photoelastic material, PSM-1), 50.8 mm in diameter and 5 mm thick. The wavelength of the three narrow-band filters used are and As in any photoelastic analysis, the specimen was calibrated and the material fringe values are determined for the three wavelengths. Figures 2(a)-2(c) show the fringe patterns of the disk under a diametral compression of 500N observed through the three filters, respectively. Using the four approaches, the fractional fringe orders were determined for each wavelength. A gray-level representation (0-255 correspond to the orders 0-0.5) of the determined fractional fringe orders of the disk using different methods is shown in Figs. 2 (d)-2(g) for A comparison of the gray-level represented fractional orders along a horizontal line across the center of the disk for the four methods is shown in Fig. 3. Then the total fringe orders were determined using the criteria for two-wavelength and three-wavelength, respectively. A gray-level representation (0-255 correspond to the orders 0-5) of the determined total fringe orders of the disk using two-wavelength method is shown in Fig. 4 for A comparison of the total fractional orders determined by the four methods to the theoretical result along a horizontal line across the center of the disk is shown in Fig. 5.

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4. Discussions Comparing the fringe patterns in Fig. 2, it can be observed that the fractional fringe patterns agree well to the original ones. Further examining the plots in Fig.3, it can be seen that the gray-level values do no reach 0 or 255, which correspond to the integral and the half fringe order for all methods. The maximum gray-level values

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reach almost the same level for all methods, but the minimum gray-level values show a little different for different approaches. The MIN-MAX method reaches the lowest value and the error is less than 6 gray levels, and the errors for the rest methods are almost the same, which are between 0-20 gray levels. The effect of these errors may also he observed in the total fringe orders plot shown in Fig. 5.

We observe in Fig. 4 that the gray-level (fringe order) varies in most parts of the image except for the regions close to the load point where an error is evident. The error is caused mainly by the resolution of the image processing system. Enlarging those regions would solve this problem. There is not much difference among the four methods. From Fig. 5, it can be seen that the determined fringe orders agree well with the results determined from the theory for all methods. The difference among them is less than 0.05 fringes except for some points at or near the integral or half-fringe order locations. Use of a higher resolution camera or digitally filtering the results may alleviate the errors. Averaging speaking, MIN-MAX method gives the best result for fringe orders higher than 0.6, a better result can be given by the rest methods for fringe order less than 0.5. Owing to the various noises contained in the digital values, it is noted that the total fringe orders may not be those with a minimum difference. Present studies show that the correctness of the determined total fringe orders is about 55% by using the two-wavelength criterion, and that is 70% for using three-wavelength criterion. To insure the determined total fringe orders is correct, a two-wavelength digital procedure [7] that uses 5x5 cross-shaped points can be used no matter the zero-order fringes are present or not in the fringe pattern. If zero-order fringes exist

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in the fringe pattern, they can be identified by superimposing two fringe patterns having different wavelengths. Thereby two-wavelength method is sufficient to determine the total fringe orders.

5.

Conclusions

Use of different wavelengths to determine the isochromatic fringe orders has been presented. Four approaches, a novel, two modified, and a previous developed approaches, are compared to determine the total fringe orders. The results show that an error of less than 0.05 fringes can be achieved by all methods. The MIN-MAX method gives the best result for fringe orders greater than 0.6, a better result can be obtained by the rest methods for fringe orders less than 0.5. Although the three-wavelength criterion yields better result than the two-wavelength criterion for determination of the total fringe order, both criteria cannot assure their correctness with existing noises. Therefore two-wavelength method would be sufficient to determine the total fringe orders if a total fringe order can be correctly determined.

6. Acknowledgment The author is thankful to the National Science Council of the Republic of China for supporting this research under grant NSC89-2212-E-006-116.

7. References 1. 2. 3 4. 5. 6.

7. 8. 9.

10. 11. 12. 13. 14.

15. 16. 17.

Dally, J.W. and Riley, W.F. (1978) Experimental Stress Analysis, McGraw-Hill Co.. Chen, T.Y. (1999) Digital photoelasticity, in P.K. Rastogi (ed.), Photomechanics, Springer-Verlag, pp. 197-232. Chen, T.Y. (2000) Recent advances in digital photoelasticity, in P. Rastogi and D. Inaudi (eds.), Trends in Optical Nondestructive Testing and Inspection, Elsevier Science, pp. 533-544. Chen T.Y., Taylor C.E. (1989) Computerized fringe analysis in photomechanics, Exp. Mech. 29, 323-329. Ramesh K., Ganesan V.R., and Mullick S.K.. (1991) Digital image processing of photoelastic fringes - a new approach, Exp. Tech. 15,41-46. Voloshin, A.S. and Burger, C. P. (1983) Half-fringe photoelasticity: a new approach to the whole field stress analysis, Exp. Mech. 23, 304-313. Redner, A.S. (1985) Photoelastic measurements by means of computer aided spectral-contents analysis, Exp. Mech. 25, 148-153. Patterson, E.A. and Wang, Z.F. (1991) Toward full-field automated photoelastic analysis of complex components, Strain 27, 49-56. Ajovalasit A., Barone S., and Petrucci G (1995) Toward RGB photoelasticity; full-field automated photoelasticity in white light, Exp. Mech. 35, 193-200. Chen, T.Y. (1997) Digital determination of photoelastic birefringence using two wavelengths, Exp. Mech. 37,232-236. Chen T.Y. (2000) A simple method for the digital determination of the photoelastic fringe order, Exp.Mech. 40, 256-260. Asundi, A. Liu, T. and Chai, G.B. (2000) Determination of isoclinic and isochromatic parameters using three-load method, Meas. Sci. Technol. 11, 532-537. Ramesh, K and Tamrakar (2000) Improved determination of retardation in digital photoelasticity by load stepping, Optics and Lasers in Eng. 33, 387-400. Umezaki, B., Tamaki T. and Takahashi, S. (1984) Automatic stress analysis from photoelastic fringes due to image processing using a personal computer, Proc. Soc. Photo. 504, 127-134. Plouzennec, N. and Lagarde, A. (1999) Two-wavelength method for full-field automated photoelasticity, Exp. Mech. 39, 274-277. Kihara T. (1994) Automatic whole-field measurement of principal stress directions using three wavelengths, in S.Gomes (ed.) Recent Advances in Experimental Mechanics, Balkema, pp.95-99 Buckberry, C. and Towers, D (1996) New approach to the full-field analysis of photoelastic stress patterns, Optics and Lasers in Eng. 24, 415-428.

THE APPLICATION OF SPECKLE METROLOGY TO HEART MECHANICS

G. R. GAUDETTE, E. U. AZELOGLU, J. TODARO, L. KEENE, I. B. KRUKENKAMP, F. P. CHIANG State University of New York at Stony Brook Division of Cardiothoracic Surgery Department of Mechanical Engineering Stony Brook, NY 11794-2300

Abstract Commonly used techniques for analysis of regional heart function include sonomicrometry, implanted markers, and surface markers. However, these techniques cannot offer the high spatial resolution needed to define regional abnormalities in the heart. We have recently applied a computer aided speckle interferometry technique (CASI) with high spatial resolution to measuring the deformation of heart muscle. We applied silicon carbide particles to the surface of the isolated heart and loaded the heart by increasing the intracavitary pressure. The movement of the speckle pattern was tracked with a CCD camera. This technique produces equivalent results to that of the gold standard in heart mechanics, sonomicrometry, but with three orders of mapitude higher spatial resolution. We have used this technique to determine changes in myocardial deformation of perfused, ischemic and reperfused rabbit hearts. 1. Introduction The heart tissue, myocardium, is anisotropic, inhomogeneous, and has a non-linear material properties [1-3]. The heart can experience a reduction in blood flow due to partial or total occlusion of the vessels that supply blood to the region. In general, this reduction in blood flow (ischemia) is a regional phenomenon resulting in changes in the material properties of the heart. These changes can occur over a 10-20 millimeters length during regional ischemia in hearts of dogs or pigs[4-6]. Other clinically relevant scenarios can occur which also result in abrupt changes in regional myocardial function, such as pacing from various locations [7] and tissue engineering procedures [8]. It is therefore necessary to measure the mechanical response of the heart over a large region with high spatial resolution. Currently employed techniques used in determining regional function of the heart include sonomicrometry, surface attached or implanted markers, echocardiography, and magnetic resonance imaging (MRI). Sonomicrometry is the most commonly used technique for measuring regional function in the laboratory setting. This technique involves the implantation of two (or more) transducers in the heart wall (myocardium). One transducer send an ultrasonic signal while the second transducer receives the 353 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 353–364. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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signal. This technique can only provide a one-dimensional change in length over a relatively large gauge length (generally > 10 mm). To determine 2-D deformation, investigators have placed paper markers [5, 9] or titanium dioxide particles [10] on the surface of the heart and tracked their movement with a video camera. By using 40-60 paper markers some investigators have been able to obtain a spatial resolution of approximately 5 mm[5]. Although this technique offers higher spatial resolution compared to sonomicrometery, important deformation information may be lost in the borderzone between normally perfused tissue and ischemic tissue. In addition, this resolution may not be sufficient in smaller hearts, such as that of rabbits and rats. Clinically, echocardiography is generally used to determine regional function [11, 12]. This technique can determine the velocity of sound inside the tissue at various locations and determine the approximate displacement from the sequential acquisition of the velocity. However, orientation of the echocardiography probe may generate errors and some areas of the heart are difficult to image [13]. In addition, the spatial resolution of this technique is lacking. MRIs can offer higher spatial resolution and determine 3-dimensional deformation [14, 15]. However, this technique can not acquire a full heartbeat in one data acquisition set and must rely on steady state conditions. Furthermore, it is a very expensive technique and requires extensive computational time. Computer aided speckle interferometry (CASI) is a technique which was recently developed in our laboratory [16-20]. This technique evolves from the earlier works on laser and white light speckle methods [21, 22]. It offers high spatial resolution, is nondestructive, relatively inexpensive, and is a whole field technique. In this paper we have applied this technique to determining myocardial function in hearts exposed to ischemia, which results in a decrease in heart function.

2. Materials & Methods 2.1. ANIMAL CARE All animals received humane care in compliance with the "Principles of Laboratory Animal Care" formulated by the National Society for Medical Research and the "Guide for the Care and Use of Laboratory Animals" prepared by the National Academy of Sciences and published by the National Institutes of Health (NIH Publication No. 85-23, revised 1985). Furthermore, the Institutional Animal Care and Use Committee at Stony Brook reviewed and approved the protocol followed in this study (IACUC # 0834). In this study rabbits were chosen because their hearts could easily be isolated and they are commonly used in a Langendorff apparatus, which is a well-designed model for myocardial function. For all studies involving rabbits, New Zealand rabbits, weighing 2.5-3.5 kg, were anesthetized with sodium pentobarbital (30 mg/kg) and anticoagulated (1,000 units sodium heparin) by an ear vein. The chest was opened via bilateral thoracotomy, the heart rapidly excised and placed in iced Krebs-Henseleit solution.

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2.2. CASI ANALYSIS One of the major advances of experimental strain measurement techniques is the advent of speckle interferometry (also referred to as speckle photography) [18, 19]. Heretofore, displacement/strain measurements were done by reading the change of distances between well-defined geometric markers. The speckle technique on the other hand employs a pattern of random markers called speckles as displacement transducers. Instead of following the movement of each and every speckle, the movement of a cluster of speckles is monitored through a correlation calculation before and after the movement. In the earlier stages of its development coherent laser speckles (which are the natural result of multiple interference of wavelets reflected from an optically rough, laser illuminated, surface) were exclusively used [21]. Later incoherent white light speckles were employed [22]. With the rapid advance of CCD (charge coupled device) cameras and computer technology, a digital version of the method, i.e. CASI, was developed [20]. In the following section the essence of CASI, as evolved in references [17, 20, 23, 24], is briefly described. First a random speckle pattern is created on the surface of a specimen, typically using a combination of white and dark particles to obtain a high contrast light intensity distribution. This speckle pattern is recorded by a CCD camera and is digitized. Upon deformation of the specimen, the speckle pattern is digitally recorded again. The two speckle patterns are subdivided into corresponding subimages, each consisting of 32 x 32 pixels (other choices such as 64 x 64 or 16 x 16 pixels can also be used depending on the situation). Let (x,y) and (x,y) denote the intensity distributions of speckle patterns before and after deformation, respectively, and:

in which u and v are the average displacement components along the x and y directions, respectively, as represented by the cluster of speckles within the subimage. By maximizing the cross correlation coefficient of the two speckle patterns,

the displacement components, u and v, can be obtained. In the earlier days, correlation was obtained optically by either pointwise or full field spatial filtering techniques [21]. Direct numerical calculation of the cross correlation coefficient [25, 26], however, is very time consuming. Instead a Fourier Transform is applied to the speckle functions and

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In the spectrum domain an interference function is introduced:

where and are phase angles of and * denotes the complex conjugate. It can be shown [20] that:

respectively, and

By performing another Fourier Transform, we obtain the so-called halo function, G:

which is an expanded impulse function located at (u,v). By detecting the crest of this function, the displacement vector between the two speckle patterns is uniquely determined. Strains can be calculated by taking derivatives of the displacement components (u and v) and using an appropriate strain-displacement relationship (be it linear or non-linear). The resolution of the speckle technique largely depends on the size of speckle employed, the resolution of the camera, and the magnification factor. In this study we typically used 40 µm particles with a 1,024 x 1,024 pixel resolution, and 20X magnification. 3. Validation of CASI with Sonomicrometry 3.1. INRODUCTION In order to establish CASI as a useful technique to measure myocardial deformation, the results obtained with CASI must be compared to those obtained with an accepted technique, such as sonomicrometry. The use of sonomicrometry in myocardial mechanics involves the implantation of two ultrasonic transducers. One transducer sends an ultrasonic signal and the second transducer receives the signal. The time it takes for the signal to travel between the two transducers is proportional to the distance between them. This technique is essentially the “gold standard” in myocardial deformation measurements. Urheim and coworkers have used sonomicrometry to validate determination of regional myocardial strain using echocardiography[27], similarly, Omens and colleagues have used sonomicrometry to validate their use of epicardial markers to determine regional myocardial deformation[28]. Herein, sonomicrometry will be used to validate the CASI technique.

3.2. EXPERIMENTAL PROTOCOL In this study CASI was used to measure deformation in the epicardium of rabbit hearts (n=3). After the heart was isolated, a latex balloon was placed in the left ventricle of the heart, and the pressure in the ventricle was adjusted by changing the pressure head of a static column of water attached to the intracavitary balloon. Small silicon carbide


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particles (approximately 40 microns in diameter) were dispersed in an area on the anterior surface of the heart to serve as the speckle pattern. A schematic diagram of the test arrangement is shown in Figure 1. Two ultrasonic transducers (Iowa Doppler Products, Iowa City, IA) were implanted into the epicardium at the boundary of the speckled region. Three hearts were subjected to various protocols to determine the relationship between results obtained with sonomicrometry and CASI. All hearts were conditioned three times prior to protocols by inflating the intracavitary balloon to approximately 25 mmHg, and then deflating. Subsequently the heart was subjected to incremental intracavitary pressure increases of 10 mmHg, from 0 to 30 mmHg. A second heart was allowed to passively contract, with data acquired at 0, 2, 4, and 6 minutes after conditioning. In the final heart, the strain determined by sonomicrometry was incremented from 0 to 1% and 2% strain (based on the original length between the sonomicrometry transducers at intracavitary pressure being equal to zero). Both sonomicrometry data and the speckle image were acquired at each load and digitally stored for analysis.

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The use of silicon carbide particles provided for a good speckle image. These particles adhered well to the myocardium. Immediately after application, any loose particles were removed by gently bathing the heart in Krebs-Henseleit solution or by applying gentle air currents over the surface of the heart. The speckle patterns before and after the heart deformation were recorded using a high resolution charge coupled device (CCD) camera (Kodak Megaplus camera Model 4.2) with a 2029 x 2048 pixel array and 8 bit gray level resolution. The images were stored on a workstation for subsequent analysis using CASI. 3.3 STRAIN DETERMINATION For the purpose of comparison, normal strains were determined from displacement data obtained by both sonomicrometry and CASI using the following equation:

In the case of sonomicrometry, and are the initial and final distances between the sonomicrometry transducers. For corresponding strains determined by CASI, displacements from the two subimages closest to each sonomicrometry transducer were used. Because CASI gives both the u and v displacements (i.e. displacement components in the x and y axes, respectively), the magnitudes of these displacement components resolved along the axis of the sonomicrometry transducers were used to calculate the CASI strain data The algebraic sum of the resolved displacement components is simply and is the distance between the centroids of the two subimages most adjacent to the transducers.

3.4. RESULTS For the three hearts, the region of interest between the transducers was covered by 8001200 pixels in the x direction, and 200-400 pixels in the y direction. With a 32 x 32 subimage size and a shift of 16 pixels between subimages, matrices between 50 x 12 “points” and 75 x 24 “points” representing 600-1800 displacement vectors were generated. This is compared to only one value (the displacement component along the direction of the transducers) determined by sonomicrometry. In addition, with displacements vectors two-dimensional strain tensor distribution can be calculated using appropriate strain-displacement relationships. Figure 2 shows approximately 25% (for reasons of visual clarity) of the displacement vectors obtained from heart #1 using CASI in the region between the two transducers. The strains obtained from CASI and sonomicrometry from all three hearts are shown in Figure 3. A solid line with slope of one is also shown. These data clearly show that the two methods produce equivalent results. However, the spatial resolution of CASI is some three orders of magnitude higher than sonomicrometry.


4.

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Deformation in the Ischemic Heart

4.1. INTRODUCTION Although CASI has been established as a technique that can be used to measure 2-D strain distribution in the myocardium, its ability to measure deformation in clinically relevant scenarios must be established. One scenario that occurs during cardiac surgery is global ischemia in the arrested (non-beating) heart. This results in no blood flow being supplied to the entire heart. We therefore used CASI to determine regional differences in myocardial deformation that results from global ischemia. 4.2. MATERIALS & METHODS In addition to the procedures previously described in the rabbit model, a cannula was inserted into the aorta to perfuse the myocardium with Kreb’s Heinselett solution via the coronary arteries. The coronary arteries were perfused at a constant perfusion pressure head equal to 75 mmHg. An intracavitary balloon was inserted into the left ventricle. The balloon was filled with water and attached to a vertically adjustable reservoir used to change the intracavitary pressure (see Figure 1). Small silicon carbide particles were dispersed randomly on the left ventricle to create a speckle pattern. After a 15-minute equilibrium period, the hearts were then infused with Krebs-Henseleit solution plus high potassium (approximately 20 mM) to stop the heart from beating. The intracavitary load was adjusted from 0 to 20 mmHg, and data acquired at 0, 10, and 20 mmHg. The flow was then discontinued to the hearts (global ischemia). After 5, 10, 15 and 30 minutes of global ischemia, at which the speckle data were recorded, the left ventricle was again incrementally loaded by increasing the intracavitary pressure. The hearts were then reperfused for 15 minutes and incrementally loaded a final time. 4.2.1.

CASI ANALYSIS

The images of the region of the myocardium that is covered with speckles are recorded at different pressures using a 1,024 x 1,024 pixel CCD camera. The subimage size for data analysis in these hearts was 64 x 64 pixels, and the shift size was 4 pixels. The following equations were used to determine strain:

A linear fit to several data points is determined from the data obtained with CASI. As the approximate resolution of the system in this protocol was 20 µm/pixel, and the shift size was 4 pixels between subimages, a linear fit was employed over 13 subimages (slightly larger than 1 mm). Therefore, to approximate the partial differentials, a least squares linear regression was fit to 13 points in their respective directions. The principle strains and were determined and the first invariant of the 2D strain tensor was also computed using the following equation:

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These variables provided valuable information to determine the perfusion status of the heart.


4.3.

361

RESULTS

Figure 4 shows the u and v displacement contour plots of a typical rabbit heart under the status of perfusion, global ischemia and reperfusion. It is seen that more deformation is experienced by the ischemic myocardium when the heart is subjected to the same intracavitary pressure. In the small region that we investigated the strain is essentially uniform. Thus we calculated the average strain to show the difference of myocardial response between perfused and ischemic states of the heart. In Figure 5 the average maximum principle strain and the first strain invariant of the epicardium are plotted as a function of time as the heart goes through the different stages of perfusion, ischemia, and reperfusion. It is seen that when the hearts were made globally ischemic, both the maximum principle strain and the first invariant significantly increased. Upon reperfusion (restoration of flow), the maximum principle strain and the first invariant both decreased compared to the ischemic case. Indeed it is even slightly lower than the original perfused state. Clearly the deformation is significantly higher (about seven times) in the ischemic condition than in the perfused or reperfused conditions. This indicates that the

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tissue is significantly weakened by the lack of blood flow. While both and indicators of epicardium deformation, seems to be slightly more sensitive.

are good


5.

363

Conclusion & Discussion

We have demonstrated that CASI, a full field strain measurement technique that heretofore has been used in measuring strain in engineering materials, can also be used to measure deformation of myocardium. It has a spatial resolution at least three orders of magnitudes higher than that of sonomicrometry, the “gold standard” in heart mechanics. We have shown that the effect of ischemia significantly weakens the heart muscle. However, after reperfusion, the strength of the muscle is restored. In work to appear elsewhere we have also demonstrated that CASI can be applied to in-vivo studies as well. 6. Acknowledgements

The financial support of this work by the National Science Foundation through Grant number BES – 9903516 is gratefully acknowledged. 7. References 1. 2. 3.

4.

5.

Fung YC. Biomechanics, Mechanical Properties of Living Tissues. 1981, New York: SpringerVerlag. Spotnitz HM. Macro design, structure, and mechanics of the left ventricle. Journal of Thoracic & Cardiovascular Surgery, 2000.119(5): p. 1053-77. Guccione JM, O'Dell WG, McCulloch AD, Hunter WC. Anterior and posterior left ventricular sarcomere lengths behave similarly during ejection. American Journal of Physiology, 1997. 272(1 Pt 2): p. H469-77. van Leuven SL, Waldman LK, McCulloch AD, Covell JW. Gradients of epicardial strain across the perfusion boundry during acute myocardial ischemia. American Journal of Physiology, 1994. 267: p. H2348-2362. Prinzen FW, Arts T, Hoeks APG, Reneman RS. Discrepancies between myocardial blood flow and fiber shortening in the ischemic border zone as assessed with video mapping of epicardial deformation. Phlugers. Arc (Europ. J. Physiol.), 1989. 415: p. 220-29.

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6.

7.

8.

9. 10.

11.

12. 13. 14.

15.

16. 17.

18. 19. 20.

21. 22. 23. 24. 25. 26. 27.

28.


Gallagher KP, Gerren RA, Stirling MC, Choy M, Dysko RC, McManimon SP, Dunham WR. The distribution of functional impariment across the lateral border of acutely ischemic myocardium. Circulation Research, 1986. 58: p. 570-583. Prinzen FW, Hunter WC, Wyman BT, McVeigh ER. Mapping of regional myocardial strain and work during ventricular pacing: Experimental study using magnetic resonance imaging tagging. Journal of the American College of Cardiology, 1999. 33: p. 1735-1742. Chatterjee S, Stewart AS, Bish LT, Sweeney HL, Garder TJ. Gene transfer of HGF is superior to VEGF in inducing coronary angiogenesis and preservation of myocardial contractility. Surgical Forum, 2001.52: p. 67-69. Augustijn CH AT, Prinzen FW, Reneman S. Mapping the sequence of contraction of the canine left ventricle. Pflugers Arch, 1991. 419: p. 529-533. Karlon WJ, McCulloch AD, Covell JW, Hunter JJ, Omens JH. Regional dysfunction correlates with myofiber disarray in transgenic mice with ventricular expression of ras. American Journal of Physiology - Heart & Circulatory Physiology, 2000. 278(3): p. H898-906. Derumeaux G, Ovize M, Loufoua J, Andre-Fouet X, Minaire Y, Cribier A, Letac B. Doppler tissue imaging quantitates regional wall motion during myocardial ischemia and reperfusion. Circulation, 1998. 97(19): p. 1970-7. Heimdal A, Stoylen A, Torp H, Skjaerpe T. Real-time strain rate imaging of the left ventricle by ultrasound. Journal of the American Society of Echocardiography, 1998. 11(11): p. 1013-9. Castro PL, Greenberg NL, Drinko J, Garcia MJ, Thomas JD. Potential pitfalls of strain rate imaging: angle dependency. Biomedical Sciences Instrumentation, 2000. 36: p. 197-202. Moulton MJ, Creswell LL, Downing SW, Actis RL, Szabo BA, Vannier MW, Pasque MK. Spline surface interpolation for calculating 3-D ventricular strains from MRI tissue tagging. American Journal of Physiology, 1996. 270: p. H281-H297. Scott CH, Sutton MS, Gusani N, et al. Effect ofdobutamine on regional left ventricular function measured by lagged magnetic resonance imaging in normal subjects. American Journal of Cardiology, 1999. 83(3): p. 412-7. Chen DJ, Chiang FP. Optimal sampling resolution and range of measurement in digital speckle correlation 2: white speckle method. Proc SEM Annual Spring Conf on Exp Mech, 1989. Chiang FP, Wang Q, Lehman F. New development's in full field strain measurements using speckles., in Nontraditional Methods of Sensing Stress, Strains, and Damage in Materials and Structures., C.F. Lucas and D.A. Stubbs, Editors. 1997, ASTM. p. 156-169. Erf RK. Speckle Metrology. 1978: Academic Press. Chiang FP. Speckle Metrology, in Metals Handbook. 1989, SSM-International. p. 432-437. Chen DJ, Chiang FP, Tan YS, Don HS. Digital speckle-displacement measurements using a complex spectrum method. Applied Optics, 1993. 32(11): p. 1839-49. Chiang FP. A family of 2D and 3D experimental stress analysis technique using laser speckles. Solid Mechanics Archives, 1978. 30: p. 1-32. Asundi A, Chiang FP. Theory and Application of White Light Speckle Methods. Optical Engineering, 1982. 24(4): p. 570-580. Chen DJ, Chiang FP. Computer aided speckle interferometry using spectral amplitude fringes. Applied Optics, 1993.32: p. 225-36. Chen DJ, Chiang FP. Range of measurement of computer aided speckle interferometery (CASI). Proc 2nd Int Conf on Photomechanics and Speckle Metrology, 1991.1554A. Peters WH, Ranson WF. Digital imaging techniques in experimental stress analysis. Optical Engineering, 1982. 21: p. 427-31. Lu H, Vendroux G, Knauss WG. Surface deformation measurements of a cylindrical specimen by digital image correlation. Experimental Mechanics, 1997. 37: p. 433-39. Urheim S, Edvardsen T, Torp H, Angelsen B, Smiseth OA. Myocardial strain by Doppler echocardiography. Validation of a new method to quantify regional myocardial function. Circulation, 2000. 102(10): p. 1158-64. Omens JH, Farr DD, McCulloch AD, Waldman LK. Comparison of two techniques for measuring two-dimensional strain in rat left ventricles. Am. J. Physiol, 1996. 271: p. H1256-61.

5. Non-Destructive Evaluation


LINE FOCUS ACOUSTIC MICROSCOPY FOR THIN-FILM MEASUREMENTS ZHIQI GUO Sonoscan Inc. 2149 E. Pratt Blvd. Elk Grove Village, IL 60007 USA JAN D. ACHENBACH Center for Quality Engineering and Failure Prevention Northwestern University Evanston, IL 60208 USA

Abstract A thin-film coating generally is a single or multiple layered thin film deposited on a component to extend its life and performance. As discussed in this paper the V(z) effect in line-focus acoustic microscopy is a very suitable technique for the quantitative nondestructive determination of thin-film elastic constants and the bond quality at interfaces. After a brief introduction to acoustic microscopy and a discussion of calibration for frequency dependence, the modeling of multilayered configurations is summarized. The modeling results are used in a measurement model for the V(z) effect which can be used to select a frequency range suitable for a particular configuration. In combination with measurements the model is subsequently employed to determine elastic constants and interface stiffnesses. Experimental results are presented for single, multilayered, isotropic and anisotropic films. The effect of imperfect interfaces on strength considierations is also discussed. 1.

Introduction

Thin film coatings can significantly extend the life or enhance the performance of components of turbine engines and machine tools. In fact, approximately 75% of all the components in an aircraft engine are now coated with a different material [1]. For example, transition metal carbides and nitrides of the IV to VI group of the periodic table have extremely high melting points and extreme hardness, excellent hightemperature strength and good corrosion resistance. They are widely used for thin-film coatings to produce wear resistant surfaces, to provide corrosion protection against harsh environments, and for other applications in surface engineering [2]. Other coatings such as multilayered ceramic and metallic coatings can offer unique physical and mechanical properties due to the refined microstructure. Such coatings have a wide range of uses, including for optical devices, for corrosion protection, as thermal barriers, for wear resistance, and for tribological applications. Examples are the layered coatings for cutting tools [3], and the layered Diamond-Like-Carbon (DLC)/Si coatings for biomedical and aerospace applications [4]. 367 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 367–380. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Z. GUO, J. D. ACHENBACH

Basically, a coated component, here referred to as a film/substrate system, is a layered structure, with a single or multiple layered thin film deposited on a substrate. Film/substrate systems are usually prepared by deposition techniques such as physical vapor deposition (PVD) or chemical vapor deposition (CVD). The mechanical properties of the thin film and the adhesion quality between coating layers and between the thin film and the substrate determine its functional characteristics. These properties are very sensitive to a number of factors, which are determined by the deposition technique and by processing parameters. For essentially the same material, the thin-film properties may be quite different from bulk properties. The mechanical properties of thin films need to be measured and analyzed after deposition for quality assurance of the coated components and for the optimization of coating design and development. Currently, only limited destructive methods, such as the scratch test and the nanoindentation test [5], are used to measure the mechanical properties of thin films and to tune the deposition processing parameters. A suitable nondestructive technique that is capable of providing reliable and reproducible quantitative information for thin films will have obvious advantages. Acoustic Microscopy (AM) is a very suitable technique for the quantitative nondestructive determination of material properties such as elastic constants and the bond quality of thin films because it can be utilized to generate surface acoustic waves (SAW) at very high frequencies to obtain quantitative information. The generated high frequency SAWs that propagate in a thin surface layer make them ideally suitable for thin film characterization. Acoustic Microscopy has been used in various fields including biology, material science, and quantitative NDE. It can be broadly divided into two classes, the scanning mode (SAM) for imaging [6] and the quantitative mode (QAM) for material property measurements. This paper is concerned with QAM, which has the capability to measure acoustic properties with high accuracy, especially with line-focus acoustic microscopy (LFAM). This technique, which has been discussed in great detail in Ref. [7], provides for directional generation of SAWs as well as more accurate analysis of the acoustic properties, as compared with point-focus acoustic microscopy. In earlier work, LFAM has been used to determine properties, such as the thickness, the elastic constants and sometimes-even the densities of bulk materials and thin films. Generally, only one thin film layer on a substrate has been considered and it has been assumed that the interface between the thin film and the substrate is perfect. In certain frequency ranges it has been difficult to obtain a useful material acoustic signature for thick or very stiff thin films. Very little work has been done for the case of multilayered thin films. Quantitative acoustic microscopy for the evaluation of thin film interface conditions has also been relatively unexplored. This paper is concerned with these topics. 2.

Line-Focus Acoustic Microscopy

An acoustic microscope consists of four main components: the acoustic probe, the pulse-mode measurement system for transmitting and receiving signals, the mechanical system for alignment and movement of the specimen and a computer for controlling the system and processing the recorded wave forms.

ACOUSTIC MICROSCOPY FOR THIN-FILMS

369

The water-coupled line-focus acoustic lens allows the measurement of the SAW velocity in specified directions Hence line-focus acoustic microscopy (LFAM) can be used to determine the elastic constants of anisotropic materials. The usual technique measures the so-called V(z) curve, which is a recording of the voltage output of the transducer when the distance between the lens and the specimen is decreased. Figure 1 shows the configuration. For the focal line is on the upper surface of the specimen. As z is decreased, oscillations appear in the V(z) curve due to interference between the rays specularly reflected by the specimen and the rays associated with propagation of a leaky SAW on the specimen. The phase velocity of a leaky surface wave, can be determined from the period, of the periodic oscillations, using a well-known relationship based on ray considerations:

where

is the wave velocity in the coupling fluid, and

Equation (1) implies that the measured

is the operating frequency.

must satisfy

The period can be accurately extracted from the measured V(z) curve by a spectrum analysis, as discussed in detail in Ref. [8]. Therefore, by measuring V(z) curves for a specimen at different frequencies or at the same frequency but different


370

wave propagation directions, the dispersion curves or the directional variations of the leaky surface wave velocity can be measured. By numerically simulating the V(z) measurement procedure, a V(z) measurement model can be developed to calculate theoretical V(z) curves for a layered specimen of arbitrary configuration for use with a line-focus acoustic microscope. In this paper, we use the approach proposed by Achenbach et. al. for synthesizing V(z) curves [7]. In that approach, the output of the line-focus acoustic probe is expressed as a Fourier integral over the product of the characteristic functions of the acoustic lens, and and the reflectance function of the fluid-loaded specimen

where

defines the wave vector, and

As shown by Eq. (3), the theoretical measurement model for the V(z) curve contains the reflectance function of the film/substrate system as a principal component. Wellknown theoretical methods are available to determine reflectance functions. See, for example, Ref. [9]. By using the theoretical reflectance function model for parametrical studies, the phase velocities of the possible SAW modes and their corresponding mode reflection coefficients can be predicted for an arbitrary film/substrate system. Therefore the reflectance function can be used as a predictive tool to estimate in which frequency range a wave mode may be picked up by acoustic microscopy. By applying the V(z) measurement model, the phase velocities can be calculated for a film/substrate system at different frequencies. For trial values of the unknown elastic constants of the thin film, these theoretically calculated phase velocities have been used in an iterative manner together with phase velocities obtained from V(z) measurements to inversely determine the elastic constants of thin films by minimization of the deviation between theoretical and experimental results at N points. The deviation is defined as

where 3.

and

are the measured and calculated values of the leaky SAW velocities.

Calibration for Frequency Dependence

The lens of a line-focus acoustic microscope optimally operates at a fixed center frequency. The lens used in this work is designed to be used at 225 MHz. For some applications, for example for isotropic materials, measurements at a number of


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frequencies are required. The measurements at frequencies other than 225 MHz have to be corrected by the use of a calibration factor. Thus, it is necessary to calibrate the LFAM system at each used operating frequency to eliminate system errors. Using a standard specimen for calibration provides a reasonable approach. In principle, any material could be used as the standard specimen if all the elastic constants necessary for the theoretical calculation of the leaky SAW characteristics can be determined with high accuracy. However, a specimen with the following properties is preferred: (1) non-piezoelectric, homogeneous, with a small number of independent elastic constants (isotropic or cubic); (2) small attenuation at higher frequencies, such as 150~300 MHz, and (3) having a suitable leaky SAW velocity quite close to those of the materials to be investigated. A silicon single crystal wafer was taken as the standard specimen, because its properties are well known, and its direction-dependent leaky SAW velocities cover a suitable velocity range. For a silicon single crystal wafer, the directional variations of the leaky SAW velocities, displayed as curves, were measured at different frequencies in the range from 135 MHz to 255 MHz using the nominal 225 MHz LFAM system. In Fig. 2 the measured results are plotted in dotted lines with symbols and the theoretical ones in solid lines. The theoretical curves are calculated by using the known material properties of silicon. The difference between the measured and calculated curves at each frequency indicates the measurement accuracy at that frequency. The results show that the measured curve at each frequency generally shifts up or down a fixed value relative to the theoretical curve, this shift, here defined as a correction factor can be used for calibration at this frequency. By performing


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the same procedure for all the frequencies to be used, a set of correction factors corresponding to a set of operating frequencies can be obtained. For later measurements on other specimens, these correction factors are used to compensate the measured velocities to obtain more accurate velocities namely, For example, the measured curve at 225 MHz is very close to the theoretical one, thus there is no need to compensate at this frequency, The measured curve at 210 MHz shifts down 11(m / s) on average, relative to the theoretical one, thus Next a uniform aluminum specimen with a polished surface was used to check the obtained correction factors. The leaky SAW velocities at different operating frequencies for the aluminum body, which should be constant, were measured and calibrated using the obtained correction factors. After calibration, the error of the velocity measurement by LFAM is estimated to be less than ±0.5%.

4.

Modeling

Figure 3 shows the configuration of a layered medium with N layers on a substrate. A time-harmonic plane wave is incident from a coupling fluid, at arbitrary incident The response of the layered medium to the incident time-harmonic plane wave can be expressed in terms of the reflectance function. This reflectance function carries information on the layered medium, including its mechanical properties, the thickness of each layer, and the interface conditions. A general theoretical model has been developed which quantitatively relates the reflectance function to these characteristic properties of the layered specimen. See Ref. [9]. It has been shown that the reflectance function is of the general form


373

where

Here and are the density and the velocity of the coupling fluid, and and the frequency and the angular frequency of the incident wave with incident angle and is the tangential component of the wave vector, where

are

and are the elements of the global transfer matrix [T]. For the numerical calculations, the delta-matrix reformation is introduced to avoid the “precision problem,” see Ref. [10]. The reflectance function given by Eq. (4) can be used for wave mode identification. The vanishing of its denominator, i.e.,

is exactly the characteristic equation of the layered medium. Solutions for the velocity and the amplitude of the possible wave modes can be obtained by numerically solving this equation. Setting takes out the effects of the fluid loading, and thus the equation can be used to identify the free wave modes. For a given frequency, the reflectance function is a function of the incident angle, The reflectance function shows distinct behavior when the incident angle equals the critical angle associated with a possible wave mode. By virtue of the relations between the incident angle and the phase velocity c shown in Eq. (8), the reflectance function can be used to obtain the phase velocities of the wave modes for the layered system at a fixed frequency. The reflectance function undergoes a phase shift and its real part undergoes a jump decreasing its magnitude as the incident angle passes a critical angle which corresponds to a wave mode. Using this criterion to determine the critical angles, the phase velocity of a wave mode can be calculated as

As discussed by Guo et al. [11], the magnitude of the reflectance function at the critical angle indicates how strongly the wave mode will be reflected. For convenience that magnitude will be referred to as the mode reflection coefficient. A strongly reflected SAW mode with a large mode reflection coefficient can be picked up easily by acoustic microscopy. The results show that the theoretical mode reflection coefficient of a true surface wave mode is unity and that of a pseudo-SAW mode that radiates energy into the substrate is somewhere between 0 and 1.

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5.


Selection of Frequency Range

Theoretical calculations and experimental observations show that the LFAM can not pick up any SAW mode with a mode reflection coefficient smaller than around 0.15. A strongly reflected SAW mode can be easily picked up by the LFAM for generation of a V(z) curve, and its velocity can be extracted with high accuracy. The V(z) measurement will be less precise for a weakly reflected surface wave mode. There is a significant decrease in the number and magnitude of the oscillations of the V(z) curve for a weakly reflected SAW mode. Comparisons of theoretical results for the phase velocity for a slow-on-fast system calculated both directly from the reflectance function and from the V(z) curve via Eq. (1), presented by Guo et al. [11], illustrate the significance of an appropriate selection of the frequency range for modes with a sufficiently large value of the reflectance function.


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For a fast-on-slow system of an isotropic (TiN) film on an isotropic (M2-HSS) substrate the material constants have been determined from the experimental V(z) results by the iterative optimization method mentioned in Section 2. After determination of the material constants the corresponding phase velocities were recalculated to verify the procedure. Results are shown in Fig. 4. The best results are obtained in the frequency range defined by i.e. before the cut-off of the true SAW mode. In that range the magnitude of the reflectance function is unity. Based on the excellent agreement displayed in Fig. 4 it may be expected that the material constants have been obtained with good accuracy. A detailed discussion of the accuracy can be found in the paper by Guo et. al. [11]. 6.

Two-Layer Thin Film

A two-layer thin film, was deposited on an M2-HSS substrate. The earlier discussed procedure for measurement and data inversion was applied to determine the material constants, namely, Young’s moduli and Poisson’s ratios of the Titanium and Aluminum coatings layers. Known material properties of the substrate and known values of the densities, and the thicknesses, d, of the two layers were used. The determined elastic constants are listed in Table 1.

After the material constants had been determined, the corresponding leaky SAW phase velocities were recalculated to verify the procedure. The recalculated phase velocities are shown in Fig. 5 as a solid line, together with the measured values in empty circles for various frequencies.

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In order to investigate the uniqueness and accuracy of the data inversion method to determine the four unknowns, a convergence and sensitivity study was performed. The contour plots in Fig. 6(a) and Fig. 6(b) show the deviations due to variations of and respectively. The deviation is defined in terms of N-point measured and calculated phase velocities as in Eq. (5), here N=11. The elastic constants and were varied ±20% around the determined values. The individual cross-sectional variations of the deviation with respect to and shown in Fig. 6(c) and Fig. 6(d), illustrate the sensitivity of the deviation to variations of those constants. The more sensitive the deviation, the more accurate the constants can be determined. The results show that the expected accuracy of is better than that of, and the expected accuracy of is better than that of. This is so because the Aluminum layer has greater thickness and is more effective than the Titanium layer. The expected accuracy of Young’s modulus is better than that of Poisson’s ratio.


3.

377

Anisotropic Thin Film/Substrate Systems

For an anisotropic film/substrate system composed of anisotropic thin-film layers on an anisotropic substrate, the determination of material constants can be based on the directional variations of the leaky SAW velocities. For each specimen, the directional variation of the leaky SAW velocities were measured by operating the LFAM system at 225 MHz and measuring V(z) curves for different wave propagation directions on the specimen. The phase velocities were extracted by spectral analysis of the corresponding V(z) curves to determine and by subsequent use of Eq. (1). The measured phase velocities were then used together with theoretical predictions to inversely determine the elastic constants of the thin film, based on Eq. (5). Four specimens with different cubic-crystalline thin films deposited on the (100) plane of an MgO substrate were tested. These specimens include two Tungsten (W) and Molybdenum (Mo) single-layer thin films and two double-layer thin films with a Niobium Nitride (NbN) layer on a Tungsten layer or a Molybdenum layer. The material properties of the MgO substrate were determined separately on a polished MgO specimen. Known values of the densities and the thickness of the thin films were used. The values of the densities on MgO: 3,598; Tungsten (W): 18,700; Molybdenum (Mo): 10,200; and Niobium Nitride (NbN): 8,430, all in

The determined elastic constants and are listed in Table 2 together with the global errors for the specimens. The global error is the value of the deviation function when a best fit has been found for each specimen. The comparisons between the determined values and literature values (from Ref. [12]) are listed in parentheses. After the material constants had been determined, the corresponding directional variation of the leaky SAW velocity was recalculated for each specimen to verify the

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data inversion procedure. For the different specimens the recalculated phase velocities versus the wave propagation direction are shown in Figures 7(a)-(f) in solid circles with the measured values in empty circles. As shown in Fig. 7, the presence of a single-layer or a multilayered thin film generally causes the variation with direction of the leaky SAW velocity to shift and to change the shape of the curve. The determined elastic constants are consistent for different specimens and they agree with values in the literature within less than ±5%. Different deposition conditions are thought to lead to different values of the elastic constants of the coatings layers.

4.

Effect of Imperfect Interfaces

The paper is completed with some comments on the effects of imperfect interfaces. In the theoretical approach an imperfect interface is replaced by a layer of extensional and shear springs. Such a layer of springs can easily be included in the method of calculation of the reflectance function that was earlier presented in Section 4. In Ref. [13] it has been attempted to correlate calculated values of the spring constants with


379

results obtained by a destructive scratch test for strength determination. Specimens with different and interface conditions were prepared by using different lengths of time for etching of the substrate surface. For film deposition by the reactive sputtering technique, etching of the substrate for different time periods has an observable effect on the stiffness constants and of the interface, as shown in Fig. 8. The determined constants and show that for 10 minutes etching (specimen 1) they are about 2.7 times the values for the case of 20 minutes etching (specimen 2). The scratch test results indicate that the adhesion of specimen 1 is much better than that for specimen 2. These limited experimental results show better adhesion for a higher stiffness interface.

However, it is important to note that there is no theory which shows that nondestructive techniques can directly measure interface strength. At the present time, only destructive techniques, such as the scratch test and the pullout test, can provide us with a measure of the interface strength. In Ref. [13] the LFAM at high frequency was used to measure the interface stiffness. For most of the film/substrate systems studied in Ref. [13] the interface stiffness parameters seem to have a strong correlation with the destructively measured interface strength, such as the critical normal force of a scratch test. Therefore, measured interface stiffness parameters may provide an indication of the bond quality between a film and a substrate.


380

Acknowledgement

The results reported here were obtained in the course of research sponsored by the Office of Naval Research under Grant N00014-89-J1362. References 1.

2. 3. 4. 5. 6.

7. 8. 9.

10. 11. 12.

13.

Wolfe, D. E., Movchan, M B. and Singh, J.: Architecture of functional graded ceramic/metallic coatings by electron beam-physical vapor deposion, in Advances in Coatings Technologies for Surface Engineering, C. R. Clayton, J. K. Hirvonen and A. R. Srivatsa, Eds., TMS, New York (1997), 93-128. Toth, L. E.: Transition Metal Carbides and Nitrides, 7, Academic Press, New York and London, 1971. Prengel, H. G.: Coating carbide cutting tools, Manufacturing Engineering, 117 (1996), 82. Ianno, N. J., Dillon, R. O., Ali, A. and Ahmad, A.: Deposition of diamond-like carbon on a titanium biomedical alloy, (in press). Valli, J.: A review of adhesion test methods for thin hard coatings, J. Vac. Sci. Technlogy., A4(6) (1986), 3007-3014. Briggs, A.: An Introduction to Scanning Acoustic Microscopy, Oxford U. Press, New York, 1985. Achenbach, J. D., Kim, J. O. and Lee, Y.-C: Measuring thin-film elastic constants by line-focus acoustic microscopy, in Advances in Acoustic Microscopy I, A. Briggs, Ed., Plenum Press, New York (1995), 153-208. Kushibiki, J. and Chubachi, N.: Material characterization by line-focus beam acoustic microscopy, IEEE Trans. Sonics Ultrason, SU-32, (1985) 189-212. Nayfeh, A. H. and Chimenti, D. E.: Reflection of finite acoustic beams from loaded and stiffened half-spaces, J. Acoust. Soc. Am., 75, (1984) 1360-1368. Pestel, E. G. and Leckie, F.A.: Matrix methods in elasto-mechanics, McGraw-Hill Book Company, New York, 1963. Guo, Z., Achenbach, J. D., Madan, A, Martin, K. and Graham, M. E.: Integration of modeling and acoustic microscopy measurements for thin films, J. Acoust. Soc. Am., 107(5), (2000) 24622471. Anderson, O. L.: Determination and some uses of isotropic elastic constants of polycrystalline aggregates using single-crystal data, in, Physical Acoustics, W. P. Mason and R. N. Thurston, Eds., Academic Press, New York and London (1965), 43-95. Guo, Z., Achenbach, J. D., Madan, A., Martin, K. and Graham, M.E.: Modeling and acoustic micrscopy measurements for evaluation of the adhesion between a film and substrate, Thin Solid Films (2001), in press.

RECENT ADVANCES IN ACOUSTOGRAPHY-BASED NDE

J. S. SANDHU and H. WANG Santec Systems, Inc. 716 South Milwaukee Avenue Wheeling, IL 60090 Abstract Acoustography is the ultrasonic analog of radiography and photography in that a 2D area detector is used to form full-field ultrasonic images. This unique detector, called the acousto-optic (AO) sensor, is capable of directly converting ultrasound into a visual image; much like a fluorescent screen is able to convert x-rays into visual image. It offers an exceptionally high pixel resolution since ultrasound field is detected by a continuous layer of liquid crystal material with molecules that are only 20 Angstroms in size. The AO sensor can be fabricated to have a large sensing area that allows image formation either through simple shadow casting (analogous to x-ray image formation) or with acoustic lenses (analogous to photographic or video camera). In this paper, we report on recent progress that is making acoustography a viable ultrasonic testing method both for through-transmission and reflection modes. 1.

Introduction

Acoustography [1] is a full-field ultrasonic imaging process where an acousto-optic area detector (AO Sensor) is employed to convert the ultrasound into a visual image in near real time. The physical principle by which the AO sensor converts ultrasound into visual images is based upon the birefringent properties of a proprietary liquid crystal layer contained in the AO sensor [1]. The layer exhibits no birefringence in the absence of ultrasound and exhibits a uniform dark field under cross-polarizer viewing [1]. Upon exposure to ultrasound, the layer becomes birefringent, showing a brightness change (i.e. optical density change) that is related to the ultrasonic intensity by the AO sensor's acousto-optic transfer curve [1]. To date, acoustography has been used analogously to x-ray imaging where images were formed in through-transmission via simple shadow casting [2-5]. However, recently progress has been made toward developing photography analogy where images are formed using acoustic lenses [6]. Regardless of how the images are formed, geometrical and lateral resolution in acoustography, like other ultrasound techniques, is diffraction-limited and is governed by the ultrasound wavelength in the test specimen [1]. The pixel resolution, on the other hand, is not usually an issue because of the high pixel density of the AO detector. For example, the ultrasound image is converted into a visual image by countless number of minute (~ 2 nanometer) liquid crystal molecules, and correlation between molecules persists for only a few micrometers [1]. Therefore, 381 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 381–388. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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the pixel size in the AO detector can be expected to be on the order of a few micrometers. The contrast resolution in acoustography depends on two factors: 1) acoustic contrast between anomaly and the normal component structure; 2) acoustooptic transfer curve of the AO detector which expresses the relationship between ultrasonic intensity (exposure level) and the corresponding optical density (brightness level). In this paper, we will report on recent advances in acoustography including improvement in AO sensor acoustic detection sensitivity, correlation between C-scan and acoustography ultrasonic data, and development of reflection-mode acoustography. 2.

Acoustography-based Ultrasonic Testing

Acoustography has been used for ultrasonic testing of materials mostly in throughtransmission mode, where ultrasound is passed through the test component. As it propagates, the ultrasound wave is absorbed, reflected, refracted and scattered by material structure and any anomalies therein. Upon exiting the test specimen, the ultrasound wave creates a projection image of the material structure and anomalies, which is received by a wide area AO sensor that converts the ultrasound image into a corresponding visual image instantly (Figure 1). Once the image appears on the AO sensor, a video camera can be used in combination with a frame grabber to acquire and digitize the image for computer storage and processing.

3.

Recent Advances

3.1 ACOUSTIC DETECTION SENSITIVITY

Although acoustography was introduced more than a decade ago [2,3], its applicability for nondestructive testing has remained limited due to numerous technical drawbacks. One of the major drawbacks has been the low acoustic detection sensitivity (i.e. high


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threshold) of the AO sensor. This shortcoming has recently been overcome through improved understanding of the acoustic coupling mechanism in LC materials, as discussed below. A theoretical model has been developed [6] to show that the torque (Figure 2), responsible for the reorientation of LC molecules in a LC layer exposed to ultrasound is given by:

where

d=LC layer thickness =Acoustic attenuation anisotropy c= Sound velocity I=Acoustic Intensity =Incident angle of the ultrasound beam

The molecular reorientation, in the LC layer (Figure 2) due to the acoustic torque, produces an optical output (acoustic-to-optic conversion) that can be described for normal-incident viewing by [7]:

where is the angle between the polarization axis of the incident light beam and projection of the optic axis of the liquid crystal and known as the retardation, is the phase difference between the ordinary ray and the extraordinary ray and is given by:

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where and are refractive indices of the extraordinary and ordinary rays respectively, is the light wavelength, and is the LC molecule tilt angle. Using the above model as a guide, new LC materials with physical properties that enhance the acoustic detection sensitivity have been developed. Figure 3 shows the performance of new materials compared to the prior LC materials. These proprietary LC materials have improved acoustic detection sensitivity of AO sensors from prior

3.2 CORRELATION BETWEEN ACOUSTOGRAPHY AND C-SCAN To test validity of the acoustography approach, a side-by-side acoustography and Cscan study was conducted of a graphite/epoxy composite panel standard. The panel was 10" x 12" in size with a thickness of approximately 1/8 in. It contained intentionally introduced defects of various sizes, ranging from 3/8 x 3/8 sq. in to 1/8 × 1/8 sq. in. The map of the general arrangement of defects is shown in Figure 4.

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Figure 5 shows the acoustographic image and ultrasonic C-scan of the graphite/epoxy composite panel standard. Only four rows of the intentionally placed defects are shown because of the limited tank size of the acoustography system (see Figure 4). As seen in Figure 5, the ability of acoustography to detect all defect sizes is comparable to the conventional C-scan method. However, definition of the defects seemed better in the acoustography method. The fibrous nature of the composite was also more evident in the acoustography method. The superior definition of defects and the fibrous nature of the specimen may be attributed to the superior pixel resolution of the AO detector used in acoustography. We should note that the C-scan was generated with a 3.5 MHz focused transducer, where the 0.02” (0.5mm) indexing defined the lateral resolution. In comparison, lateral resolution in acoustography depends on the pixel molecules in the AO detector are on the order of 20 angstroms and the correlation between molecules persists only for a few micrometers. Therefore, the pixel size on the AO detector is on the order of a few micrometers. However, the pixel resolution of the video camera and graphic monitor is much lower compared to that offered by the AO detector. Accounting for the lower resolution of the video camera and monitor and the magnification factor, we determined the actual resolution of the acoustographic image to be around 0.23mm. We should note that the field of view of the acoustography system used was 5cm x 5cm, therefore, the part was moved to image 5cm x 5cm contiguous areas. The collaging is apparent due to the inaccuracy of the manual part manipulation. This can be overcome by employing a mechanical manipulator that allows repeatable position of the test areas. Moreover, the use of large area AO detectors (e.g. 6”× 6” or larger), currently under development, will reduce the number of images required to inspect the entire part as well as produce dramatic improvements in inspection speed (e.g. 6”× 6” area in only seconds).

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3.3 REFLECTION MODE ACOUSTOGRAPHY A major limitation of acoustography has been its through-transmission mode of operation, which does not permit the method to be applied for NDT in the field. However, this limitation has recently been overcome. Several reflection mode concepts have now been explored. The first concept to be explored is shown in Figure 6.

The sound source and the AO sensor are placed on the same side of the test part. A water-filled prism constructed from acoustically transparent materials was used to interface the sound and the AO sensor assembly to the test part. Sound was received by the test part at an incident angle, where the front face (water/part interface) reflection was much lower than that from the back face (part/air interface) reflection. Since the AO sensor is an intensity detector, and ultrasound speed is much greater than the AO sensor response time, a summation image of all the transmitted, reflected and scattered rays is displayed on the AO sensor. The transmission, reflection and scattering of ultrasound waves is affected by the presence of flaws in the test component. Therefore, an image of the defect can be made. th Figure 7 shows the plastic sheet specimen that was inspected using reflection-mode acoustography. The specimen was a l/8th in. thick plastic sheet. Three plastic pieces of different thickness (1/16th in., 1/8 in. and ¼” ) were adhesively bonded on the back surface of the plastic sheet specimen. In addition, a blob of silicone adhesive was placed on the back surface approximately in the middle of the three plastic pieces. The bonded pieces on the back surface and silicone blob are evident in the acoustographic


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image. However, the image quality is not as high as the through-transmission acoustography-based UT because of three reasons. First, the image geometrically distorted since the AO sensor is at an angle with respect to the test specimen. Second, the standoff distance between the AO sensor and the specimen is large compared with the through-transmission case. Third, the sound source was monochromatic (single frequency) causing interference effects.

To improve the image quality of reflection-mode acoustography, a new approach was explored [8]. In this approach, the AO sensor is placed directly over the test component with an acoustic coupling layer (Figure 8). Ultrasound passes through the AO sensor and on into the test component. Regions containing defects absorb, reflect and scatter ultrasound differently than the normal material, resulting in a differential attenuation that creates the defect image on the AO sensor. The acoustographic image of the plastic sheet specimen produced using this reflection mode approach is shown in Figure 9. The improvement in image quality was found to be significant. Note the sharpness with which the bonded plastic pieces and silicone blob are imaged compared to the first reflection mode approach.


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4. Conclusions The recent advancements in acoustic detection sensitivity of the AO sensor used in acoustography are reported. The correlation of data between acoustography and conventional C-scan is excellent, lending support to the efficacy of acoustography as an ultrasonic NDE method. Finally, the feasibility of acoustography to be implemented in reflection mode has been demonstrated, which could lead to significant applicability of the method for NDE in the field where access is limited only to one side. 5.

Acknowledgements

This material is based upon work supported in part by the National Science Foundation (Grant # DMI-9407745) and Department of Defense (Contract # DAAD17-00-C-0014). The authors would also like to acknowledge Ms. Noreen Cmar-Mascis, U.S. Army Research Laboratory Vehicle Structures Directorate, and Dr. Pat Johnston, NASA Langley Research Center for their assistance in performing ultrasonic C-scans of the materials tested for this effort. 6. 1.

2. 3. 4. 5. 6. 7. 8.

References J. S. Sandhu, "Acoustography," Special nondestructive testing methods, Nondestructive Testing Handbook, 2nd. Edition, Editors: P.O Moore and P. Mclntire, vol.9, pp. 278-284, American Society for Nondestructive testing, Columbus OH, 1995. J. S. Sandhu, "Acoustography: A new imaging technique and its applications to nondestructive testing," Materials Evaluation, vol. 46, No. 5, pp. 608-613, 1988. I. M. Daniel, S.C. Wooh, J.S. Sandhu and W.A. Hamidzada, “Acoustographic Nondestructive Evaluation of Composites,” Review of Progress in QNDE, Eds. D.O. Thompson and D.E. Chimenti, Plenum Press, Volume 7A, pp. 325-332, 1988. J. S. Sandhu, H. Wang and W.J. Popek, "Acoustography for rapid ultrasonic inspection of composites," Proceedings SPIE conference on NDE, vol. 2944, pp. 117-124, AZ, Dec. 1996. J. S. Sandhu, H. Wang and W.J. Popek, "Ultrasonic Inspection of tight-radii in composites using acoustography," Proceedings SPIE conference on NDE, vol. 3396, pp. 169-179, San Antonio, TX, Mar. 1998. J. S. Sandhu, H. Wang and W.J. Popek, "Liquid crystal based acoustic imaging,” in Liquid Crystal Materials, Devices, and Flat Panel Displays, Ranganathan Shashidhar, Bruce Gnade, Editors Proceedings of SPIE Vol. 3955, pp. 94-108, 2000. M. Born and E. Wolf, “Principles of Optics,” Pergamon Press, Chapter XIV pp665, 1987. J.S. Sandhu and H. Wang, “Reflection Mode Device” worldwide patents filed and in process.

EXPERIMENTAL LIMITATIONS TO GUIDED WAVE GENERATION IN ELASTIC MATERIALS D. A. SOTIROPOULOS and E. BABATSOULI College of Science, Math & Technology The University of Texas at Brownsville & TSC Brownsville, Texas 78520, USA

Abstract

In generating small amplitude guided waves in elastic materials for diverse experimental investigations, one is faced with several limitations. This paper examines the limitations arising from the inherent nature of materials not to withstand, in some cases, the existence of guided waves either in certain finite frequency ranges or in the whole frequency spectrum. The material constants as well as the stress conditions in the materials play a key role in allowing the experimental generation of guided waves. The model examined in this paper incorporates a small layer either imbedded in a host material or a small surface layer overlying a host material, both materials in general being pre-stressed. The analysis defines limitation regimes where experimental generation of guided waves would not be possible. These regimes are given explicitly in terms of material constants and stress conditions. 1. Introduction

The generation of guided elastic waves in layered structures is of practical significance both for experimental investigations and for in-situ diagnostics. The areas of applications include the ultrasonic non-destructive evaluation and characterization of materials, their interfaces and stress conditions, the design of vibration isolators, and the processing of electronic signals. Of interest, here, is the generation of guided waves both in anisotropic layered materials and in stressed non-linear elastic layered materials. Of particular interest is the existence of frequency regions in which guided elastic waves cannot be generated as an inherent characteristic of the structure and its elastic properties. Previous studies, see for example [1-3], have demonstrated this. However, from the practical point of view a simple, if possible, rule is needed as an experimental guide to wave generation limitations. A simple rule that defines the structural elastic properties which permit guided wave generation in layered structures. In the present paper, simple formulae are presented which define structural properties regions in which guided waves can be generated. This is done for structures that consist of a small layer either surrounded by a host material or as a surface layer under-laid by a host material. Several cases are considered depending on whether the materials are compressible or incompressible, isotropic or anisotropic, and stressed or unstressed before the waves are generated. The limitations of guided wave generation are obtained by considering the dispersive wave characteristics. The simplicity of the 389 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 389–396. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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limitations, as far as their dependence on the structural properties is concerned, depends on the state of the underlying stress as well as on the direction of propagation of the generated waves with respect to material axes of symmetry and geometry of the structure. In the present paper the underlying stress is homogeneous in both materials with common principal strain axes, one axis being normal to the planar interfaces and guided wave propagation along a principal axis. 2. Dispersive Characteristics

The structure considered in the present paper consists of a small layer either imbedded in or overlying a host material. Guided waves are considered propagating along the planar surfaces separating the layer from the host material. Therefore, the host material can be considered infinite without loss of generality. Since the layer considered is small, it suffices to take kh small, where k is the wave number and h is the layer thickness. The generic structure consists of a pre-stressed layer of non-linear elastic material imbedded in an infinite pre-stressed different non-linear elastic material. The underlying stress is homogeneous in the two materials with the finite strains having common principal axes, one axis being normal to the planar interfaces. For the case that both materials are compressible, the dispersive characteristics are described by the dispersion equation which can be derived (the details of the derivation are outside the scope of the present paper) as

where

in which the elastic constants are given by

GUIDED WAVE GENERATION

where

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are the principal Cauchy stresses and are given as

with being the strain energy density function after deformation. The elastic constants satisfy the conditions:

In equation (2), the star refers to the layer properties which are defined analogously to the above. The bars are defined as

Also, in equations (1, 2) the following definitions are pertinent:

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3. Limiting Conditions Most cases of practical interest (in this is also included the case of no pre-stressing) fall under the category and Then, for small kh, can be written as where is obtained from equation (1) as

By definition, must be positive in order for the guided wave to decay away from the interface. Therefore, guided waves cannot be generated (or exist) when

or, equivalently, when

Necessary conditions to satisfy (16) or (17) are

The above simple material and pre-stressed parameter conditions define the limitations to experimental investigations as far as the generation of guided waves is concerned. For equibiaxial in-plane (along and perpendicular to the layering) deformations as well as for materials that are not pre-stressed, so that R > 1 or r > 2 guarantee the feasibility of generating guided waves. When there is no pre-stressing the material constants become

and J = 1. Then, we conclude that materials with being the shear moduli of the host material and layer respectively), or allow the propagation of guided waves. The limiting condition (16) for the non-existence of guided elastic waves is shown graphically in Figure 1 that follows. In the figure, for simplicity, is represented by whose value affects the existence of condition (16) for small r depending on whether it is smaller, equal or larger than one.


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Naturally, the value of (in the figure) depends on the type of material and its prestress condition. For example, for Varga or Blatz-Ko materials where and are the principal stretches of the interlayer along and perpendicular to the interface respectively. 4.

Incompressible Materials

The limiting condition in the case of incompressible materials for the non-existence of guided waves can be obtained from (16) by setting and The resulting condition is:

where and The constants are related to the strain energy density function of the material, W, and the principal stretches by

The limiting condition (20) for the non-existence of guided elastic waves is shown graphically in figure 2 that follows. In the figure, for simplicity, is represented by and by The value of (in the figure) affects the existence of condition (20) for small r depending on whether it is smaller, equal or larger than one. This value (of is given by equations (21) as independent of the type of incompressible material. 5.

Incompressible Layer in a Compressible Material

For the case that the layer is incompressible and the host material is compressible, the limiting condition for the non-existence of guided elastic waves becomes

where From the generic cases described in sections 3, 4, and 5 follow results for a specifically stressed surface layer overlying the host material as well as for orthotropic materials. These will be discussed following Figure 2.


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A Thin Surface Layer Over a Host Material

The results obtained above as far as material properties not withstanding guided wave generation are applicable also to the case that a surface layer is stressed perpendicular to the interface by a Cauchy principal stress equal to / J. In other words, effectively, this stress replaces the host material above the layer in the cases considered in sections 3, 4 and 5. The limiting materials region is, then, given by (16), (20) or (23) depending on whether the surface layer and the host material are compressible or incompressible. 7.

Anisotropic Layer and Host Material

When both the layer and the host material are not stressed, the results obtained above can be used to obtain limiting conditions for orthotropic materials. In other words, the stressed isotropic materials possess a stress-induced anisotropy, consequently, including results for stress-free orthotropic materials. The limiting condition for orthotropic materials with the direction of guided wave propagation along a material axis of symmetry is obtained from (16) as

where The constants are the shear moduli for the host material and layer respectively in the plane of guided wave propagation, i.e. along the interface and perpendicular to it. 8.

Composite Layer and Host Materials

The layer, considered in this paper, is small as compared to the guided wave length. For this reason, the experimental limitations derived above for generating guided elastic waves are applicable to the layer and host material being composite materials as well. In this case, the composite materials can be effectively represented by homogeneous materials. For more details on this, the reader is referred to [4]. 9.

References

1.

Ogden, R. W. and Sotiropoulos, D. A.(1995) On interfacial waves in pre-stressed layered incompressible elastic solids, Proceedings of the Royal Society of London A, Vol. 450, pp. 319341 Sotiropoulos, D. A. and Sifniotopoulos, C. G. (1995) Interfacial waves in pre-stressed incompressible elastic interlayers, J. Mech. Phys. Solids 43, 365-387 Sotiropoulos, D. A. and Sifniotopoulos, C. G. (2001) The effect of stress on interfacial waves in elastic compressible interlayers, in Solid Mechanics and Its Applications Vol. 92: Mechanical Waves for Composite Structure Characterization, D. A. Sotiropoulos (ed.) Kluwer Academic Publishers, 169-185 Daniel, I. M. and Ishai, O. (1994) Engineering Mechanics of Composite Materials, Oxford University Press

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NONDESTRUCTIVE TESTING USING SHEAROGRAPHY MICHAEL Y. Y. HUNG Department of Building and Construction City University of Hong Kong Tat Chee Avenue, Kowloon, Hong Kong The author is presently on leave from Oakland University Rochester, Ml 48309. Abstract This article reviews shearography and its applications in nondestructive testing. Shearography is an interferometric technique for full-field and non-contacting measurement of surface deformation (displacement or displacement derivatives). It was invented to overcome some limitations of holographic interferometry by eliminating the reference beam, resulting in having much higher tolerance to environmental disturbances. Consequently, shearography can be practiced in a typical industrial setting, and it has already received considerable industrial acceptance, in particular, for nondestructive testing. One major difference of shearography from other NDT techniques is the mechanics of revealing flaws. Shearography reveals defects in an object by identifying defect-induced deformation anomalies which are more relevant to structural weakness. Other applications of shearography include strain measurement, material characterization, residual stress evaluation, leak detection, vibration studies and 3-D shape measurement.

1. Introduction Shearography is an optical method for measuring surface deformation. Unlike traditional measurement techniques, shearography does not require the laborious task of mounting a large number of strain gages or transducers. It is a noncontacting method that yields full-field information about surface displacement or displacement derivatives. Shearography was developed to address several limitations of holography. Its significant advantages include (1) not requiring a reference light-beam, thus leading to simple optical setups, reduced coherence length requirement of the laser, and lax vibration isolation; and (2) direct measurement of surface strains (first-order derivatives of surface displacements). These distinct advantages have rendered shearography as a practical measurement tool and it has already gained wide industrial acceptance in nondestructive testing. For instance, the rubber industry routinely uses shearography for evaluating tires, and the aerospace industry has adopted it for nondestructive testing of aircraft structures, in particular, composite structures. Other applications of shearography include: measurement of strains, material properties, residual stresses, 3-D shapes, vibrations, as well as leakage detection. 397 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 397–408. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Three versions of shearography are in existence which are based on different recording media : photographic[1]; thermoplastic[2]; and digital[3]. In this paper, only the digital version is presented.

2. Principles of Digital Shearography

Description of shearography.

A schematic diagram of digital shearography is shown in Fig. 1. The object to be evaluated is illuminated with laser light radiating from a point-source, and it is imaged by an image-shearing camera connected to a microcomputer for recording and processing. The camera comprises a CCD image sensor, a lens and an image shearing device. The image shearing device consists of a double-refractive prism and a polarizer. The function of the image shearing device is to produce a pair of laterally displaced (sheared) images, and hence the technique is named as shearography.

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The principle of image-shearing device using a doubly-refractive prism is illustrated in Figure 2. A light beam passing through a doubly refractive prism such as a Wollaston prism is split into two angularly separated beams. Conversely, as shown in the figure, two non-parallel beams of light scattered from two different object points are combined by the prism and become nearly collinear. Since the spatial frequency of the interference fringe pattern formed by two beams is proportional to the sine of the half angle between the interfering beams, two nearly collinear beams produce a very low frequency interference pattern that is comfortably resolved by a video image sensor such as CCD. The two sheared wavefronts transmitted by the two axes of the image-shearing device, however, are orthogonally polarized, hence they will not interfere with each other. To enable interference, a polarizer with its polarization axis oriented at an azimuth of 45° is required. As an object surface is generally optically rough, interference of the two sheared wavefronts will result in a speckle pattern embedded in the shearographic image. The speckle pattern is slightly altered when the object is deformed. Two speckle patterns of the test object, one before and another after it is slightly deformed, are digitized into a microcomputer via a frame grabber. As will be shown below, the difference of the two speckle patterns will enable reconstruction of a visible fringe pattern that depicts displacement-derivatives with respect to the direction of image-shearing. Fig. 3 shows a fringe pattern depicting the deflection derivatives of a rectangular plate clamped along its boundaries and loaded by uniform pressure.

Principles of Fringe Formation A shearographic image contains a speckle pattern that may be mathematically represented as follows.

where I is the intensity distribution of the speckle image received at the image plane of the camera; is the intensity of the laterally-sheared images (which may be perceived as the d.c. term); is the amplitude of modulation of the speckle pattern; and is a random phase angle that accounts for the random interference

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pattern. After the object is deformed, the intensity distribution of the speckle pattern is changed slightly to I’, which is described by the following equation.

where denotes phase-change due to surface deformation. Numerically, the difference of intensities of the two speckle patterns (Eqs. (1) and (2)) yields the following equation.

where

manifests as a fringe pattern in which the dark fringes correspond to with n = 0,1,2,3,... being the fringe orders, and the bright fringes correspond to half fringe orders. The fringe pattern of Fig. 3(left) is for a fully clamped rectangular plate under uniform lateral load when image-shearing is along the x-direction, and Fig. 3(right) shows the corresponding fringe pattern when image-shearing is along the y-direction. Note that the absolute value of is used in the display of the fringe patterns, as an image cannot have negative value.

Fringe Interpretation The phase-change in Eq. (2) is induced by the change in the relative optical path length of light scattered from two neighboring object points, P(x,y,z) and considering that the direction of image-shearing is parallel with the x-axis and the amount of shearing is is related to the relative displacement of the two neighboring points as follows.

where (u,v,w) and are, respectively, the displacement vectors of P(x, y, z) and is the wavelength of the laser, and A, B, C are sensitivity factors that are related to the positions of the illumination point and the camera in the following manner.

with It is noted that z = z(x,y) describes the object surface and is not an independent variable when surface points are considered. Equation (4) may be rewritten in the following form.


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If the amount of shearing, is small, the relative-displacement terms in Eq. (6) may subsequently be treated as the partial derivatives of u, v and w with respect to the direction of image-shearing x. Thus, Eq. (6) becomes

Rotating the image-shearing device (Figure 1) about its optical axis will alter the direction of image-shearing, resulting in fringe patterns depicting displacement derivatives with respect to the current direction of image-shearing. Hence, if imageshearing is along the y-axis by an amount Eq. (7) is modified to the following.

The fringe patterns in Figures 2(a) and 2(b) for a laterally deflected plate therefore represent, respectively, displacement derivatives, namely,

and

Equation (7) contains three and

so that three measurements

with different sensitivity factors A, B, C, are required to separate these displacement derivatives. In NOT applications to be presented below, only the derivative of out-of-plane displacement is needed. In this case, the setup parameters A and B are selected to be zero.

Automated Fringe Phase Determination Traditional methods of shearographic analysis rely on the reconstruction of visible fringes and the correct identification of fringe orders (Eq. (3)). Therefore, only at locations where the dark and bright fringes appear can the fringe orders be correctly determined; elsewhere the fringe orders are generally estimated using linear interpolation. Because of the page limit, readers are referred to Ref (3) for an automated process in which the phase-change is directly determined at every digitized points without having to rely on fringe reconstruction and fringe order identification. Fig 4 shows a 3-D plot depicting the deformation of Fig 3 obtained by the automated fringe phase determination technique

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3. Applications Shearography is a full-field optical sensor based on the principles of optical interference. It measures the relative phase change between interfering wavefronts. The phase change can be induced by surface displacements, strains and refractive index change. Applications of shearography that have been explored so far include measurement of strains, material properties, residual stresses, 3-D shapes, vibrations,leakage detection, and nondestructive evaluation. This paper reviews its application for nondestructive evaluation, in particular for composites structures.

How does shearography detect flaws? When an object containing a flaw is loaded, strain concentration at the vicinity of the defect is induced. If the flaw is not too remote from the object surface, the induced strain concentration would cause anomalies in the surface strain distribution. Subsequently, these anomalies are translated into fringe-anomalies if two speckle patterns, taken one before and another after loading, of the object are compared. Thus, shearography reveals flaws, both surface and internal, through identification of anomalies in the fringe pattern. Shearographic nondestructive inspection is full-field and non-contacting.

Shearography vs Ultrasound Shearography has found wide applications in nondestructive testing and it has already received industrial acceptance as a useful NDT tool, particularly for composite structures such as tires and honeycomb structures. It is particularly effective in revealing delaminations in laminated composites. Fig 5 shows a comparison of test result obtained using digital shearography and C-scan ultrasonic testing on a composite sample. Both techniques readily reveal an edge pullout and a delamination. With digital shearography, the flaws are revealed in less than one second, whilst the ultrasonic technique requires point-by-point scanning along the test surface and at the same time, proper fluid coupling between the transducer and the test surface is to be ensured. A limitation of shearography is the need to apply suitable stress-increments to the test object during inspection, since the underlying principle of this technique is based upon the response of defects to stresses. However, the mechanics of revealing flaw in shearography yields information more relevant to the structural weakness.

Methods of stressing The development of shearographic NDT procedures has therefore essentially become the development of a practical means of stressing the object that would readily reveal flaws. Ideally, the stress-increment should be similar to the service stresses so that flaws that are critical and detrimental to the service life of the object would be revealed, and cosmetic flaws that do not undermine the structural integrity of the test object can be ignored. This would minimize unnecessary rejects during inspection. In this regard, shearography has an advantage over ultrasonic testing, as the latter detects flaws by identifying inhomogeneities in the object and does not provide direct information on the criticality of the flaws. Exact duplication of the actual stress-increment for shearographic testing, however, is generally difficult. Therefore, various practical means of stressing the object must be developed. In


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developing these methods, an important precaution to be taken is restricting rigidbody motion of the object during stressing, as excessive rigid-body motion would cause speckles de-correlation, resulting in degradation of fringe quality. The methods that currently used that do not cause intolerable rigid-body motion of the test object include the use of pressure, vacuum, thermal, and acoustical and mechanical excitations. The use of microwave that excites water molecules is particularly effective for detecting moisture in plastics and non-metallic composites, but safety precautions must be taken seriously. The following figures illustrate some examples of the shearographic NDT application.

Samples of NDT applications Some examples of nondestructive testing using shearography are illustrated in Fig 5 through Fig 13.

4. Conclusion A review of shearography and its applications in nondestructive testing is given in this paper. Unlike holography, shearography does not require a reference beam and hence it is more tolerable to environmental disturbances. Indeed, shearography can be employed in field/factory settings. This technique is still relatively young and its full capability awaits further exploration.

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5. Acknowledgement The work presented in this paper was supported by a National Science Foundation grant (9601778).


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6. References 1. 2.

3.

Hung, Y.Y. “Shearography: A New Optical Method for Strain Measurement and Nondestructive Testing”, Optical Engineering, pp.391–395, May/June, 1982. Hung, Y.Y. and J.D. Hovanesian, “Fast Detection of Residual Stresses in an Industrial Environment by Thermoplastic-based Shearography”, Proceedings of the 1990 SEM Spring Conference on Experimental Mechanics, pp769-775, Albuquerque, New Mexico, June4-6, 1990. Hung, Y.Y. “Digital Shearography and applications” Trends in Optical Non-destructive Testing and Inspection, pp287-308, Elsevier, 2000 (Editors: Pramond K. Rastogi and Daniele Inaudi).

THEORETICAL AND EXPERIMENTAL STUDY OF LASER-ULTRASONIC SIGNAL CHARACTERISTICS ENHANCED BY WETTING THE SURFACE SHI-CHANG WOOH Department of Civil and Environmental Engineering Massachusetts Institute of Technology Cambridge, Massachusetts

Abstract This paper deals with a theoretical and experimental study to predict the characteristics of ultrasonic waves irradiated from a wet surface illuminated by a laser beam. The theoretical model is developed by assuming normal surface tractions distributed elliptically over the line-focused illumination source. The solutions for both longitudinal and shear waves are obtained using Fourier analysis, in which the transform integral is evaluated asymptotically. The theory is evidenced by the experimental study which allows for quantitative comparisons between the theoretical and experimental directivity patterns. The effects of beam width and frequency of the propagating wave, which are the most influential parameters on the characteristics of the generated wave, are investigated. The experimental results are in strong agreement with the theoretical predictions.

1 Introduction The work described in this article is inspired by the need for developing a testing method enabling non-contact detection of rail flaws, particularly those which are not easily detectable by other methods. In order to avoid catastrophic failure due to the growth of defects, the running rails should be inspected periodically using one or more nondestructive evaluation (NDE) techniques. Among the many types of flaws introduced to rail tracks from a variety of sources, the most common and pernicious kinds are transverse defects (TD) which may cause derailment if they go undetected [ 1 ]. These flaws are mostly detectable using high-angle ultrasonic wheel transducers. However, they are sometimes masked by other defects, such as longitudinal shells. Traditional ultrasonic techniques (e.g., those that impinge ultrasound from the top surface of the head) are intrinsically blind in dealing with such situations because the propagating beam is blocked off by the shell. The increased number of undetected defects reduces the overall reliability of detection. In order to resolve this problem, it may be necessary to access the rail specimen on its sidewall. In this way, it is possible to detect the hidden TD because the area 409 E.E. Gdoutos (ed), Recent Advances in Experimental Mechanics, 409–420. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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underneath the shell is directly insonified. However, it is not practical to use traditional wheel transducers because the accessible space on the side is t y p i c a l l y less than 5 cm. The use of laser generation and detection of ultrasound allows for easy access to the sidewall and enables totally non-contact operations. This paper deals with the theoretical and experimental study which may be used to find the optimum design of laser ultrasonic rail defect detection. Figure 1 shows a laser-ultrasonic system scanning a rail to detect internal defects. In principle, acoustic wave energy is generated by illuminating a point on the surface of the railhead using a laser beam, and the traveling wave arriving at another location on the head is recorded. The use of optical laser beams for both the generation and detection of ultrasound could completely eliminate the need for contact. If there exists a discontinuity that blocks the acoustic beam pathway, it may change the characteristics of the propagating wave and the signals may be interpreted to identify and characterize the discontinuity or rail defect. Figure 2 shows the laser-ultrasonic rail flaw detection scheme, in which an excitation laser (Q-switched Nd:YAG laser) shoots a pulsed laser beam at a point (Point A) on the front wall of the specimen. At the same time, a detection laser beam (Photo-EMF detector with a continuous probe laser) is aimed at another point (Point ) on the same surface, separated by a fixed distance Both lasers are operating in synchronization to generate an ultrasonic pulse and measure the surface displacements at the corresponding points. The waves irradiated from the source point A travels forward into the rail covering a wide range of angles and they are reflected off after hitting the back wall. As shown in Fig. 2(a), one or more rays of the reflected waves may continue traveling all the way to the detection point if no flaw exists along its pathway. By contrast, if there is a discontinuity (Fig.2(b)). the rays cannot reach the detection point , since they are blocked by the

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flaw. This results in the formation of a shadow zone behind the flaw created by a bundle of such rays. The shadow pattern may be used not only to detect the defect but also to measure its size and location. While maintaining the distance between the generation and detection lasers at the system moves together along the specimen (from Point A to D) to scan the area of interest. Figure 3(a) shows a typical waveform obtained from an unflawed zone. As noted in the figure, the laser source beam produces both body waves (L and T-waves) and strong surface waves (R-waves). In Fig. 3(b), a noticeable reduction of shear wave amplitude is observed at a shadowed region. These observations give us a clue that the most efficacious wave mode is the shear wave propagating strongly at high angles, which promotes good interaction between the wave and the flaw. The characteristics of lasergenerated ultrasound are dependent on various factors such as the size and shape of the illumination spot, frequency, elastic and thermal properties of the target materials, and the surface condition. It is known that a surface modified by grease, water or other liquid can significantly boost the signal amplitude [2,3]. Since the body waves must travel relatively long distances in the specimen, the signal amplitudes produced by the laser irradiation on a dry

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surface may not be high enough to be detected. Thus, it is preferred in our case to wet the surface before activating the generation laser, which can be achieved in-situ using water sprays. It is the objective of this paper to study the behavior or ultrasound radiated from a wet surface.

2 Theory The basis of our theory and experimental work taking into account the wet surface condition was reported in our previous articles [4, 5], in which a laser beam of circular cross section was modeled. In this paper, we extend our study by considering a line-focused laser source. The fundamental difference between the circular and line sources is that the former produces axisymmetric conical waves, whereas the latter produces cylindrical waves. Most of the derivations are similar to those for circular sources except for the choice of coordinate systems and the use of the double Fourier transform (FT) instead of the Fourier-Bessel transform in order to take into account the rectangular source geometry.


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Excitation Source As shown in Fig. 4(a), we consider an infinitely long line-focused source of width loaded by a normal stress traction. It is reasonable to assume that the rapid evaporation of fluid will cause high reaction pressure acting in the direction normal to the surface. This assumption is supported by [3], in which it is shown that the experimental directivity patterns for a wet surface are close to those of the ablation source. Studies involving the range of laser intensity below the evaporation level lie beyond the scope of this paper. The problem is treated using Cartesian coordinates where the is normal to the surface and the is parallel to the lateral direction of the line strip. Due to the cylindrical symmetry of the propagating wave, the problem is simplified as a two-

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dimensional elastic half-space excited by the surface traction, as illustrated in Fig. 4(b). It is also assumed that the vertical displacement component and its derivatives vanish due to the uniform loading along the

Boundary Conditions Fora line source of width . the boundary condition can be written by prescribing the time harmonic surface tractions distributed arbitrarily within the source width as:

where is the position of a point on the surface measured from the source origin is the excitation frequency, and is the unit imaginary number. It is assumed that the shear traction is zero everywhere on the surface because we only consider the reaction forces acting normal to the surface. We are interested in obtaining the field equations and studying the directivity patterns to study the angular distribution of acoustic wave energy propagating in the medium. Since the directivity can be directly obtained from the steady-state solution, it is sufficient to consider only the amplitude in the analysis. We consider two ideal tractions representing dry and wet surfaces (Case I and II) shown in Figs. 5(a) and 5(b), respectively. For dry surfaces, the traction is idealized as a vertical load of magnitude distributed uniformly over the illuminated area. On the other hand, the wet surface is modeled by assuming elliptical distribution of traction with an aspect ratio of The tractions can thus be written in the following forms:


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The elliptical distribution model is based on the same physical insights that we explained in our previous articles [4, 5]. If a wet surface is struck by a pulsed laser beam, it is likely that the reaction forces at the center, resulting from the evaporation of water particles, are significantly greater than those near the boundary of the illuminated area, which is surrounded by colder neighboring molecules. In order to represent this physical phenomenon, an elliptical function is chosen primarily for mathematical convenience.

Field Equations The following equation of motion for elastic isotropic solids governs the behavior:

where u is the displacement vector, is the mass density, and are the Lame constants of the medium, and is the gradient operator. Note that this equation is written in terms of displacements and should be solved to satisfy the stress boundary conditions given earlier. By defining the scalar and vector potentials and the wave equations, Eq. (3), can be written as

so that

where

and are the time-harmonic displacements, W is the component of W in the and and are the wavenumbers of the longitudinal and shear waves, which are respectively given as

Using Hooke’s law, the stress boundary conditions can be written in terms of potentials as:

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within the region where The Fourier transformation can be used to eliminate from these equations. For example, the FT of the elliptical loading given in Eq. (2) can be simply written as

After carrying out the transforms of Eqs. (4), (5), (6), (7), (9) and (10), the FT of the potentials are obtained. Then, the displacements and are obtained by solving for and and asymptotically evaluating their inverse FT [4, 6]. The longitudinal and transverse displacement components can be derived by decomposing the displacement vector using the relationship

The subscript is used to distinguish the line source from the circular source. The radial and tangential displacement amplitudes for a line source of width can thus be written for the two difference loading cases as follows:

Case I. Dry surface (Uniformly distributed load) The solutions for the corresponding displacement amplitudes for this case are already derived by Miller and Pursey [6] as

where

The functions and denote the longitudinal and transverse displacement amplitudes in the far-field for a hairline source, in which the width approaches zero. They are respectively given as


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where

Note that the displacement amplitudes hairline sources.

and

are the same for both point and

Case II. Wet Surface (Elliptically distributed load) The solutions for wet surfaces are obtained as follows, using the aforementioned elliptical loading model:

where and are the far-field displacement amplitudes of the compressional and shear waves radiated from a hairline source given by Eqs. (17) and (18).

3 Experimental Results and Discussion

Experimental Setup As we have done in our previous work [5] for convenience, the out-of-plane displacements are measured in this study to validate and compare the models, instead of measuring the directivities directly. Figure 6 illustrates the schematic of the experimental setup used to measure the displacements and their angular dependence. First, a laser beam is fired to illuminate a source point on the aluminum test block of thickness Then, an ultrasonic ray irradiated at an angle from the source point reaches the point on the opposite surface of the specimen. The displacement in the direction normal to the surface is then detected by either a laser interferometer or a polyvinylidene fluoride (PVDF) piezo-polymer transducer. By scanning the surface with the excitation laser beam, it is possible to obtain a displacement profile along the or as a function of corresponding angle for the known thickness The detailed experimental

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test procedure can be found in our earlier article [5]. The basic difference between the two studies is the laser illumination shape. While using the same laser source of circular beam cross-section, the beam shape was transformed into a line segment using an assembly of spherical and cylindrical lenses. With this setup, the rectangular illumination shape of the line segment is provided by the cylindrical lens while its illumination width was controlled by the optical characteristics of both lenses.

Results Figure 7 shows the displacements of transverse waves plotted as a function of propagating angle The circular symbols represent the normal (out-of-plane) displacements measured experimentally, while the solid lines represent the corresponding theoretical displacements for the same experimental conditions. The curves displayed on the left-hand side of the figures are the displacements for dry surface (Case I); the ones on the right-hand side are those for wet surface (Case II). The signals for wet surface were obtained by wetting the illumination surface using a water spray gun for each measurement. The tests were carried out for four different conditions: and 3.5 mm and and 5 MHz. The line width was varied by changing the distance between the lens assembly and the specimen surface so that the beam is defocused to the desired degrees. For this, a calibration paper was used to measure the dimensions. On the other hand, the different frequencies were obtained by using different detection techniques: A Photo-EMF laser detection system was used to attain signals of 1 MHz


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and a PVDF film transducer was used for 5 MHz. The validity of using these techniques are again discussed in our previous article [5] where the frequency spectra of the signals are shown.

4 Conclusions The models for both dry and wet surfaces are considered for various conditions. From a theoretical point of view, the two solutions are similar, except that the equations are modified by different terms. In describing the directivities for a line source loading, the corresponding modification terms for wet and dry surfaces are and respectively. The experimental results are surprisingly consistent and predictable by the theory, even with the assumption that the source is infinitely long in the From the results, we can observe that the propagation angles where the amplitude reaches its maximum are almost the same for both wet and dry surfaces (approximately at 35°), which can be used for optimum operation. For the lower frequency of 1 MHz, the change of beam diameter does not alter the characteristics of transverse waves much at low angles (below 30°). But the curves at high angles are noticeably squeezed to the left. On the other hand, the changes due to the beam size for the higher frequency (5 MHz) are dramatic in both high and low angles. It is interesting to note that wetting the surface introduces stronger shear waves propagating at low angles. For the wet condition for a large beam at high frequency, the amplitude shows more uniform and periodic patterns.

5 Acknowledgment This work was supported by the National Science Foundation under the Contract Number CMS-9733157. We are grateful for their support and encouragement.

6 References 1. Sperry Rail Service (1989) Rail Defect Manual. 2. Fox, J. A. (1974) Effect of water and paint coatings on laser-irradiated targets, Applied Physics Letters, 24(10), pp. 461–464. 3. Hutchins. D. A.. Dewhurst, R. J., and Palmer, S. B. ( 1 9 8 1 ) Laser generated ultrasound at modified metal surfaces, Ultrasonics, 19(3), pp. 103-108. 4. Wooh, S. C. Wooh and Zhou, Q. (2001) Behavior of laser-induced ultrasonic waves radiated from a wet surface. Part I: Theory. Journal of Applied Physics, 89(6). pp. 3469–3477. 5. Wooh, S. C. Wooh and Zhou, Q. (2001) Behavior of laser-induced ultrasonic waves radiated from a wet surface, Part II: Experimental Work, Journal of Applied Physics, 89(6), pp. 3478–3485. 6. Miller, G. F and Pursey, H. (1954) The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid, Proc. Royal Society (London), A223, pp. 521–541.

DEFECT DETECTION BY THE SCATTERING ANALYSIS OF FLEXURAL WAVES PAUL FROMME and MAHIR B. SAYIR Institute of Mechanical Systems ETH Zürich - Swiss Federal Institute of Technolog, CH-8092 Zürich, Switzerland

Abstract The propagation and scattering of flexural waves by obstacles in plates is studied experimentally and theoretically. When the guided wave hits a discontinuity like a hole, a typical scattered displacement field is obtained. Defects like a notch or a fatigue crack at the hole boundary change the scattered field significantly. The first antisymmetric Lamb wave mode is excited selectively by means of a piezoelectric transducer. The scattered field around undamaged and damaged holes is measured on a grid around the hole using a heterodyne laser interferometer. The out-of-plane displacement is measured with good spatial resolution, accuracy, and repeatability. Applying fast Fourier transform, the amplitude and phase values of the scattered field are extracted, overcoming the typical problems associated with the measurement of dispersive waves. The experimental results are compared with theoretical calculations. The wave propagation is studied using classical plate theory and Mindlin’s theory of plates. The scattered field around the circular hole is calculated analytically and good agreement is found for the range of validity of the used theories. The scattering at the hole with a defect is calculated numerically, using a finite difference scheme. The method can be applied for the detection of cracks at rivet holes in aircraft fuselage, an important nondestructive testing application. Guided waves allow a fast inspection of large areas, reducing the need for time-consuming scanning. The minimum detectable crack length and the sensitivity of the method are discussed.

1. Introduction Technical machinery, systems, and components, e.g., airplanes, automobiles, and pipes, are subject to varying or cyclic service loads and environmental influences. Such operation conditions can lead to wear, corrosion, and damaging of the components. The problem is relevant in aircraft industry, where a common maintenance problem is the development of fatigue cracks during the service life span of the aircraft. The fuselage and wings of planes consist of large plate-like parts, connected with lines of rivets and fasteners. Due to the stress concentration at these connections, fatigue cracks start to grow from the fastener holes, and must be detected before reaching a critical length (Fig. 1). 421 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 421–432. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Mechanical waves, as used in ultrasonic testing (UT) have a well-established performance for the detection of such defects. However, in classical UT the used wavelength of the mechanical waves is in the order of the crack length one aims to detect and thus small compared to the thickness of the structure. Waves at such rather high frequencies dampen out quickly and good signal strength of either transmitted or reflected wave can usually only be achieved working through the thickness of the structure. Therefore, classical UT involves manual scanning around the area of suspected defects, which is time-consuming and therefore cost-intensive, as it increases the downtime of the plane. An alternate, more elegant and promising approach is the use of guided waves, resulting in a propagation direction along the structure and performing such checks automatically over large parts of the structure. Placing a transducer on the specimen, a guided wave can be excited, that travels along the plate and interrogates a whole line of rivets. From the measurement of the guided wave at a few points on the surface of the structure, it is possible to detect defects in a large area with a fast and cost-effective method [1]. Rose and Soley [2] showed the practical applicability of guided waves for a variety of problems in aircraft components, e.g., crack detection in helicopter blades. In complicated systems, as in an aircraft, there are many difficult to access places like the interior of the fuel tanks in the wings. Automated built-in measurement systems will reduce the laborious and hazardous checking by personnel. Using low cost piezoelectric transducers for the excitation of the guided waves, they can be integrated into the structure. This way a smart structure is fabricated, where the damage checks can be performed automatically while being in service. The problem arising from the use of guided waves is the fact that the wavelength is usually of the order of magnitude or larger than the thickness of the structure and hence large compared to the typical flaw size one aims to detect. The large ratio between wavelength and defect length reduces the sensitivity and makes an accurate study of the propagation and scattering characteristics necessary, to determine experimental constraints and gain a well-founded theoretical understanding of the interaction between wave and flaw.

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Background

The propagation of guided waves in isotropic, homogeneous plates is described by the theory of Lamb waves [3]. The first antisymmetric mode is a flexural wave, studied in the work reported here. In contrast to the bulk waves used in UT, the propagation of the guided wave modes is dispersive, i.e., wavelength and propagation velocity depend on the frequency. This distortion of the signal poses some experimental challenges, but can be overcome by using either narrowband excitation or advanced signal processing [4]. Due to the two-dimensional propagation of waves in plates, the amplitude of the wave decreases with distance from the source. This decrease in amplitude can lead to problems with the signal to noise ratio, as the wave travels from the excitation transducer to the defect and then further on to the measurement spot. Therefore, measurement methods that are sensitive to small excitations like laser interferometry are advantageous. Flexural waves were successfully employed for the measurement of the material properties of composite plates [5]. The displacement of the flexural wave is primarily a bending of the plate. For low frequencies, i.e., when the wavelength is large compared to the plate thickness, the propagation can be described approximately using classical plate theory (CPT), taking only bending stiffness and inertia into account (see e.g. [6]). For higher frequencies, shear and rotatory inertia have to be considered according to the theory of Mindlin [7]. The model system investigated here is a circular through hole in an aluminum alloy plate with a notch at an arbitrary angle. This interesting theoretical problem has only partially been studied in literature. Analytical solutions can be found for the geometrically simpler problem of the scattering of flexural waves at a circular cavity [8]. Only few, mostly numerical, studies exist on the scattering characteristics of flexural waves at a notch or a crack in a plate [9, 10].

3.

Experimental setup

Flexural waves can be excited rather easily by means of a piezoelectric transducer and measured using a heterodyne interferometer, but are highly dispersive in the frequency range of interest here, between 20 kHz and 200 kHz. Applying Fourier transform for the data evaluation and studying amplitude and phase variations instead of time of flight measurements, this problem can be overcome. The dispersive nature of the excited pulse can be further used to measure material properties. Applying the known dispersion properties, a desired signal shape in the measurement area can be achieved, e.g., a contraction of the signal in the time domain. The geometry of the setup (Fig. 2) used in this investigation and further described in Ref. [11] is not a conventional C-scan, as used in UT. The scattered field of the flexural is measured on a grid around the hole, while the excitation transducer is fixed in one position. Therefore, as assumed in the analytical calculation, the incident wave for each measuring point is the same and the scattered field can directly be compared with the calculations. The first antisymmetric Lamb wave mode is excited selectively, working in a frequency range below the cutoff frequencies of the higher Lamb modes. The velocity of the out-of-plane displacement is measured point-wise on a grid around the hole using a heterodyne laser interferometer. At each measuring point, a time signal of the velocity of the plate surface is stored. Amplitude and phase of the scattered wave are extracted by fast Fourier transform (FFT) and analyzed.

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The specimen is a thin aluminum alloy plate (1 mm thick), having a size of 1000 mm by 1000 mm. The plate material is a commercially available aluminum alloy with a Young’s modulus E of 6.9 and a density of 2700 Poisson’s ratio v is assumed as 0.31. A circular hole with a radius of 10 mm is drilled through the plate. Static bending of the plate is avoided by suspending it vertically. A piezoelectric disc, polarized for thickness extension mode, of 10 mm diameter and 1 mm thickness is glued to the plate 300 mm away from the hole using a two-component epoxy glue. For the simulation of a defect at the hole boundary, a through thickness notch of about 0.2 mm width, 2 mm length, and a blunt tip is cut into the hole boundary at an angle to the vertical (see Fig. 2), using a fine saw blade. The excitation signal is chosen as a sinusoid multiplied by a Hanning window to achieve a short time, narrowband signal with the energy concentrated around the center frequency The short time duration of the excitation pulse allows a time gating between the measurement signal (the interference of incident wave and wave scattered at the hole and the notch) and the reflection from the plate boundaries, arriving some time later at the measurement spot. The arrival times of the different pulses are calculated from the theoretical group velocities. The narrow bandwidth avoids extensive signal distortion due to the dispersive character of the flexural wave. A typical measured time signal with a center frequency of 50 kHz and 20 cycles is shown in Fig. 3. The excitation signal is generated in a programmable function generator and amplified to 200 V peak to peak. Applying the voltage to the piezoelectric transducer, the disc contracts and expands. This way a vertical force to the plate surface is generated and primarily the first anti-symmetric Lamb wave mode is excited. For the frequencies range of this study, the energy transferred to the longitudinal mode and the shearhorizontal mode SH is negligible. Working below the cutoff frequencies for the


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higher wave modes, only the desired mode is excited, as no selection between different modes is necessary. The used frequencies of up to 200 kHz are well below the eigenfrequencies of the piezoelectric disc (ca. 1 MHz), thus working in the frequency range of the transducer with a linear transfer curve. However, the amplitude of the excited wave is rather small, below 1

The resulting flexural wave propagates radially outwards from the piezoelectric disc. The distance between the transducer and the hole is selected as large compared to the hole radius. Therefore, the wave front can be assumed to be planar when it reaches the hole, simplifying the theoretical simulation. The incident wave is scattered at the stress-free boundaries of the hole, and a scattered wave is generated. The scattered wave consists of three parts: a flexural wave propagating radially outwards from the hole, a flexural boundary layer close to the hole, and a shear boundary layer. Near the hole, incident and scattered wave overlap in time. Therefore, only a single pulse is visible in Fig. 3. The specimen size is selected large enough that the reflections from the plate boundaries arrive at the measurement area some time later, allowing a time gating of the measurement pulse. The scattered field on a measurement grid around the hole is recorded using a commercially available heterodyne laser interferometer. The demodulator output is a voltage signal proportional to the velocity of the out-of-plane component of the displacement of the plate surface. The measurement spot is less than 0.1 mm in

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diameter, defined by the laser beam. This achieves a point-wise measurement of variations in the scattered field, as no implicit average over a rather large surface of the measuring transducer is made. The laser interferometer is moved parallel to the plate using a positioning system. A radial measurement grid is used, moving the laser beam spot on concentric circles around the hole and recording a time signal with 10’000 values at every measurement point. The measurements are very repeatable, with the largest cause of variation due to an inaccurate positioning of the laser beam relative to the hole center, which could only be achieved with an accuracy of about 0.1 mm. The voltage signal is band pass filtered around the center frequency and averaged over 25 measurements in a digital storage oscilloscope. The function generator triggers the oscilloscope, so that excitation and measurement start at the same time. The measured time series are evaluated on the computer. Fast Fourier transform (FFT) is applied and the amplitude and phase values at the center frequency are calculated. These values are the equivalent of the theoretical results, derived in section 4, where an infinite sinusoid is assumed for the incident wave.

4. Theoretical calculations The propagation and scattering of flexural waves in isotropic, homogeneous plates can be described by various approximate solutions. To achieve a fast computation, different degrees of simplification have been used. The simplest approach is using classical plate theory (CPT), taking only inertia and bending stiffness into account according to

in which is the specific mass, h the plate thickness, and w the out-of plane displacement of the plate. The flexural rigidity D is given by

where G is the shear modulus and v Poisson’s ratio. This approach is valid only at low frequencies, when the wavelength is large compared to the plate thickness. This was shown by Niordson [12], using an asymptotic expansion. For higher frequencies, corresponding to shorter wavelengths, shear and rotatory inertia have to be considered. Therefore, the theory of Mindlin [7] can be used

The term denotes a dimensionless correction factor. One can choose the value of this factor so that waves at very high frequencies propagate with the Rayleigh velocity of surface waves. According to this condition, the value of the correction factor is obtained as a function of v, given by Mindlin [7]. With the theories described above, the scattering of a plane incident flexural wave by a circular cavity with radius is studied [11]. The incident wave is taken as an infinite sinusoid, planar wave, propagating in the y direction, and is given by


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is the wavenumber of a propagating flexural wave. The origin of the coordinate system is chosen at the center of the hole, cylindrical coordinates are introduced. To satisfy the stress-free boundary conditions at the hole

a scattered wave must occur. This problem was studied by Norris and Vemula [13], and Staudenmann [14] using CPT. Following their approach, the scattered wave is assumed in the form of

The first part describes a wave propagating outwards (Hankel function of the second kind), and the second part a boundary layer around the hole (Hankel function of the first kind). The angular dependence is given by the sum over the cosine functions, is the wavenumber of the nonpropagating flexural mode. With this approach, the boundary conditions can only be fulfilled in the Kirchhoff approximation

satisfying a combination of vertical force and the derivative of the twisting moment. Using Mindlin’s theory and following the work of Pao and Chao [8], the boundary conditions can be fulfilled as an average over the plate thickness with

The scattered wave consists of the same flexural wave as in Eq. (6) and an additional shear boundary layer with the two components

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Evaluating the boundary conditions in polar coordinates, the coefficients of the scattered waves are calculated. is the wave number of the shear wave, and give the relation between the wave numbers, as defined in Ref. [8]. Finite difference methods (FDM) are used to calculate the propagation and scattering characteristics of flexural waves in a plate with a more complicated geometry, i.e., the scattering at a hole with a notch. Mindlin’s equations of motion are discretized on a Cartesian, staggered grid, as described in Ref. [15]. To implement the scattering at a circular hole in the plate, the hole is approximated by a right-angled contour and the stress-free boundary conditions are implemented. A crack or notch at the hole boundary, through the thickness of the plate, is implemented in a similar way. The boundary conditions are applied to two parallel lines and one point at the tip of the notch.

5. Measurement of the scattered field A typical measured scattered field around a hole in an aluminum alloy plate for an excitation with a center frequency of 50 kHz can be seen in Fig. 4a. The measurement is made on a circular grid around the hole, with a signal recorded every 5 degrees on radii 0.5 mm apart. The incident wave with a nearly straight wave front propagates from the direction of the positive y-axis and is scattered at the hole boundary. Behind the hole, an area with low amplitude can be seen, the socalled shadow area. Directly at the hole, a high amplitude results from the scattering at the free surface. Further outwards a characteristic hill and valley pattern develops due to the constructive and destructive interference of incident and scattered waves. In Fig. 4b the measured and calculated amplitude of the scattered field on a circle around the hole is shown. Measurement and analytical description using Mindlin's theory show good quantitative agreement. The calculation using CPT, neglecting the effects of shear and rotatory inertia, has a slight deviation around 180°, but the main features of the scattered field are described accurately enough. After the first measurement (Fig. 4a), a notch of 2 mm length is cut at an angle of 90° on the hole boundary using a fine saw blade. The scattered field is measured again and the difference in amplitude due to the notch is shown in Fig. 4c. At the free surfaces of the notch an additional scattered wave is generated, which changes the scattered field significantly up to about 30% in amplitude and allows the detection of such a notch from an amplitude measurement. The calculation of this difference in amplitude with FDM (Fig. 4d) shows good agreement with the measurement, and it can be concluded that we can accurately model the scattered field around a hole with a notch. More complicated geometries, like a line of holes in a plate, have also been studied with this method [16].


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6. Application for nondestructive testing The growth of fatigue cracks in tensile specimen was monitored as a nondestructive testing application in cooperation with the fatigue engineering laboratory of RUAG Aerospace, Emmen, Switzerland. Tensile specimen 3.17 mm thick, 40 mm wide, and 500 mm long, made of Al-7075 PL-T3 were used. A fatigue grown crack at the hole boundary was generated by cyclic tensile loading in a servohydraulic testing machine (Fig. 5). A small starter notch was cut to prescribe the position of the crack. The setup described in section 3 was slightly modified to automatically monitor the motion of a single spot close to the crack during the cyclic tensile loading. The laser interferometer was affixed to the testing machine and the measurement was triggered on the maximum tensile force. This assures a measurement at always the same spot and when the crack faces are open. An excitation frequency of 40 kHz was used, resulting in a wavelength of the flexural wave of 26 mm. The amplitude of the measured movement of the specimen surface in the vicinity of the crack was extracted from the time series by FFT and plotted. Additionally the crack length was measured optically using a microscope.

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After the crack initiation of several thousand cycles, the crack started to grow as a quarter-elliptical crack before reaching through the thickness and increasing further in length. Monitoring measurements during crack growth were made at several specimens. Typical measured signals can be seen in Fig. 6, where the crack started to grow around 100’000 cycles and went through the thickness of the specimen at 124’000 cycles. The amplitude measured at an excitation frequency of 40 kHz (Fig. 6a) starts to rise around 122'000 cycles, or a crack length of ca. 2 mm, and increases smoothly with the crack growth. A significant increase in amplitude could be seen in all measurements for a crack length of ca. 2.5 mm, and therefore a crack of this length can be securely detected. The variation among the measured amplitude curves is rather small and the curves agree well with the FDM calculation, allowing a sizing of the crack length.

7. Conclusions Flexural waves in isotropic, homogeneous plates are generated using piezoelectric transducers. Driven below the cutoff frequencies of the higher wave modes, the first antisymmetric Lamb wave mode is excited selectively. The scattered field on a measurement grid around a hole is measured pointwise using a heterodyne laser interferometer. Amplitude and phase of the scattered field are extracted from the measured time series using FFT. Good agreement between the measurements and analytical calculations of the scattered field at a circular cavity is found. Mindlin’s theory of plates describes the propagation and scattering very accurately, while a slight divergence can be seen for CPT, neglecting the influence of shear and rotatory inertia. The measured scattered field around the hole changes significantly with the presence of a notch or a crack, much smaller than the wavelength. The change in the measured amplitude is modeled using finite difference methods and good agreement is found. The significant change in amplitude due to a defect at the hole allows the use of the proposed method for the nondestructive testing of large structures. A flexural wave traveling along the structure can be excited, and its scattering measured. From changes in the measured amplitudes, the development of defects, like fatigue cracks in aircraft structures, can be detected. As a first application, the monitoring of fatigue cracks in tensile specimen is shown. Significant changes in the measured amplitude of the flexural wave can be seen for crack lengths smaller than the specimen thickness.

8. Acknowledgements We would like to thank Daniel Gsell and Tobias Leutenegger for their help on the FDM coding, and Bernard Masserey for his help on the analytical calculations. Our thanks go to the fatigue engineering center of RUAG Aerospace, Emmen, Switzerland for the possibility of conducting the cyclic tensile loading in their laboratory. References 1. 2.

E.V. Malyarenko, M.K. Hinders, Ultrasonic Lamb wave diffraction tomography, Ultrasonics 39, 269-281 (2001) J.L. Rose, L.E. Soley, Ultrasonic guided waves for anomaly detection in aircraft components, Mater. Eval. 58(9), 1080-1086 (2000)

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3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16.


H. Lamb, On waves in an elastic plate, Proc. R. Soc. London, Ser. A 93, 114-128 (1917) D. Gsell, D. Profunser, J. Dual, Measurement of the dispersion relation of guided nonaxisymmetric waves in filament-wound cylindrical structures, Ultrasonics 38, 517-521 (2000) M. Veidt, M.B. Sayir, Experimental evaluation of global composite laminate stiffnesses by structural wave propagation, J. Composite Mater. 24, 688-706 (1990) K. F. Graff, Wave motion in elastic solids, Dover Publications, New York (1991) R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech. 18, 31-38 (1951) Y.-H. Pao, C.C. Chao, Diffractions of flexural waves by a cavity in an elastic plate, AIAA J. 2(11), 2004-2010 (1964) R. Paskaramoorthy, A.H. Shah, S.K. Datta, Scattering of flexural waves by a crack in a plate Eng. Fract. Mech. 33(4), 589-598 (1989) G.C. Sih, J.F. Loeber, Flexural waves scattering at a through crack in an elastic plate, Eng. Fract Mech. 1, 369-378 (1968) P. Fromme, M.B. Sayir, Measurement of the scattering of a Lamb wave by a through hole in plate, J. Acoust. Soc. Am. 111(3), (2002) F.I. Niordson, An asymptotic theory for vibrating plates, Int. J. Solids Struct. 15(2), 167-1 (1979) A.N. Norris, C. Vemula, Scattering of flexural waves on thin plates, J. Sound Vib. 181(1), 1: 125 (1995) M. Staudenmann, Structural waves in nondestructive testing, PhD. Thesis, Diss. ETH No. 113 (1995) P. Fromme, M.B. Sayir, Monitoring of fatigue crack growth at fastener holes using guided Lamp waves, to appear in Review of Progress in Quantitative Nondestructive Evaluation 21, ed. by D. Thompson and D.E. Chimenti P. Fromme, M.B. Sayir, Experimental detection of cracks at rivets using structural waves propagation, in Review of Progress in Quantitative Nondestructive Evaluation 20 B, ed. by D. Thompson and D.E. Chimenti, AIP Conference Proceedings 557, New York, 1626-1633 (2001)

EVALUATION OF FIBER WAVINESS IN THICK COMPOSITES BY ULTRASONIC TEST

H.-J. CHUN School of Electrical and Mechanical Engineering Yonsei University 134, Shinchon-dong Seodaemun-gu, Seoul, 120-749, Korea

Abstract An analytical model, based on the ray and plane wave theories, was proposed to understand the ultrasonic wave propagation in thick composites with uniform fiber waviness. In the analysis, the composites were assumed to have continuous fibers with sinusoidal fiber waviness in a matrix and were modeled as stacks of infinitesimally short off-axis subelements with varying fiber orientation along the lengthwise direction. From the analysis, it was found that the converted wave of the same mode as that of incident wave carried most energy and this tendency was sustained throughout the thick composites. The path and traveling time of ultrasonic wave in the subelement were obtained by computing the velocity and direction of group wave. The predicted results showed that the ray paths of quasi-longitudinal and quasi-transverse waves were highly affected by the degree of fiber waviness and had tendencies to trace toward the fiber direction and to converge to the adjacent peak of fiber waviness. Some parameters that showed strong effects on the wave propagation in the composites were identified for both insonified longitudinal and transverse waves. The experiments were also conducted on the specially fabricated thick composite specimens with various degrees of uniform fiber waviness using the conventional through-transmission method to verify the predicted results. Then, the analytically determined values were compared with the actual measurements obtained from the test specimens. Good agreements were observed among them. 1.

Introduction

Fiber waviness is one of the manufacturing defects frequently encountered in thick composite structures. It results from local buckling of prepreg or wet hoop-wound filament strands under pressure exerted by the overwrapped layers during the filament winding process or from lamination residual stress built up during curing. Its characteristic can be represented by the through thickness undulation of fibers within thick composite laminates. According to a number of studies on the behavior of thick composites with fiber waviness, fiber waviness causes degradation of strength and stiffness significantly[12]. Therefore, nondestructive evaluation technique that can detect fiber waviness of thick composite is very important for the integrity of structures. For this purpose, there have been investigations on the thick composites with fiber waviness by means 433 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 433–442. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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of ultrasonic waves. Wooh and Daniel[3] attempted to explain how ultrasonic wave propagates in thick composites with fiber waviness by using discrete ray tracing method. Mclntyre et al. [4] studied amplitude of ultrasonic wave using finite difference method. Kim et al. [5] investigated ray paths and traveling time using geometrical acoustics. In this paper, both theoretical and experimental investigations were conducted to evaluate uniform fiber waviness in thick composites nondestructively. Efforts were made to fully understand ultrasonic wave propagation in thick composites with fiber waviness by considering various parameters that represent degree of fiber waviness. The ray path, time of flight and refracted waves at imaginary interface between subelements in the composites with fiber waviness were calculated. The experiments were also conducted to prove predicted traveling time and wave paths. 2.

Analysis

Figure 1 shows the geometry of a representative volume of uniform fiber waviness. It is assumed that all the fibers are parallel to each other and have sinusoidal curvature along one spatial coordinate direction (x axis).

The angle between the tangent to fiber and the x axis is given by,

where a and are the amplitude and wavelength of fiber waviness, respectively. is the maximum amplitude of fiber waviness. is defined as the angle between the tangent to fiber and the x axis. Off-axis stiffness referred to the xyz coordinates is obtained from the transformation relation with on-axis stiffness and fiber

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orientation If ultrasonic wave propagates in the x-y plane, Christoffel's equation is express as follows

and

where is wave normal vector. For a given wave normal direction, phase velocities v and its polarization vector are obtained as below from Eq 2.

Due to shear coupling in anisotropic media, the wave polarized in the x-y plane has pure-mode only if wave normal is parallel to the principal axes of materials. Hence, in the case of wave normal lie in the x-y plane, the longitudinal and transverse waves polarized in the x-y plane have quasi-mode coupled each other. But, the transverse wave vertically polarized to the x-y plane has pure-mode. Waves that have quasi-mode actually propagate along the direction of group wave with group velocity. The group velocities of quasi-longitudinal and quasi-transverse waves can be determined by known phase velocity and polarization vector [7].

where is slowness vector. Because the elastic properties change continuously as wave propagates, an analytical model is proposed to consider the refraction of wave as below.

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For incident wave with the elastic properties of the ray point, two refracted waves and two reflected waves at the ray point are considered to calculate the propagation direction and energy of wave. Based on Snell's law, the wave normal of refracted or reflected wave is obtained from Christoffel's equation in terms of slowness[8]. The group waves of all waves are also obtained using Eqn 3 at the ray point with each phase velocity and its polarization vector. Finally, the ray path is given by

where superscript i is the step number and is a discrete ray increment. angle of group velocity determined by fiber orientation angle. 3.

is the

Experimental Procedures

The materials investigated in this study were DMS 2224 graphite/epoxy composites (Hexcel, Inc.). Ultrasonic inspections were conducted on specially fabricated thick composite specimens with uniform fiber waviness. The fiber waviness ratios were 0.011, 0.034 and 0.059. The standard composite specimens without fiber waviness were fabricated for comparing experimental results with those with fiber waviness and also used for the characterization by conventional wave velocity measurement method. The elastic constants of standard specimen listed in Table 1 were used for input data for the numerical ray tracing model as on-axis stiffnesses.

Pulser/reciver (5072PR, Panametrics Inc.) and longitudinal transducers (10MHz central frequency, Panametrics Inc.) were operated in through-transmission mode as shown in Figure 3. The captured ultrasonic signals in oscilloscope (ScopeStation 140, LeCroy Inc.) were stored into the personal computer via GPIB bus.


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For a given position of transmitting transducer, waveforms were recorded as receiving transducer moved along x axis at 1mm interval. The energy P of received wave was obtained by integrating the magnitude of signal s(t) as below.

For a particular position of transmitting transducer, received energy of wave at various positions were represented as relative values to the maximum value of P.

4.


The numerical simulations were conducted for the waviness model with various fiber waviness ratios. Incident wave was normal to the surface of model and wave source was considered as a point-like source. The ray paths were traced for each wave mode. Numerical results are shown in Figures 4 and 5. As the results of simulation, insonified waves at different locations propagate toward adjacent concave region of fiber waviness.

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Waves insonified at various bottom positions were detected at the opposite side of the specimen. It was observed that the energy of wave converged to the adjacent peak of fiber waviness. For the standard specimen without fiber waviness, waves with the maximum energy were detected at the same location in the x-axis as that of transmitting transducer, and energy distribution of received energy was symmetric to it. The location where maximum energy of wave was detected moved toward the concave region of fiber waviness as the degree of fiber waviness increased. It is expected from the predicted results. This tendency is obviously shown in the contour plots of energy distribution of received wave. The regions checked by ellipse in Figure 6 are the locations where maximum energy of wave is detected for various positions of transmitting transducer.


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The predicted traveling time and measured traveling time of wave are shown in Figure 7. The traveling time of wave was measured at the location where the maximum energy was detected. As the degree of fiber waviness increased, variation of traveling time was also increased. Some discrepancies were found between the experiments and prediction, but tendency of traveling time obtained by the experiments was similar to that of predicted results. These discrepancies might be caused by assumption that wave-source is point-like source.

5.

Conclusions

Wave propagation in thick composites with fiber waviness was studied by considering wave path and traveling time. From the numerical simulations, it was predicted that insonified waves at different locations propagated toward the adjacent concave region of fiber waviness. This tendency was confirmed by ultrasonic tests in through-transmission operation. If the maximum energy of received waves is detected at the same locations in the x-axis for both transmitting and receiving transducers, it indicates convex or concave peak of fiber waviness. The wavelength of fiber waviness is determined quantitatively by the relative distance between the peaks. By

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considering the energy distribution of received waves for a position of transmitting transducers, concave or convex region of fiber waviness is distinguished. The predicted traveling time also showed good agreement with the experiments and can be correlated with the degree of fiber waviness. This study of wave propagation in thick composites with fiber waviness can suggest possibilities to detect the presence and degree of fiber waviness in composites nondestructively. 6.

References

1.

H. -J. Chun, J. -Y. Shin and I. M. Daniel : Nonlinear Behaviors of Thick Composites with Uniform Fiber Waviness, AIAA Journal, Vol. 38, No. 10 (2000), 1949-1955. Heoung-Jae Chun : Flexural behavior of thick composites with fiber waviness, proc. of first Asian-Australasian Conference on Composite Materials(ACCM-1) (1998), 421-424. Shi-Chang Wooh, Isaac M. Daniel : Wave propagation in composite materials with fibre waviness, Ultrasonics, Vol. 33, No. 1 (1995), 3-10. Joseph S. McIntyre, Charles W. Bert, Ronald A. Kline : Wave propagation in a composite with reinforcing fibers, Review of Progress in Quantitative Nondestructive Evaluation, Vol. 14 (1995), 1311-1318. Kwang Yul Kim, Wei Zou, Wolfgang Sachse : Wave propagation in a wavy flber-epoxy composite material : Theory and experiment, J. Acoust. Soc. Am., Vol. 103, No. 5 (1998), 22962301. Isaac M. Daniel, Ori Ishai: Engineering mechanics of composites materials, Oxford University Press, 1994. Fedor I. Fedorov : Theory of elastic wave in crystals, Plenum Press, New York, 1968. Edmund G. Henneke II : Reflection-refraction of a stress wave at a plane boundary between anisotropic media, J. Acoust. Soc. Am., Vol. 51 (1972), 210-217. S. I. Rokhlin, T. K. Bolland, Laszlo Adler : Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media, J. Acoust. Soc. Am., Vol. 79, No. 4 (1986), 906-918.

2. 3. 4.

5. 6. 7. 8.

9.

DEVELOPMENT OF A DRY-CONTACT ULTRASONIC TECHNIQUE AND ITS APPLICATION TO NDE OF IC PACKAGES H. TOHMYOH and M. SAKA Department of Mechanical Engineering Tohoku University 01, Aoba, Aramaki, Aoba-ku, Sendai 980-8579 Japan

Abstract

A dry-contact technique for transmitting higher frequency components of ultrasonic wave is proposed to accomplish an inspection which requires higher resolution without wetting the tested parts in water. In this technique, a plastic film is inserted between the water and the tested parts, and the interfacial continuity is improved by decreasing the pressure between the film and the tested parts. From both the experiment and the calculation used acrylic resin, we verified that the dry-contact technique achieves higher resolution similarly to the conventional immersion technique. The dry-contact technique was applied to the inspection of a delamination in IC package, and the delamination was imaged clearly without wetting the package. 1. Introduction

Sensitive nondestructive testing methods that can achieve higher resolution are strongly demanded in the electronic industry with miniaturization of the integrated circuit (IC) packages [1]–[3]. In point of higher resolution, ultrasonic method is superior to other nondestructive testing methods. However, conventional ultrasonic testing is performed by immersion of tested parts in water, using various liquid and semiliquid couplants [4]. The use of such couplants is not allowed for IC packages that can absorb moisture and be contaminated. By this reason, X-ray method is widely used to the nondestructive testing of the 1C packages [5]–[8]. Although the X-ray method enables us to test with non-contact, it is difficult to detect a plane defect, such as crack, delamination and so on. On the other hand, laser ultrasonic method [9]–[11] and microwave method [12] also enable us to test with non-contact. But the laser ablation with the generation of the bulk acoustic waves is destructive in the laser ultrasonic method, and the microwave method is not available for the test object through the metallic conductor. Recently, air-contact ultrasonic technique that transmits an ultrasound by the medium of air has been studied, but higher frequency components of ultrasound are not available by the technique under the existing conditions [13]. 443 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 443–454. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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In this study, a new dry-contact technique for transmitting higher frequency components of ultrasound is proposed. In this technique, by inserting various thin plastic films, such as polyvinylidene chloride, polyvinyl chloride and polyethylene, between the water as the medium and the tested parts, ultrasonic testing is performed without wetting the tested parts in water. In this technique, it is a key to remove an air gap on the contact area between the plastic film and the tested parts to achieve higher resolution. The air gap is formed by the surface roughness of the tested parts and the elastic deformation of the film and the tested parts due to ultrasonic transmission. The air gap was removed by decreasing the pressure between the film and the tested parts with a vacuum pump. At last, we succeeded in transmitting higher frequency components of ultrasound into acrylic resin by the new dry-contact ultrasonic technique, similarly to the conventional immersion technique. Furthermore, the lateral resolution on the back-wall of the acrylic resin was estimated by calculating the beam intensity formed at the focal plane by the dry-contact technique. The calculation results and the experimental ones showed that the drycontact technique has high lateral resolution as the conventional immersion technique. The dry-contact ultrasonic technique was applied to nondestructive evaluation (NDE) of the IC package. The delamination in the IC package was detected clearly without wetting the IC package. 2. Proposal of a Dry-Contact Ultrasonic Technique

To carry out ultrasonic testing without wetting the tested parts, some attempts have been made [14]. In these studies, by inserting various elastic membranes, such as rubber sheet, between the water as the medium and the tested parts, ultrasonic testing was performed [15]–[17]. However, signal intensity was not sufficient, and the testing which requires high resolution is not yet accomplished. The main reason is considered to be due to the ultrasonic attenuation in the elastic membrane and disagreement of the acoustic impedance between the water and the membrane. In addition to these two factors, we took note of that the interfacial continuity between the membrane and the tested parts must be kept during ultrasonic transmission to achieve higher resolution. Based upon the above-mentioned knowledge, a new dry-contact ultrasonic technique (DCUT) is proposed. Figure 1 illustrates the principle of DCUT. In this technique, by inserting various thin plastic films, such as polyvinylidene chloride (PVDC), polyvinyl chloride (PVC) and polyethylene (PE), between the water as the medium and the tested parts, ultrasonic testing is performed without wetting the tested parts in water. Furthermore, to improve the interfacial continuity between the film and the tested parts, the pressure of the interface between the film and the tested parts was decreased by a vacuum pump. In the vacuumizing process, it is important on the uniform interface formation to control the exhaust path of the air. The exhaust path was controlled with a path control layer between the film and the tested parts. Details of the layer are shown in Figure 2. The path control layer has eight radial holes on the circumference and eight perpendicular holes, and the exhaust path was controlled by the radial holes. The tested parts were kept under the path control layer by the perpendicular holes.

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3. Selection of Plastic Film

3.1. CHARACTERISTICS OF PLASTIC FILMS

To achieve higher resolution by DCUT, selection of plastic film inserted between the water and the tested parts is important. Optimum plastic film was considered from the viewpoint of resolution. A PVDC, a PVC and two kinds of PE films were used. One PE film was not added additive, and another was added additive which promoted cohesion between the film and the tested parts. In the following, the former is denoted by PE1, and the latter is denoted by PE2.

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On the selection of plastic film, two experiments were carried out. Firstly, a reflective plate of stainless steel was placed in a water bath, and the frequency spectrum of the surface echo from the reflective plate, was recorded in the case of inserting a plastic film between the transducer and the reflective plate. Measurement setup used in this experiment is shown in Figure 3. Broadband ultrasonic transducer V390-SU/RM (Panametrics, Inc.) was used, where it had the nominal center frequency 50.0MHz, the diameter of the piezoelectric element and the focal length The frequency spectra of the surface echo with inserting PVDC, PVC, PE1 and PE2 films, and the frequency spectrum without inserting plastic film are shown in Figure 4. In addition, the characteristics of the plastic films are summarized in Table 1. Figure 4 shows that the decreasing in spectrum intensity caused by inserting PVDC film is remarkable in comparison with the cases of inserting the other plastic films. Here, signal loss caused by inserting plastic film, is calculated as a function of the frequency, by the following equation:

The quantity represents the sum of the signal loss caused by the ultrasonic attenuation in plastic film and the signal loss caused by the difference of the acoustic impedance between the film and the water. Figure 5 shows for various plastic films. The absolute value of of two PE films and PVC film are smaller than that of PVDC film in available frequency range, and that of PE1 film is the smallest in the four films.


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Next, back-wall echo of the acrylic resin with 2.0mm in thickness was recorded by DCUT. Back-wall echoes of the acrylic resin received by DCUT using various plastic films, and its frequency spectra, are shown in Figures 6 and 7, respectively. Figure 6 shows that the signal with sufficient intensity, which is similar to the conventional immersion technique, is received by DCUT using various plastic films. However, fairly large discrepancy is notable among the frequency spectra for every plastic film. Here, actual signal loss at the back-wall of acrylic resin, [dB], is calculated by the following equation:

The quantity represents the actual signal loss in case of DCUT. Figure 8 shows for various plastic films. Especially, the decrease of higher frequency components in PE1 film whose absolute value of was the smallest in the four films is remarkable. The fact indicates that the plastic film which is superior on acoustical characteristics, for example, ultrasonic attenuation and acoustic impedance, cannot necessarily be superior on transmitting higher frequency components of ultrasonic wave to the tested parts in DCUT. It is suggested that the condition of the interface between the plastic film and the tested parts affects significantly the transmission of higher

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frequency components. The decrease of higher frequency components may be called frequency filter effect. Figure 9 illustrates the frequency filter effect. The ultrasonic waves can well pass through the continuous interface between the plastic film and the tested parts [Figure 9 (a)], but the waves hardly pass through the interface with air gaps. The air gaps are formed by the surface roughness of the tested parts [Figure 9 (b)] and the elastic deformation of the film due to ultrasonic transmission [Figure 9 (c)] and so on. The frequency filter effect between the film and the tested parts is considered due to a disorder of the interfacial continuity. In fact, Figure 8 shows that the absolute value of of PE2 which is added additive for promoting cohesion is much smaller in higher frequency range than that of PE1 without additive, and it is clear that the disorder of the interfacial continuity between the plastic film and the tested parts is improved by the additive.


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3.2. LATERAL RESOLUTION OF DRY-CONTACT ULTRASONIC TECHNIQUE

By using the experimentally recorded frequency spectra, let us now consider on the lateral resolution on the back-wall of the acrylic resin theoretically, and select a plastic film to achieve higher resolution. The lateral resolution is estimated by calculating the beam intensity formed at the focal plane. In case of pulse echo mode, the normalized beam intensity formed at the plane focused by a spherical lens to a spot, is given by [18]

where is Bessel function of the first kind and first order, the wavenumber, C the ultrasonic velocity ( in water), the radial distance from the center axis of the lens and calculated from Equation (3) is valid when In case of 50MHz transducer, at a plane focused in water is shown in Figure 10. On the other hand, by using lateral resolution, is defined as the separation distance between the two sound sources which can be distinguished clearly, and is given by the following equation empirically:

where is the wavelength, and In case of using a broadband transducer, the transformation of the frequency components in ultrasonic waves at the focal plane must be considered. This was done by multiplying by the frequency spectrum at the focal plane [19], The improved beam intensity at the focal plane, is given by the following equation:

where is the lower frequency limit and the upper frequency limit. As an example, in case of DCUT using PVC film is shown in Figure 11. On the other hand, according to the definition of must satisfy the following equation:

The response of ultrasound to two sound sources in case of DCUT using PVC film is shown in Figure 12. When the separating distance between two sources equals to each sources is barely resolved [Figure 12 (b)], while the separation distance less than makes it impossible to resolve each sources [Figure 12 (a)]. If the separating distance is greater than each sources is clearly resolved [Figure 12 (c)].

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The lateral resolution on the back-wall of the acrylic resin in case of using the broadband transducer was estimated by Equations (5) and (6). In addition, by inserting the peak frequency of the frequency spectrum at the focal plane into Equation (4), empirical resolution for reference was calculated, too. Both values of calculated lateral resolution are shown in Table 2 respectively. The lateral resolution achieved by DCUT using PVDC and PVC films are 68 and 66µm, respectively. Each lateral resolution is near to that of the conventional immersion technique (53µm), and enough high for applying to NDE of the IC package.


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3.3. CONFIRMATION OF LATERAL RESOLUTION

To confirm the lateral resolution calculated from Equations (5) and (6), the groove pattern introduced by a slicing machine to the back-wall of the acrylic resin with 2.0mm in thickness was imaged by DCUT. The dimensions of the groove pattern at the backwall of the acrylic resin and the optical view are shown in Figures 13 (a) and (b) respectively. C-scan mode image of the groove pattern obtained by the conventional immersion technique is shown in Figure 14 (a), and C-scan mode images of the groove pattern by DCUT using PVDC and PVC films are shown in Figures 14 (b) and (c), respectively. The scan pitch was 0.01mm.

Figure 14 (a) shows that the narrowest projection that is clearly resolved by the conventional immersion technique is of 60µm in width. Figures 14 (b) and (c) show that the same projection as the above is clearly resolved by DCUT using PVDC and PVC films. From the experimental facts, it was confirmed that DCUT has high lateral resolution as the conventional immersion technique. In addition, the value of 60µm in width resolved by DCUT and the conventional immersion technique agrees with the lateral resolution estimated in the previous section, and the validity on the estimation of lateral resolution using Equations (5) and (6) is confirmed. Based on the discussion made in the previous and this sections, PVDC and PVC films were selected for the use in DCUT to NDE of IC package. These films are two which can achieve higher resolution in four films examined.

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4. Application to NDE of IC Package DCUT was applied to the inspection of a delamination in a quad flat package (QFP). The examined QFP contained a delamination between the silicon chip and the metallic lead frame in the plastic encapsulant. The delamination was inspected from the side of silicon chip in C-scan mode imaging. C-scan mode image of the delamination in the QFP obtained by the conventional immersion technique is shown in Figure 15 (a), and C-scan mode images of the delamination by DCUT using PVDC and PVC films are shown in Figures 15 (b) and (c), respectively. The scan pitch was 0.20mm. Figure 15 (b) and (c) show that the delamination part is clearly discriminated from the part without delamination. Especially, in the C-scan mode image by DCUT using PVC film shown in Figure 15 (c), the lead frame pattern is also imaged clearly, and the image closely resembles the one by the conventional immersion technique shown in Figure 15 (a). At last, by DCUT achieved higher resolution, we succeeded in testing an IC package without wetting the package in water. On the other hand, while DCUT using PVDC film discriminates the delamination part from the part without delamination, the lead frame pattern is not so clear. The cause is considered due to a disorder of the interfacial continuity between the PVDC film and the package.


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5. Conclusions A dry-contact technique for transmitting higher frequency components of ultrasonic wave was proposed. In this technique, a plastic film was inserted between the water and the tested parts, and the interfacial continuity was improved by decreasing the pressure between the film and the tested parts. From both the experiment and the calculation used acrylic resin, it was confirmed that the dry-contact technique achieves higher resolution similarly to the conventional immersion technique. Finally, the dry-contact technique was applied to the inspection of a delamination in IC package, and the delamination was imaged clearly without wetting the package in water. 6.

Acknowledgements

The authors wish to acknowledge Mr. M. Mikami of Tohoku University for his help in the experiment. This work was partly supported by Japan Society for the Promotion of Science under Grant-in-Aid for Scientific Research (B)(2) 12450041. 7.

References

1. Tummala, R.R. (2001) Fundamentals of Microsystems Packaging, McGraw-Hill, New York, 44–79. 2. Baba, S., Tomita, Y., Matsuo, M., Matsushima, H., Ueda, N. and Nakagawa, O. (1998) Molded chip scale package for high pin count, IEEE Trans. Comp., Packag., Manufact. Technol.–Part B, 21, 28–34. 3. Amagai, M. (1999) Characterization of chip scale packaging materials, Microelectronics Reliability, 39, 1365–1377. 4. Gilmore, R.S. (1996) Industrial ultrasonic imaging and microscopy, J. Phys. D: Appl. Phys., 29, 1389–1417. 5. Ikeda, Y., Mizuta, Y. and Hirashima, R. (1999) Radiography testing for micro defects with a psudo-3D X-ray TV system, Proceedings of the Fifth Far-East Conference on Nondestructive Testing, 361–368. 6. Rooks, S.M., Benhabib, B. and Smith, K.C. (1995) Development of an inspection process for ball-gridarray technology using scanned-beam X-ray laminography, IEEE Trans. Comp., Packag., Manufact. Technol.–Part A, 18, 851–861.

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7. Neubauer, C. (1997) Intelligent X-ray inspection for quality control of solder joints, IEEE Trans. Comp., Packag., Manufact. Technol.–Part C, 20, 111–120. 8. Sankaran, V., Kalukin, A.R. and Kraft, R.P. (1998) Improvements to X-ray laminography for automated inspection of solder joints, IEEE Trans. Comp., Packag., Manufact. Technol.–Part C, 21, 148–154. 9. Yamanaka, K. (1997) Precise measurement in laser ultrasonics by phase velocity scanning of interference fringes, Jpn. J. Appl. Phys., 36, 2939–2945. 10. Nishino, H. and Tsukahara, Y. (1993) Excitation of high frequency surface acoustic waves by phase velocity scanning of a laser interference fringe, Appl. Phys. Lett., 62, 2036–2038. 11. Lafond, E., Coulette, R., Grand, C., Nadal, M.-H., Dupont, B., Lepoutre, F., Balageas, D. and Petillon, O. (1998) Application of a two-layer semi-analytical model for the improvement of laser-ultrasonic generation, NDT&E International, 31, 85–92. 12. Ju, Y., Saka, M. and Abé, H. (2001)NDI of delamination in IC packages using millimeter-waves, IEEE Trans. on Instrum. Meas., 50, in press. 13. Manthey, W., Kroemer, N. and Magori, V. (1992) Ultrasonic transducers and transducer arrays for applications in air, Meas. Sci. Technol., 3, 249–261. 14. Rogovsky, A.J. (1991) Development and application of ultrasonic dry-contact and air-contact C-scan systems for nondestructive evaluation of aerospace composites, Materials Evaluation, 1491–1497. 15. Liaw, P.K.., Hsu, D.K.., Yu, N., Miriyala, N., Saini, V. and Jeong, H. (1996) Investigation of metal and ceramic-matrix composites moduli: experiment and theory, Acta mater., 44, 2102–2113. 16. Im, K.-H., Hsu, D.K. and Jeong, H. (2000) Material property variations and defects of carbon/carbon brake disks monitored by ultrasonic methods, Composites: Part B, 31, 707–713. 17. Roth, D.J., Riser, J.D., Swickard, S.M., Szatmary, S.A. and Kerwin, D.P. (1995) Quantitative mapping of pore fraction variations in silicon nitride using an ultrasonic contact scan technique, Res. Nondestr. Eval., 6,125–168. 18. Canumalla, S. (1999) Resolution of broadband transducers in acoustic microscopy of encapsulated ICs: transducer selection, IEEE Trans. Comp. Packag. Technol., 22, 582–592. 19. Bechou, L., Ousten, Y., Tregon, B., Marc, F., Danto, Y., Even, R. and Kertesz, P. (1997) Ultrasonic images interpretation improvement for microassembling technologies characterisation, Microelectron. Reliab., 37, 1787–1790.

6. Neutron Diffraction and Synchrotron Radiation Methods


HIGH-RESOLUTION NEUTRON DIFFRACTION TECHNIQUES FOR STRAIN SCANNING P. MIKULA, M. VRANA, P. LUKAS Nuclear Physics Institute 250 68 Rez, Czech Republic V. WAGNER Physikalisch-Technische Bundesanstalt Bundesallee 100, 38116 Braunschweig, Germany Abstract Several high-resolution neutron diffraction techniques using Bragg diffraction optics based on cylindrically bent perfect crystals which have been tested with the aim of their efficient exploitation for strain/stress are summarized. Presented experimental results demostrate their experimental abilities for nondestructive testing and evaluation of a phase and deformation response of polycrystalline materials on residual stress or deformation loading. Due to the unique resolution and good luminosity achieved by focusing in real and momentum space, besides the macrostrain scanning, the presented techniques are also qualified for microstrain/stress studies by an analysis of the diffraction profiles. At the medium flux reactor LVR-15 in NPI there are operating two focusing double axis strain/stress diffractometers. Good luminosity of the diffractometers and a sufficiently high-resolution (FWHM of the instrumental - profile can be less than at nm) permit investigations of both the macro- and microstrains in the sample gauge volumes of several cubic millimetres with a sensitivity in the relative peak shift of several rad.

1. Introduction Residual stresses or their development under applied external force can have a strong influence on their basic mechanical properties. They displace atoms from their original positions in a crystalline material and neutron diffraction along with X-ray diffraction can nondestructively reflect the resulting change of lattice constants with a sufficiently high precision by using the Bragg Diffraction Angle Analysis (BDAA) and/or by Energy-Dispersive Neutron-Transmission Diffraction (EDNTD). Conventional neutron strain/stress scanners using the BDAA method (based on Bragg condition lattice distance, angle, wavelength) are in fact powder diffractometers equipped with a mosaic monochromator and a system of Soller collimators and for high-resolution measurements they usually suffer from required luminosity. However, the dedicated diffractometers using Bragg diffraction optics provide luminosity by one order of 457 E. E. Gdoutos (ed. ), Recent Advances in Experimental Mechanics, 457–466. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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magnitude higher with a substantially better resolution (FWHM of the instrumental can be of the order of [1-7]. The EDNTD method is based on the measurement of a decrease of a beam intensity transmitted through a sample in the dependence on the wavelength in a -range in the vicinity of the Bragg cut-off. The edge in this case can be observed when passing through the limit ( is the lattice spacing), below which particular reflection planes (hkl) begin to scatter neutrons. It means that no angular dependence of scattering is measured and only integrated intensity of this particular reflection can be determined for the transmitted beam for each value of Mathematically, edge profile can be described by a Gaussian cumulative function (FWHM of the instrumental can be of the order of As will be presented, using the Bragg diffraction optics the EDNTD measurements, usually carried out only at the pulsed neutron beams [8], can be effectively carried out even at a medium power reactors [3, 5, 9-12]. The excellent properties of focusing strain scanners permits one investigations of both the macrostrains with the sensitivity to changes down to as well as microstrains by a peak profile or Bragg edge analysis in the gauge volumes of several mm 3 . In principle, the substructural changes of e. g. dislocation density and average grain size in engineering components after use can be investigated in situ, under loading by an external force. The stresses are not measured directly by diffraction techniques, but one measures strains defined as corresponds to the unstrained material), which are then converted to stresses using appropriate moduli. Then, the shift in the Bragg angle or of the position of the Bragg edge with respect to the strain-free material permits one to determine the average macrostrain over the irradiated gauge volume. Information on the microstrain (in the form of as well as dislocation density and grain size present in the irradiated gauge volume can be determined from a change of the width and the form of the diffraction peak or edge profile [10, 13-17]. During the last years we tested several modifications of the high resolution BDAA and EDNTD performances based on Bragg diffraction optics, which were developed within the collaboration between NPI Rez and PTB Braunschweig. In what follows, their advantages, drawbacks and properties are introduced.

2. Two axis performance, BDAA method The most spread high-resolution neutron diffraction two-axis performance (see Figure 1) consists basically of the following steps and properties [7, 18]: a. Monochromatic neutrons selected by a bent monochromator from the white spectrum are focused on a sample (real space focusing) making the luminosity of the device high. b. The monochromatic beam is diffracted by a chosen volume element (determined by a pair of input and output slits) into the angle c. There is a strong correlation between the divergences and as

which all can be easily manipulated by changing the monochromator radius is the total change of the angle of incidence (exit) over illuminated crystal

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length and value of the radius

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is the dispersion parameter). By setting a proper

and a quasiparallel and highly luminous detector signal can by adjusted for a chosen scattering angle Contrary to the conventional powder diffractometers, the minimum resolution e. g. at can be achieved even for monochromator take off angles far below 90°. d. The quasiparallel diffracted beam is directly analyzed by using a 1D-PSD. There are small resolution uncertainties and influencing the instrumental resolution which come from a nonnegligible thickness of the monochromator and from the finite width w of the irradiated volume of the sample determined by the input and output slits. Thus, for a point like sample, the monochromator of the thickness and the width of the sample w,

Both of them bring about the diffracted beam slightly divergent (quasiparallel) and directly determine the instrumental resolution. It is clear that it can be easily estimated and adjusted according to the experimental requirements [5]. e. No Seller collimators are required.

Of course, the total resolution of the instrument is strongly dependent on the spatial resolution of the 1D-PSD. The employment of the Bragg diffraction optics resulted in an improvement of the luminosity and/or resolution of our dedicated strain scanners several times in comparison with the conventional counterparts. Moreover, recently, one of our two axis strain scanners has been equipped with a two-crystal-slab sandwich monochromators (combinations of Si(111)+Si(220) or Si(111)+Ge(311) reflections) which permit us to work simultaneously with two neutron wavelengths of 0. 27 nm + 0. 165 tun or 0. 27 nm + 0. 143 nm, respectively [19]. This fact enables us to investigate more sample-reflection profiles under the same experimental conditions within a range of covered by one angular setting of the PSD. Thanks to a remote control of the bending device and a remote

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manipulation of the crystal curvature, the optimum focusing conditions can be achieved practically for any scattering angle As a result of a compromise between the resolution and the luminosity, the two axis performances used in our laboratory work with the resolution derived from the FWHM of the diffraction profiles taken with a well annealed standard samples.

3. Three axis performance, BDAA method The three axis performance is along with the cylindrically bent perfect monochromator equipped also with a cylindrically bent analyzer set either in symmetric/slightly asymmetric diffraction geometry (see Figure 2) [1-3, 5, 6] or in the fully asymmetric diffraction geometry in combination with 1D-PSD (see Figure 3) [2, 9, 20]. The analysis of the beam diffracted by the irradiated sample volume is in these cases performed in momentum space.

In the performance sketched in Figure 2 a high-resolution can be achieved when the analyzer is optimally curved, thus, all correlated rays coming within from the sample fulfil the Bragg condition on the analyzer simultaneously (focusing in momentum space). Then, the dispersion over the whole diffraction device is minimized. In case of the slit-like sample it can be simply expressed in the simple form [6]

where is the monochromator-sample distance and is the sample-analyzer distance. If the three-axis setup is optimized for the slit-like sample a non-negligible width of the sample Y introduces a resolution uncertainty


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In comparison with the two axis performance, the resolution of the detector has no influence on the resolution device and a high resolution can be achieved with the analyzer situated relatively close to the sample (about 30-40 cm). Thanks to a high peak reflectivity of the bent perfect crystals luminosity of the performance is still high. The drawback of the performance consists in a necessity of the step-by-step scanning (by rocking the analyzer) of the diffraction profiles. The performance sketched in Figure 3 overcomes the problem of the step-bystep scanning. A slightly divergent output beam from the sample is analyzed also by the optimally chosen bent analyzer [9]. However, thanks to the special asymmetric diffraction geometry of the crystal, the angular scale is transformed on the linear scale of the 1D- PSD. Such analysis in the momentum space makes possible to place the analyzer close to the sample and to cover relative large area of the Debye Scherrer cone without affecting the angular resolution which results in a favourable detector signal. Contrary to the two axis counterpart, in the case of three axis performances, there are much smaller blurring effects influencing the instrumental resolution. It has been demonstrated that the three axis performances can work with the resolution even better than For optimization of this three axis performance a more detailed treatment of individual rays coming from the sample with respect to the two dimensional lattice deformation in the cylindrically bent crystal is required. A detailed procedure can be found in refs. 2, 9 and 21.

4. One axis performance, EDNTD method The performance sketched in Figure 4 is the simplest one and by adjusting the monochromator take-off angle it can be used in a large range of the neutron wavelengths. A strong angular-wavelength correlation in the beam passing through the slit can be expressed in the form as [12, 18]

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where

is an angular deviation of a neutron trajectory from the central beam and is the value of the edge position (on the related to the lattice planes (hkl). The resolution uncertainties and which come from the effective mosaicity of the bent perfect crystal and the width of the slit w, respectively, can be derived in the form [12]

They bring about the correlation imperfect and thus have an influence on the total smearing of the Bragg edge as seen by PSD. The parameter was derived for the case of the monochromator crystal thickness For an effective slit width should be introduced. The parameter was derived for the case of an unlimited collimation of the incident polychromatic beam. However, in practice its impact can be considerably reduced for well collimated beam in a guide tube having a divergence For we arrive at the estimation Using a well annealed Fe(321) standard sample, a 4 mm thick monochromator and 2 mm wide slit, the instrumental resolution (represented by the width of the edge profile) was about of

5. Two axis performance, EDNTD method Similarly to the case of BDAA method sketched in Figure 5 the analyzer in fully asymmetric diffraction geometry can be also effectively used for EDNTD scanning [5, 11, 15]. Its advantage consists in a possibility to place the analyzer close to the sample and also in minimizing the influence of the slit width w and spatial resolution of the PSD on the FWHM of the edge profile. The drawback of this performance is in the necessity of using a special crystal cut for each mean wavelength For obtaining maximum amplitude we investigated the Bragg edge corresponding to reflection with a high multiplicity. To achieve a sufficiently high resolution with a reasonable counting rate the reflection combination Si(220)/polycrystalline working in the vicinity of nm was chosen. Using the standard sample and the 3 mm slit


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width, for crystals of the thickness of 5 mm and bending radii of m and m the instrumental resolution was of about Recently performed Monte Carlo simulations indicate that using properly chosen bent crystals, the -resolution of can be easily reached. High resolution and good luminosity of this performance permits us macrostrain scanning derived from the of the Bragg edge with the sensitivity of 10-5 [11].

The principal drawback of the previous performance can be overcome by using the dispersive double-bent-crystal setting as schematically sketched in Figure 6 [3, 22]. It can be adjusted practically for any mean and the Bragg edge scanning is performed either by rocking the analyzer scan) or by simultaneous rocking of both crystals with respect to the incident polychromatic beam [3, 22]. However, step-by-step scanning makes the measurement rather long.

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In all cases of the EDNTD method the following characteristics have been observed: a. The amplitude of the Bragg diffraction edge depends on the thickness of the sample. b. The steepness and the shape of the Bragg diffraction edge is determined dominantly by the distribution. c. Steepness of the edge can change with the thickness of the sample due to the multiple reflection effect. d. The resolution of the instrument permits to study macrostrain effects even below the value of in as well as some of the microstrain characteristics from the edge profile analysis. e. The EDNTD techniques are limited to measurements of only one strain component parallel to the axis of the incident monochromatized beam.

6.

Discussion

This paper briefly reports about achievements of neutron diffraction groups of NPI and PTB Braunschweig in the field of residual stress/strain instrumentation developments. All presented performances are based on Bragg diffraction optics with cylindrically bent perfect single crystals. In comparison with the conventional strain scanner equipped with mosaic monochromator and a system of Soller collimators its employment enabled us to improve the instrumental resolution and increase the detector signal, simultaneously. Thus, at present, the strain/stress measurements can be efficiently carried out even at the medium power reactors. The only parameters which influence the instrumental resolution are the effective mosaicity (linearly dependent on the thickness) of the bent perfect monochromator, widths of the slits and the spatial resolution of the position sensitive detector. Therefore, the resolution measured by FWHM of the diffraction profile or Bragg edge is a matter of a choice of these parameters with respect to the flux of the neutron source and the resulting detector signal. For further increase of the detector signal, in some cases a doubly-bent monochromator using also vertical focusing (which has no influence on the resolution) can be used [23]. As can be seen from presented sketches the BDAA as well as EDNTD performances can be easily adapted to any conventional neutron diffractometer providing the appropriate wavelength. Monte Carlo simulations of resolution function and a detector signal can provide a great help in the course of the optimization of the experimental performance [24]. Common feature of all performances is a sufficiently high resolution permitting evaluation of microstrains in plastically deformed polycrystalline samples. Of course, information about the strain state of art averaged within the irradiated sample volume is provided. Depending on the reactor power and the sample thickness, the area of the sample slits can be reduced to less than 10 (eventually to less than 1 for high flux neutron sources) still keeping reasonable counting times. Bragg diffraction optics investigations and instrumentation developments are supported by the Grant Agency of the Czech Rep. No. 202/97/K038, Grant Agency of CAS No. A1048003 and the TMR-Network ERB FMR XCT 96-0057.


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7. References 1. 2.

3.

4. 5. 6.

7.

8. 9. 10. 11.

12. 13. 14.

15. 16.

17. 18. 19.

Kulda, J., Mikula, P., Lukas, P., and Kocsis, M.: Utilisation of Bent Si Crystals for Elastic Strain Measurements, Physica B 180&181(1992) 1041-1043. Mikula, P., Lukas, P., Vrana, M., Klimanek, P., Kschidock, T., Macek, K., Janovec, J., Osborn, J. C., and Swallowe, G. M.: Advanced Neutron Diffraction Techniques for Strain Measurements in Polycrystalline Materials, Journal de Physique IV, Collogue C7, 3 (1993) 2183-2188. Vrána, M.., Mikula, P., Lukas, P., Saroun, J., and Strunz, P.: High-Resolution Diffraction Techniques for Strain/Stress Measurements at a Steady State Reactor, Acta Physica Hungarica, 75 (1994) 305-310. Mikula, P., Vrana, M., Lukas, P., Saroun, J., and Wagner, V.: High-Resolution Neutron Powder Diffractometry on Samples of Small Dimensions, Mat. Science Forum 228-231 (1996)269-274. Vrana, M., Klimanek, P., Lukas, P., Mikula, P., Saroun, J., and Wagner, V.: Neutron Optics for Micro- and Macrostrains Investigations, Proc. of the 4th European Conf. on Advance Materials and Processes, EUROMAT 95, 25. -28. September 1995, Padua/Venice, p. 35-40. Vrana, M., Lukas, P., Mikula, P., and Kulda, J.: Bragg Diffraction Optics in High Resolution Strain Measurements, Nucl. Instrum. Methods A 338 (1994) 125-131. Mikula, P., Vrana, M., Lukas, P., Saroun, J., Strunz, P., Ullrich, H. J., and Wagner, V.: Neutron Difftactometer Exploiting Bragg Diffraction Optics - A High Resolution Strain Scanner, Proc. of the 5th Int. Conf. on Residual Stresses ICRS-5, June 16-18, 1997, Linköping, eds. T. Ericsson, M. Oden and A. Andersson, Vol. 2, p. 721-725. Priesmeyer, H. G.: "Transmission Bragg-Edge Measurements", Measurement of Residual and Applied Stress Using Neutron Diffraction, Proc. NATO ARW Oxford, eds. Hutchings, M.T., and Krawitz, A. D., Kluwer, Dordrecht, 1992, p. 389-394. Lukas, P., Kouril, Z., Mikula, P., Saroun, J., Strunz, P., Vrana, M., and Wagner, V.: Bent Crystal Analyzer in Fully Asymmetric Diffraction Geometry for Neutron Scattering Instrumentation, J. Phys. Soc. Japan 65 (1996)241-244. Lukas, P., Kouril, Z., Strunz, P., Mikula, P., Vrana, M., and Wagner, V.: Microstrain Characterization of Metals Using High-Resolution Neutron Diffraction, Physica B 234-236 (1997) 956-958. Wagner, V., Kouril, K., Lukas, P., Mikula, P., and Vrana, M.: Residual Strain/Stress Analysis by Means of Energy Dispersive Neutron Transmission Diffraction, Proc. of 5th Int. Conf. on Applications of Nuclear Techniques "Neutrons in Research and Industry", 9. 6. -15. 6. 1996, Crete, SPIE 2867 (1997) 168-171. Mikula, P., Wagner, V., and Vrana, M., Bragg Diffraction Optics for Energy-Dispersive Neutron-Transmission Diffraction, Physica B, 283 (2000) 403-405. Klimanek, P., Kschidock, T., Mikula, P., and Vrana, M.: Substructure Analysis in Polycrystalline Materials by Means of Neutron Diffraction, Physica B 234-236 (1997), 965966. Lukas, P., Tomota, Y., Harjo, S., Ono, M., Vrana, M., Strunz, P., Kouril, Z., and Mikula, P.: Neutron Diffraction Investigation of Residual Strains Fe-Cr-Ni Alloys, Proc. of the 5th Int. Conf. on Residual Stresses Stresses ICRS-5, June 16-18, 1997, Linköping, eds. Ericsson, T., Odén, M., and Andersson, A.: Vol. 2, p, 880-885. Strunz, P., Lukas, P., Mikula, P., Wagner, V., Kouril, Z., and Vrana, M.: Data Evaluation Procedure for Energy-Dispersive Neutron-Transmission-Diffraction Geometry, Ibidum p. 688-693. Lukas, P., Zrnik, J., Ceretti, M., Vrana, M., Mikula, P., Strunz, P., Neov, D., and Keuerleber, J.: Neutron Diffraction as a Tool for Microstrain Characterization of Polycrystalline Ni-Base Superaloys after Thermal Fatigue, Mat. Science Forum 321-324 (2000) 1048-1050. Neov, D., Sittner, P., Lukas, P., Vrana, M., and Mikula, P.: In Situ High Resolution Neutron Diffraction Study of Stress Induced Martensitic Transformation in CuAlZnMn Shape Memory Alloy, Mat. Science Forum, 347-349 (2000), 334-339. Mikula, P., Kulda, J., Lukas, P., Ono, M., Saroun, J., Vrana, M., and Wagner, V.: Bragg Diffraction Optics in Neutron Diffractometry, Physica B 283 (2000) 289-294. Vrana, M., Mikula, P., Lukas, P., and Wagner, V.: Two Wavelength Sandwich Monochromator for Materials Research Experiments, Physica B 241-243 (1998) 231-233.

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20. Vrana, M., Mikula, P., Lukas, P., Dubsky, J., and Wagner, V.: New Developments in 21. 22. 23.

24.

Instrumentation for Strain Scanning in NPI Rez, Mat. Science Forum 321-324 (2000) 338344. Saroun, J., Lukas, P., Mikula, P., and Alefeld, B.: Optimization of a Double Bent Crystal Diffractometer for Neutron Small-Angle Neutron Scattering Experiments, J. Appl. Cryst. 27 (1994) 80-88. Mikula, P., Vrana, M., Lukas, P., Saroun, J., Strunz, P., Wagner, V., and Alefeld, B.: Bragg Optics for Strain/Stress Measurement Techniques, Physica B 213-214 (1995) 845-847. Wagner, V., Mikula, P., and Lukas, P.: A Doubly Bent Si-Monochromator, Nucl. Instrum. Methods in Phys. Research A 338 (1994) 53-59. Saroun, J., and Kulda, J.: RESTRAX - A Program for TAS Resolution Calculation and Scan Profile Simulation, Physica B 234-236 (1997) 1102-1104, and http: //omega. ujf. cas. cz.

DRAFT STANDARD FOR THE MEASUREMENT OF RESIDUAL STRESSES BY NEUTRON DIFFRACTION G. A. WEBSTER Mechanical Engineering Department, Imperial College Exhibition Road, London, SW7 2BX, UK A. G. YOUTSOS Joint Research Centre, Institute for Energy, High Flux Reactor Unit PO Box 2, 1755 ZG Petten, The Netherlands C. OHMS Joint Research Centre, Institute for Energy, High Flux Reactor Unit PO Box 2, 1755 ZG Petten, The Netherlands R. C. WIMPORY Mechanical Engineering Department, Imperial College Exhibition Road, London, SW7 2BX, UK Abstract Residual stresses can have important consequences for the load carrying capacity and safety of engineering components. Neutron diffraction is a non-destructive method for determining residual stresses in crystalline materials. It is a relatively new technique and no standard is currently available for making these measurements. This paper gives the background to research that has been carried out to develop a standard. It outlines the main findings and indicates the precautions that are required to achieve accurate positioning and alignment of specimens (and components) in a neutron beam and the analysis required to obtain reliable results. It also shows that special attention is needed in dealing with near-surface measurements because of surface aberration. It is demonstrated that, provided the recommended procedures are followed, a positional tolerance of ± 0.1mm can be achieved with an accuracy in strain of to give a resolution in residual stress of ± 7 to 20 MPa in most materials of practical interest.

1. Introduction Residual stresses can be introduced into engineering components during manufacture as a result of, for example, forging, bending and welding processes. They can also be caused by the forces and thermal gradients imposed during operation. These stresses can affect the load carrying capacity and resistance to fracture of components. In order to quantify their effect it is necessary to know their magnitude and distribution [1, 2]. Several destructive and non-destructive techniques are available for determining residual stresses. They include X-ray diffraction [3], neutron diffraction [4, 5], hole drilling [6], slicing [7] and boring [8] methods. In all cases, strains are measured and stresses calculated. However, only the neutron diffraction method is capable of obtaining these stresses non467 E. E. Gdoutos (ed. ), Recent Advances in Experimental Mechanics, 467–476. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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destructively within the interior of components. It is similar to the X-ray technique for surface determinations, but because neutrons are not charged, neutron diffraction can be used to obtain residual stresses non-destructively to a depth of several centimetres in most materials of practical relevance. There is wide spread interest throughout the world in exploiting the technique to examine relatively large components where the consequences of failure could be catastrophic. Neutron diffraction is a relatively new technique and no standard or code of practice is available for making stress measurements by this method. As a consequence two international projects were initiated to carry out the under-pinning research necessary to develop a standard. The first, under the auspices of VAMAS (Versailles Agreement on Advanced Materials and Standards), Technical Working Area 20 (TWA 20) was initiated in January 1996. The second study was supported by a European (EU) project RESTAND (Residual Stress Standard using Neutron Diffraction) which was started in December 1997. The objectives of the two projects were to: -establish accurate and reliable procedures for making non-destructive residual stress measurements by neutron diffraction, -examine a selection of samples in which residual stresses had been introduced by different techniques, -conduct inter-laboratory comparisons to establish reproducibility, -assemble the necessary information for preparing a draft standard for making the measurements. The VAMAS TWA 20 activity involved most of the neutron sources worldwide, which are capable of making the measurements. A series of ‘roundrobin’ specimens including a shrink-fit aluminium alloy ring and plug assembly, a ceramic matrix composite, a nickel alloy shot-peened plate and a ferritic steel weldment were examined. An additional objective of RESTAND was to demonstrate the usefulness of the technique to a range of practical applications and to develop confidence in the method for industry. This paper presents the findings of these studies.

2. Principles of the Technique Neutron diffraction can be used to measure components of strain from changes in lattice spacings in crystalline materials. When illuminated by radiation of wavelength similar to interplanar spacings, crystalline materials diffract this radiation as distinctive Bragg peaks. The angle at which any given peak occurs can be calculated using Bragg’s law of diffraction,

where is the wavelength of the radiation, d is the lattice plane spacing responsible for the Bragg peak and is the Bragg angle. The peak will be observed at an angle of from the incident beam, as shown in Figure 1. Neutrons can be generated in a reactor by fission or at a spallation source. Normally at reactor sources, a continuous monochromatic beam of neutrons is used to determine lattice spacings whereas at spallation sources usually a pulsed polychromatic beam is employed.

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When a monochromatic beam of neutrons of constant wavelength is used, the lattice strain in the direction of the scattering vector Q (Figure 1) from Bragg’s law is given by,

where is the change in lattice spacing from its strain free value and and is the change in Bragg angle from its strain free value When a pulsed polychromatic beam of neutrons is used, measurements are made at a constant angle (usually 90°) and the time of flight of the neutrons is recorded. From de Broglie’s relation,

where h is Planck’s constant, m is the mass of a neutron and L is the path length of the neutrons. Therefore substitution of this equation into eq. (1) and differentiation at constant gives lattice strain as,

where is the change in time-of-flight from its strain free value Consequently either eqs. (1) or (4) can be employed for determining lattice strains.

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As neutron diffraction only measures elastic strains, stresses can be obtained from the elastic properties of materials. Since both stress and strain are tensors, in general measurements in six directions at a point are required to completely define the stress state there. However, when the principal stress directions are known 3 orientations will suffice. In this case, when the principal directions coincide with the coordinate measurement directions x, y and z the principal stresses and in terms of the coordinate strains become,

where E is the elastic modulus and v Poisson’s ratio. In practice, it is often possible to infer the principal directions from a knowledge of the fabrication procedures or loading conditions imposed on a component.

4. Round Robin Measurements The round robin samples were chosen to represent a range of situations of practical importance. Ring and plug sample: The ring and plug sample (Figure 2) was chosen as the first to be measured because of its axial symmetry and the fact that residual stresses could be produced elastically by the adoption of an interference fit. It also had the attraction of introducing a relatively simple residual stress pattern that has an analytical solution. Two nominally identical samples were prepared, one for distribution predominantly through Europe #1 and the other #2 elsewhere. Measurements were made at both reactor and pulsed neutron sources to investigate, respectively, the benefits of using a monochromatic or polychromatic beam of neutrons. In all cases an experimental protocol was specified for obtaining the strains and eqs. (5) used for calculating stresses from the strain measurements. The findings are contained in VAMAS report no. 38 [9] and are summarised here. The studies on the ring and plug assembly established the basic procedures that should be followed. Measurements were made across the diameter of each sample as shown in Figure 2. It was found that it is essential to ensure accurate positioning and alignment of a specimen in the neutron beam for reliable results to be obtained. A suitable shape and size of ‘gauge volume’ over which individual measurements should be made to achieve adequate resolution in regions of strain gradients has been identified. It is recommended that a minimum of is adopted to encompass sufficient grains and to give neutron count times that are not excessive. For the measurements in the axial orientation, it has been found that a cube shaped gauge volume is required whereas a ‘match-stick’ shape can be employed for the hoop and radial orientations (as indicated in Figure 2) because of the lack of a strain gradient in the axial direction. A Bayesian analysis has been used to interpret the results [9]. From this analysis, it has been established that equally reliable data are obtained from a continuous monochromatic neutron beam source or from a pulsed polychromatic

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source. The best estimates of the three principal strains determined from the Bayesian analysis are shown in Figure 3. The corresponding residual stress distributions calculated from eqs. (5) are presented in Figure 4. It is evident from these figures that excellent agreement is attained with theory when allowance is made for averaging of the stress over the whole of the volume of material sampled for an individual measurement. The results were obtained for sampling volume cross-sections which ranged from From this study it has been found that, provided the experimental procedure specified is followed, strains can be determined to an accuracy of which corresponds to a resolution in stress of between ± 7 to 20 MPa in most engineering materials (depending on the elastic modulus for the material being measured).

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Ceramic Matrix Composite: In this investigation three separate specimens were examined. Each specimen was in the form of a disc, 30mm in diameter and nominally 3mm thick. Two of these specimens were provided to enable the strain free lattice parameter values for the base materials to be obtained. One of the specimens consisted of sintered alumina only and the other of silicon carbide powder only. The third specimen was the ceramic matrix composite (CMC) which contained a mixture of 25% silicon carbide powder in a matrix of alumina. The aim of the neutron diffraction measurements was to establish whether the strains in each component of the CMC could be determined reliably using the individual base material specimens as references. Each participant was allowed to choose their own (hkl) crystallographic planes on which to make measurements. They were also able to select their own gauge volume sizes and shapes. This study high-lighted a number of potential problems. In some instances, positioning uncertainties resulted in the gauge volume not being totally immersed in a specimen, which could lead to undesirable surface aberration complications. Also any divergence of the neutron beam could exaggerate this effect. It was found that the main problem was experienced with overlapping diffraction peaks as illustrated in Figure 5. In this case the background and peak positions become more difficult to estimate once there is more than one peak to fit. This affects the value of uncertainty estimations produced by peak fitting routines. A criterion has been developed for dealing with this situation. It provides guidance on how close interfering peaks can be for reliable results to be obtained. Shot-Peened Plate: Shot-peening introduces steep stress gradients close to surfaces. In order to determine these gradients, it is necessary to traverse the gauge volume through a surface as indicated in Figure 6. This results in a change in shape and size of volume of material being irradiated for each measurement. To deal with


this situation, it has been found necessary to define two gauge volumes. One termed the instrumental gauge volume (IGV) is the volume in space which is illuminated by the neutron beam (taking into account any beam divergence). In Figure 6 this corresponds with the square cross section shape. The other volume is called the sampled gauge volume (SGV) which is defined as the volume in a material over which a strain measurement is being made and is averaged. For partial immersion the SGV will always be less than the IGV. For complete immersion the two gauge volumes are identical. In all cases it is recommended that the position at which the average strain is recorded corresponds to the centroid of the neutron distribution in the SGV. This will be nearer to the surface than the centroid of the cross-section for highly absorbing materials.

Results that have been obtained through the thickness of a shot-peened plate of a nickel base alloy that had been subjected to different peening intensities on opposite faces are shown in Figure 7. A gauge volume with a ‘match-stick’ shape was used. It has been found that when the above precautions are taken into account residual stress gradients as steep as 2000MPa/mm can be determined reliably.

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Ferritic Steel Weldment: The ferritic steel weldment examined is shown as an insert in Figure 8. It was produced by a manual metal arc (MMA) process. Twelve passes were used which introduced a 7° angular distortion. The weldment consisted of parent material, heat affected zone (HAZ), and weld metal. This example was chosen to investigate the feasibility of making neutron diffraction residual stress measurements through regions of variable microstructure and chemical composition. The main concern was to establish the appropriate strain free crystallographic lattice spacing reference for use in making the residual stress calculations. To aid in this process a thin slice was removed from the end of the weldment into which cuts were made to produce 'stress-free' prongs. These were situated in the parent material, HAZ and weld metal regions to enable the values of to be obtained for each zone. It has been shown elsewhere [11] that by ignoring the spatial and directional variation of which is exhibited throughout the weld pool and the heat affected zones, erroneous strain data can be derived leading to non self-equilibrating internal stress estimates. This is particularly true for austenitic and austenitic/ferritic welds. An illustration of the longitudinal, transverse and normal residual stresses measured through the weld centre line using the relevant values is shown in Figure 8. These results indicate approximately zero normal residual stresses as is to be expected, and transverse residual stresses which satisfy force equilibrium. Taken in combination, these observations demonstrate that reliable estimates of had been obtained. Also presented for comparison in Figure 8 are the residual stresses that were measured in a thin slice (without the cut prongs). These show no change from those in the plate in the transverse and normal stresses but relaxation of the longitudinal stresses to zero. This observation indicates that slices can be removed from samples without the in-plane residual stresses being affected.

4. Discussion The results of an investigation that was carried out to develop a draft standard under the auspices of VAMAS TWA 20 and RESTAND have been presented. In the RESTAND study measurements have been made additionally on felt and fibre


reinforced composites for heat insulation and thermal shock resistance, on deep rolled crankshafts to represent complex shapes, a quenched component and through a variety of welds for power generation and aerospace applications. It has been demonstrated with complex shapes that care is needed to avoid orientations, which involve long neutron path lengths to minimise attenuation. Similarly it has been found that curved surfaces can exaggerate surface aberrations. Nevertheless, it has been shown that neutron diffraction can be used to measure residual stresses reliably in components for a wide range of industrial applications.

Based on the outcome of VAMAS TWA 20 and of RESTAND, an International Standard Organisation Technology Trends Assessment document (ISO/TTA) [12] has been issued which provides guidance for making reliable measurements of residual stresses by neutron diffraction. When these procedures are followed it is anticipated that positional tolerances to ± 0. lmm can be achieved and that strains can be measured to an accuracy of giving a resolution in residual stress of ± 7 to 20 MPa in most materials of practical interest. This document is now being used as a draft for the preparation of an international “technical specification” within a joint CEN/TC 138- ISO/TC 135 group of experts (AHG 7). Finally, ASTM E28. 13 is concerned with the preparation of this standard.

5. Acknowledgements The authors would like to acknowledge the contributions of their colleagues from the following institutions and neutron facilities around the world: Chalk River (Canada); ISIS (UK); ILL, LLB (France); JRC (Netherlands); HMI, GKSS (Germany); NPI (Czech Republic); RISOE (Denmark); Studsvik (Sweden); KUR, JAERI (Japan); KAERI (Korea); AEC (South Africa); LANSCE, ORNL, NIST and MURR (USA); Salford and Manchester University (UK). They would also like to thank the European Commission for the financial support provided for RESTAND.

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6. References 1.

Webster, G. A.: Role of residual stress in engineering applications (Eds Bottger, A. J., Delhez, R. and Mittermeijer, E. J. ) Matls. Sci. Forum, trans tech publications. Switzerland, 347-349 (2000), 1-9. 2. Youtsos, A. G. and Ohms. C.: Towards an evaluation of residual stress analysis based on neutron diffraction, Proceedings of the 3rd Size-Strain Conference “Analysis of microstructure and residual stress diffraction methods” (SS-III), Trento, Italy, December 2001. 3. Hughes, H..: X-ray techniques for residual stress measurements, Strain 3 (1967) 26-31. 4. Allen, A., Andreani, C., Hutchings, M. T., and Windsor, C. G.: Measurements of internal stress within bulk materials using neutron diffraction, NDT International, 14 (1981), 249254. 5. Stacey, A., Macgillivray, H. J., Webster, G. A., Webster, P. J. & Ziebeck, K. R. A.: Measurement of residual stresses by neutron diffraction, J. of Strain Analysis 20 (1985), 93100. 6. Beaney, E. M. and Proctor, E.: A critical evaluation of the centre hole technique for the measurement of residual stresses, Strain 10 (1974), 7-14. 7. Prime, M. B.: The contour method: Simple 2-D Mapping of Residual Stresses, Proceedings of ICRS-6, Oxford, UK, July 2000. 8. Sachs, G.: Der nachweis immerer spannungen in stangen und rohren, Zeits. Metall 19 (1927), 352-357. 9. Webster, G. A. (Ed)., Neutron Diffraction Measurements of Residual Stress in a Shrink-fit Ring and Plug, VAMAS Report No. 38 ISSN 1016-2186, January 2000. 10. Withers, P. J and Pang, J. W. L (private communication). 11. Ohms, C. Youtsos, A. G., Idsert, P. v. d., and Timke, Th., Residual Stress Measurements in Thick Structural Weldments by Means of Neutron Diffraction (Eds Bottger, A. J., Delhez, R. and Mittermeijer, E J. ) Matls. Sci. Forum, trans tech publications, Switzerland , 347-349 (2000), 658-663. 12. Polycrystalline materials - Determination of residual stresses by neutron diffraction ISO/TTA 3, ISO, Geneva, Switzerland September 2001.

MICROSTRESSES DETERMINED BY NEUTRON DIFFRACTION AND SELFCONSISTENT MODEL

A. BACZMANSKI WFTJ, Akademia Górniczo-Hutnicza, al. Mickiewicza 30, 30-059 Kraków, Poland C. BRAHAM LMMM, URA-CNRS 1219, Ecole Nationale Supérieure d’Arts et Metiers, 151, Bd de l’Hopital, 75013 Paris, France R. LEVY-TUBIANA LLB, CEA-CNRS, CEA Saclay, 91191 Gif-sur-Yvette, France A. LODINI LACM Université de Reims Champagne Ardenne UFR Sciences 51100 Reims, France LIB, CEA-CNRS, CEA Saclay, 91191 Gif-sur-Yvette, France K. WIERZBANOWSKI WFTJ, Akademia Górniczo-Hutnicza, al. Mickiewicza 30, 30-059 Kraków, Poland

Abstract Neutron diffraction was successfully applied to measure the lattice strain distribution in polycrystalline materials. The experimental data were studied with help of selfconsistent model. Using a non-standard method of analysis the values of macro and microstresses in textured ferritic steel were estimated. The validity of self-consistent model for prediction of mismatch stresses in two phase materials was examined. The theoretical results were successfully compared with the diffraction data for "in situ" tensile test. The parameters characterising elastoplastic deformation of polycrystalline grain were determined.

1. Introduction Diffraction technique is commonly used for the determination of the lattice elastic deformation and distortion i. e., macro and microstrains from the displacement and broadening of diffraction peaks [1, 2]. This method allows the measurement of stresses and elastic properties of polycrystalline materials. The stress state can be examined for near-surface volume by X-ray radiation [1] and inside the sample up to several 477 E. E. Gdoutos (ed. ), Recent Advances in Experimental Mechanics, 477–486. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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centimetres in depth, by synchrotron beam [3] or neutron radiation [4]. The diffraction methods have several advantages such as non-destructive character, the possibility of macro and microstress analysis for composites and anisotropic materials. Different methods can be used for the interpretation of diffraction measurements. The standard method of stress determining is based on measurement of interplanar spacing for various directions of the scattering vector which orientation is characterised by the and angles [1]. Using the simple linear regression procedure the macrostress field can be studied. Additionally, the analysis of diffraction peak broadening allows the observation of the microstrains evolution caused due to dislocations inside polycrystalline grains [2]. In the present work the non-conventional methods for interpretation of the experimental data are described. The methods, based on the Eshelby type model [5], are used for the study of mismatch stresses created due to incompatibilities of grains in their boundary regions. The model calculations are based on the prediction of interaction of ellipsoidal inclusion with a homogeneous matrix. The behaviour of different groups of grains (inclusions) can be directly compared with the diffraction data. The neutron diffraction and self-consistent model [6-8] of elastoplastic deformation were applied for analysis of the anisotropic mismatch stresses in steel sample subjected to plastic deformation. As the result both the macro and the mismatch stresses were found. Another analysis was performed for two phase materials for which the lattice strains were measured independently for each component. Thus the evolution of mismatch stresses (between different) phases were directly observed. Neutron diffraction experiments were performed on the G5. 2 spectrometer at the LLB, Saclay, France, while the X-ray measurement was done at the LMMM, Ecole Nationale Supérieure d’Arts et Metiers, France.

2. Self-consistent model The elastoplastic model [6-8] was used for interpretation of stress measurements obtained using diffraction technique. This model treats a given number of polycrystalline grains having different lattice orientations. The calculations are based on the modelisation of the processes occurring inside and between grains. As the result, the internal stresses for different grain orientation and for the whole sample are predicted. The deformation models, used in this work, are based on the prediction of the elastoplastic behaviour of a crystal grain inside the polycrystalline material under applied external stress If the local stress (at the particular grain) is large enough the plastic deformation occurs due to slip phenomenon. According to the Shmid’s law, the slip can be activated only on this slip system [uvw] (hkl) (the slip direction and plane are specified) for which the resolved shear stress exceeds some critical value

During plastic deformation, the multiplication of dislocations and evolution of their spatial distribution inside a grain leads to the hardening of slip systems ( increases with deformation). If we are interested only in the kinetic description of the active slip systems behaviour, their latent hardening can be described with some approximation by

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a matrix reflecting the interaction between the slip systems (the work hardening matrix). Consequently, the rate of the critical shear stress on the g-th system is equal to [6-8]:

where: is the rate of the critical stress in the g-th system, is the rate of the plastic glide in the h-th system and dot denotes the time derivative. The interaction between two active slip systems depends on their relative geometrical orientation. The weak hardening relations are represented by terms in the work hardening matrix, while the strong hardening relations are given by terms. Hence, the hardening matrix can be constructed using two independent parameters, i. e., To predict the plastic deformation, all the mentioned physical phenomena and quantities should be considered at the grain-scale. Moreover, the influence of the macroscopic quantities ( and rates characterising the sample) on the behaviour of the grain ( and rates) must be established. According to the self-consistent approach [6-8], the polycrystalline material is approximated by a macro homogeneous medium at the macroscopic scale and micro heterogeneous medium at microscopic scale. The relation between the stress rate and strain can be written for both scales:

where: is the local elastoplastic tangent modulus for the grain and is the macroscopic tangent modulus for a fictionally assumed average homogeneous medium. In general the properties of the grain and assumed matrix are different and the local stress rates can be obtained from the relations:

where while T is the interaction tensor calculated assuming the polycrystalline grain as an ellipsoidal inclusion embedded into the homogeneous matrix represented by L [6-8]. However, the A tensor can be calculated only if the L tensor is known, and inversely. To find both tensors the self-consistent procedure must be used [6-8].

3.

Anisotropy of plastic mismatch stresses

The average elastic lattice strain (noted by direction of scattering vector can be defined as:

measured by diffraction in

where the measured interplanar spacing are averaged only for the reflecting grains which possess the scattering vector normal to the crystallographic

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planes is measured and is the interplanar spacing for {hkl} planes in the stress-free material. On the other hand, the strain can be expressed through residual macrostresses and mismatch microstrains, i. e.:

where: are the macrostresses, are the diffraction elastic constants and is the elastic strain caused plastic incompatibilities of grains, defined for the reflection hkl and characterises orientation of the scattering vector. The aim of this part of work is to present a method for experimental determination of the macrostresses and the mismatch plastic incompatibility microstresses for textured polycrystalline sample. Using the self-consistent model the theoretical values of stresses can be predicted. Moreover, knowing the elastic constants for single grain, the average strain can be calculated. Since now, the model-predicted (theoretical) quantities will be denoted by bar. Let us assume that the second term in the Eq. 6 can be approximated by:

where: is calculated from the model and q - is a constant scaling factor. The factor q has been introduced to find the real magnitude of "plastic incompatibility" strains, assuming that their variation with the and angles is correctly described by models. Hence, for the hkl reflection, Eq. 6 takes the following form (see also Eq. 5):

For known diffraction elastic constants theoretically predicted strains and measured spacings the other quantities from Eq. 8 can be determined using a fitting procedure [9, 10]. The unknowns are q and the macrostresses Knowing the value of the q parameter from Eq. 8 (by applying a fitting procedure) the plastic incompatibility stresses can be determined for all grain orientations g:

where are the model predicted values. Thus, the macrostresses the mismatch microstresses can be determined simultaneously. The neutron diffraction experiment has been performed for two cold rolled samples i. e.: ULC90 (90% of reduction) and LC (70 % of reduction) [10]. The spacings have been measured inside the samples. The theoretical values of the strains were predicted by the self-consistent model [9, 10], Applying Eq. 7 and fitting the results obtained from the models to the experimental data, the


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macrostresses the interplanar spacing for a stress-free material and the q-factor have been found. In Figs. 1 and 2 the results of fitting are compared to the results of measurement. The theoretical curves approaches experimental points when the influence of plastic incompatibility strains (stresses) is taken into account in calculations (i. e., assumption in Figs. 1 and 2).

To show the level of the second order stresses, the average effective residual stress was calculated:

where the effective stress calculated according to the von Mises formula is integrated over the whole orientation space E.


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The results are given Tables 1. For neutron diffraction in the interior, the three dimensional stress state must be considered. In fitting procedure the and and components are not independent. Consequently, the absolute values of the principal stresses can not be determined if the is not known precisely. Hence, at this stage of study we show only the and differences instead of and absolute values (see Table 1).

4.

Mismatch stress in two phase materials

4.1. DUPLEX STEEL Duplex stainless steel with equal fractions (50%) of austenite and ferrite (50%) was studied [11]. The X-ray diffraction method was used to determine the internal stresses for the “in situ” deformed samples (tensile test). The interplanar spacing were measured separately for each phase, i. e.: using Cr radiation for phase, using Mn radiation for phase. From the experimental strains the surface stresses were calculated using the standard analysis [1]. The measured initial stresses of the both phases were an input data for the self-consistent model [6-8]. In calculations, the optimal model parameters of elastoplastic deformation were chosen in order to fit to the experimentally determined stresses for the and phases respectively (Fig. 3a) as well to the macrostress obtained from mechanical test (Fig. 3b). The mechanical tensile curve was obtained as the function of surface macrostress vs. macrostrain measured by the electrical gauge. It should be stated that the surface macrostress is taken as the measured stress applied to the sample shifted down by the value of initial residual stress (average for both phases) obtained from X-ray diffraction. In Fig. 3, the predicted stresses are compared with the experimental ones. The parameters of elastoplastic deformation are presented in Table 2. The results presented in Fig. 3a show that the model calculations agree very well with the experimental points for the deformation lower than 3%. For larger deformation the values obtained from X-ray diffraction are over the predicted curves because of relaxation of the surface compressive macrostress due to plastic deformation of the sample. This effect is not seen in the Fig. 3b where the experimental macrostress


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corresponds to the value applied to whole cross section of the sample. The initial parts of the curves obtained from X-ray diffraction are sufficient for determination of critical resolved shear stresses and amplitude of hardening matrixes for the both phases. The self-consistent calculations performed for these parameters agree very well with the result of mechanical test in which the relaxation of the surface stress does not occur.

4.2. METAL MATRIX COMPOSITE (Al/SiCp) The material was a 2124 Al alloy reinforced with 17% vol. SiC particles, produced by Aerospace Metal Composites, by a powder metallurgy route. A plate of composite was solution heat treated at 505°C for 2 hours and next quenched in cold water [12].

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The neutron diffraction technique was applied for lattice strains measurement. The sampling volume of mm was placed 1mm below the upper surface of the "in situ" bent bar. The tensile strains, induced by bending deformation, were measured along direction parallel and perpendicular to the applied stress As the reference value the measured interplanar spacing of initial (not deformed) sample was used for calculations, i. e.:

where: is the interplanar space measured for the initial sample, i. e. quenched but not deformed, is measured for the “in situ” bent bar and ph indicates the phase for which the interplanar spaces were determined (Al or SiC) Four points bending applied different deformations, while the surface strain was controlled using the stain gauge connected to the top of the bar. The total strain of the material in the region of neutron measurement (1 mm below the surface) was calculated from the surface strain assuming symmetrical bending. Simple tensile test was simulated and the lattice elastic strains and see Eq. 10) were calculated independently for the Al and SiC phases. The average strain values corresponding to those measured by diffraction were determined from elastoplastic model:

where the average (or ) lattice strain is calculated for the phcrystallites having the scattering vector parallel to the [111] crystallographic direction and being rotated by the angle about it. The theoretical results were compared with the phase strains measured by diffraction for the "in situ" bent samples (Fig. 4). In modelling the single crystal elastic constants were used for the both phases of the composite (Table 3). Simultaneously, the calculated function of total stress vs. total strain was compared with the results of mechanical tensile test (Fig. 5). In this case the macroscopic quantities were calculated as the average over volume of all composite grains. Purely elastic properties were assumed for the SiC component, while the plastic properties of the Al matrix were changed in order to find the best agreement of the measured and theoretical strains (see Figs. 4 and 5). The optimal model parameters of plastic deformation for Al (i. e.: critical shear stress, H - rate of work hardening and A- hardening anisotropy [6-8]) are given in Table 3. It should be stated that the theoretical values of and H could be effected by the residual stresses created due to quenching. In this case, both parameters should be treated as the effective ones defined for equivalent stress-free material representing Al matrix, which can not be directly compared to the thermally treated Al 2124 alloy. An excellent agreement between experimental data and model results was obtained for the neutron diffraction as well as for simple tensile test (Figs. 4 and 5). The agreement was obtained simultaneously for three different curves when plastic


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parameters of only one component (Al) were modified. It proves that the elastoplastic self-consistent model predicts very well the relation between macro stresses and the elastic strains (and stresses) measured for two phases in the elastic and elastoplastic ranges of deformation. Moreover, in the elastic range the single crystal elastic constants (Table 3) were used as the input data of model and any free parameters were optimised.

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5. Conclusions The diffraction methods and self-consistent model are particularly convenient tools for determining of the incompatibility stresses in polycrystalline materials. Using both methods the stress state for different groups of grains can be analysed. The comparison of theoretical prediction with experimental data allows us to understand the phenomena occurring during elastoplastic deformation. The advantage of the methods presented in this work is that the qualitative as well as quantitative analysis of stress evolution can be performed. The first presented method enables to evaluate the magnitude of the macrostress tensor and the residual plastic incompatibility stresses using some additional information from theoretical models. The variation of the microstress in function of the crystal orientation is supposed to be known from a theoretical model; its magnitude, however, is fitted from experimental results. Finally, the level of mismatch stresses for the plastically deformed sample can be estimated. The study of microstresses created in the two phase polycrystalline material can be easily done using diffraction methods. The stresses are measured independently for each phase and compared with self-consistent prediction. As the result the model parameters describing behaviour of crystallites can be estimated. For the optimal parameters the model stresses fit very well to the experimental data obtained from the mechanical test as well as measured by diffraction.

6. Acknowledgements The authors thank to the Komitet Bada Naukowych (Polish Committee of Scientific Research ) for financial support and the LLB, CEA Saclay (France) for enabling them to use neutron diffraction. 7. References 1. Noyan, I. C., Cohen, J. B. (1987) Residual Stresses - Measured by Diffraction and Interpretation, Springer-Verlag, Berlin, 1987.

2. Warren, B. E. (1969) X-ray diffraction, London: Addison-Wesley. 3. Daymond, M. R. & Withers, P. J. (1996) Scripta Mater. 35, 1229-1234. 4. Allen, A. J., Hutchings, M. T., Windsor, C. G & Andreani,C. (1985) Adv. Phys., 34, 445-473.

5. Eshelby, J. D. (1957) Proc. Roy. Soc. A241, 376-396. 6. 7. 8. 9.

Lipinski, P. and Berveiller, M. (1989) Int. Journ. of Plasticity, 5, 149-172. Lipinski P., Berveiller M., Reubrez E. and Morreale J. (1995) Arch of Appl. Mech. 65, 291-311. Zattarin, P., ., and Wierzbanowski K. (2000) Arch. Metall. 45, 163-184. Baczmanski, A., Wierzbanowski, K., Lipinski, P., Helmholdt, R. B., Ekambaranathan, G., & Pathiraj, B. (1994) Phil. Mag., A 69, 437-449. 10. Baczmanski, A., Wierzbanowski, K., Tarasiuk, J., Ceretti, M. and Lodini, A. (1997) Rev. de Metall., 94, 1467-1474. 11. Baczmanski, A., Wierzbanowski, K.., Braham Ch. and Lodini, A. (1999) Arch. Metall., 44, 39-50. 12. R. Levy-Tubiana, A. Baczmanski, M. Ceretti, M. Fitzpatrick, A. Lodini and K. Wierzbanowski (2000) Mat. Sci. Forum, 347-349, 510-515.

RESIDUAL STRESS MEASUREMENTS AT THE METAL/CERAMIC INTERFACE USING MODELLING OF NEUTRON DIFFRACTION SPECTROMETER ADELE CARRADO L. L. B. - Laboratoire Léon Brillouin, CEA-Saclay 91191 Gif sur Yvette, France L.A.C.M. - Université Reims Champagne Ardenne France

JEAN-MICHEL SPRAUEL LM3 UPRESA-CNRS 8006, IUT, 2 av. Gaston Berger F-13625 Aix en Provence LAURANT BARRALLIER E.N.S.A.M. - Laboratoire MécaSurf, 2, Cours des Arts et Métiers 13617 Aix en Provence, France ALAIN LODINI L.L.B. - Laboratoire Léon Brillouin, CEA-Saclay, 91191 Gif sur Yvette, France L.A.C.M. - Université Reims Champagne Ardenne, France Abstract The aim of this work is to improve some experimental techniques dedicated to the evaluation of residual stress (RS). These methods have been applied to a sample consisting of a glassy ceramic coating moulded on a metallic substrate (palladium-silver alloy) used in dental applications. Knowing the residual stress distributions is very important to determine the lifetime of the sample. We will describe a new approach to evaluate the RS at interfaces and in the bulk of materials, using neutron diffraction techniques. The RS in a glassy ceramic coated on metallic substrate are generated by the manufacturing process. They are present in the two materials. The RS depend principally on the thermal treatments imposed to the sample and they could have a very strong influence to the lifetime of the sample and in particular for the existing bounding between the metal and the ceramic. Moreover, the mechanical proprieties of glassy ceramic are known to be affected by proprieties of particle dispersed throughout the glassy matrix. The difference in the coefficients of thermal expansion (CTE) that exists between a ceramic and a metallic material could also be at the origin of a stress field. The first part of this paper concerns a development of a numerical simulation [1], [2] of whole two-axis neutron spectrometers. This programme allows correcting systematic errors due to the great parasitic peak shifts which appear when the measurements are carried out at the interface between two different materials [3]. 487 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 487–494. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The second part of this paper leads on the experimental determination of RS by neutron diffraction in the ceramic and metallic materials. These measurements were performed using different European facilities: D1A spectrometer at ILL (Grenoble, F) and E3 at BENSC-HMI (Berlin, D) for the analysis of the palladium substrate, and G5. 2 spectrometer at LLB (CEA- Saclay, F) for the characterisation of the glassy ceramic coating. Key Words: Neutron Diffraction, Residual Stress, mechanical behaviour, and numerical simulation.

1. Introduction The control of residual stress obtained during the manufacturing of a ceramic coating applied to a metallic substrate is very delicate. The PFM (Porcelain- fused- to- metal) technology of dentistry industry is well adapted for this aim [4]. For a successful operation, the coupled materials are selected in relation to their physical propriety and their biocompatibility. Alloys intended for use as base for porcelain have special requirements, because of the need to develop and maintain strength at the temperature involved in porcelain applications and to provide a firm bond to the applied porcelain. The PFM is the manufacturing process most widely used in commercial laboratories. Rational selection of a casting alloy for porcelain- fused- to- metal restorative should be based on the following: 1. Physical proprieties 2. Chemical propriety 3. Biocompatibility 4. Laboratory workability 5. Porcelain compatibility [5]. Thermal expansion and bond strength are also important proprieties to consider when choosing among PFM restoration alloys. These characteristics determine the porcelain / metal compatibility. For our ceramic, it was chosen Leucite which is a reaction product of potassium feldspar and glass. It is a particularly important component in dental porcelain because it affects the optical properties, thermal expansion, strength and hardness of the porcelain [5]. Due to its relatively inert behaviour in aggressive environments, it has a high hardness and a good wear resistance. The ability of ceramic materials, to resist higher temperatures than metals, offers the potential for major improvements in component design for a wide range of applications. However, ceramics typically exhibit statistically variable brittle fracture, environmentally enhanced sub critical crack growth, sensitivity to machining damage, and creepdeformation behaviour at elevated temperatures. The leucite crystals serve to increase the thermal expansion of the porcelain to bring it closer to that of the metal substrate. The leucite prevents stresses occurring, due to a thermal mismatch, which could lower the strength. An opaque ceramic such as a titanium oxide glass frit has to be applied as the first layer of veneer, to mask the metallic substrate in the PFM systems. Although the PFM systems have high strength, the opacity of the metal substructure has encouraged the

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development of all-ceramic core materials containing crystalline components which are stronger than the traditional (predominantly glassy amorphous) feldspathic porcelain. This type of core material can then be veneered with a more translucent ceramic material. The CTE of the porcelain must be suitably matched with that of the alloy and the melting range of the alloy must be raised sufficiently above the fusion temperature of the porcelain for a successful enamelling operation. Dental alloys must be inert, rigid and strong. They must be rigid in thin section to resist bending and they assist in the formation of a chemical bond with the porcelain. They have a similar CTE to the porcelain, so that when the metal/porcelain combination cools high interfacial stresses (which could dislodge the porcelain) are not created.

2. Material and Methods Sample A porcelain/palladium alloy was prepared following standard routines practiced by dental technicians in construction of PFM crowns. This particular ceramic/metal combination has low thermal mismatch, avoiding the generation of significant RS in the coating. Palladium alloy casting ingot was casted at 1450°C into a substrate block of 20 x 60 x1.8 mm (TAB LE 1).

The block was then ground flat on opposite faces, and sandblasted with alumina to to provide good bounding for the ensuing coating and finally oxidised 10 minutes at 980°C at 101325 Pa. A first coating was prepared by opaque powder, which was mixed in slurry and applied to the whole surface of the substrate with a paintbrush. It gives a regular layer of opaque (approximately 100 after heating). The layer was sintered according to the following firing cycle: heat to 980°C in vacuum and hold for 1 minute, it produces the real first layer of opaque that will hide the alloy. The top surface of the porcelain coatings was then sandblasted (alumina, A second opaque layer is normally optional. It has been used for this sample and its role is to complete to hide the metallic effect of the substrate, as the size of the samples was particularly large. The second opaque layer was heated to 980°C for 1 minute under vacuum. Concerning the dentin layer, this layer was just heated to 950°C for 1 minute. First Part: The Model The simulation program accounts for all the major components of the neutron spectrometer: the characteristics of the neutron guide or tube, a double curved mosaic monochromator, up to hundred primary and secondary slits, a classical or position sensitive detector. An integrated feature oriented CAD drawing module defines the

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sample. This allows the description of very sophisticated specimens (Figure 1). The software accounts for the horizontal and vertical divergence of the incident and diffracted beams, for the local conditions of diffraction in the monochromator and the sample, and for the absorption by the monochromator and the sample.

The simulation program first computes the distribution of intensity and wavelength across the incident beam. The precise shape and size of the probe volume is then calculated. We have also defined the centre of diffracting volume, which has to account for the absorption in the material To define precisely the position of the neutron probe close to the surface we have performed a scanning along the direction normal to the surfaces. So we can plot the evolution of diffracted intensity versus the position Z of the geometric centre of the neutron probe (Figure 2).


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Classically, it is considered that at half of maximum of this curve, exactly half of the neutron probe is immersed in the material. In our case, this assumption is not valid, because of the strong absorption of palladium. A non-negligible shift exists for the determination of the true analysed depth. In addition, the program allows also computing the intersection between the neutron probe and the sample, thus defining the diffracting volume. A theoretical diffraction peak is finally calculated through a Monte Carlo simulation method. The whole simulation program allows thus to optimise the experimental conditions and to predict all parasitic shifts of the diffraction peak.

Second Part: Experimental Methods and Measurements Stress evaluation Method The diffraction method for the determination of residual stress is based on the measurement of the interplanar spacing in various directions (Figure 3).

In a diffraction experiment, the mean interplanar spacing be measured (Figure 4). The mean value of the strain scatting vector Q is equal to:

of the reflecting grains can in the direction of the

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where is the lattice spacing corresponding to the {hkl} planes of a stress free specimen, is the local strain in the direction of the scattering vector. V is the whole volume of the reflecting grains and the brackets denotes, the average over the diffraction grains of the hkl reflection [7]. Due to the big difference between the lattice parameters of the two materials, it was not possible to do all the measurements on one single instrument (TABLE 2). For that reason, neutron diffraction experiments were performed at LLB (CEA-Saclay, F), on G52 two-axis diffractometer using cold neutrons to evaluate the RS in the glassy ceramic coating. For the palladium, the measurements were realised on D1A spectrometer at ILL (Grenoble, F) and on E3 equipment at HMI-BENSC (Berlin, D). The experimental conditions of these neutron diffraction measurements are reported in TABLE 3 [8]. In figure 4, the sample geometry and the co-ordinate systems are shown.


3.

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Results

The RS profiles are represented as a function of the distance to the metal- ceramic interface. In figure 5, for both the materials, the residual stress distributions are presented. - The residual stresses observed in the palladium alloy substrate are tensile but their magnitude is very low. - In the ceramic coating, the stresses are mainly compressive. Due to the low diffracted intensity, linked to a high amount of amorphous phase, only the component was determined. The stress profile in the ceramic shows low compressive values, but significant stress may exist near the interface.

4. Discussion and Conclusions In this paper, we have presented residual stress profiles at the interfaces of a glassy ceramic coated onto a palladium alloy substrate. Measurements by neutron diffraction techniques have been realised in different zones of the sample. In particularly, we have employed the neutron diffraction to analyse the surface and the bulk of glassy ceramic and the core of the metallic substrate. This technique, largely employed, leads to very

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good results. It has permitted us to obtain a precise and non-destructive evaluation of the residual stress in the core of the materials. We have also improved the experimental techniques dedicated to the determination of residual stress. This has required a precise and refined data processing, which accounts for different physical phenomena and for geometrical aberrations, which appear in the measurements. New data treatment procedures, based on a numerical simulation of the neutron instruments, have also been developed. These methods allow accurate and non-destructive evaluations of the in-depth residual stress profiles of surface or interface layers. In particularly, we have employed the neutron diffraction to analyse the surface and the depth of the glassy ceramic, and the bulk of the metal. It would however be very interesting to evaluate the stress profile starting from the metal/ ceramic interface and going inside the first of the ceramic (opaque interface ceramic).

5.

Acknowledgements

The experiments at BENSC in Berlin were supported by the European Commission through the TMR/LSF Access Programme (Contract: ERBFMGE CT950060). Sincere thanks are due to the staff of E3 at HMI Berlin, of D1A at ILL (Grenoble, F) and of G52 at LLB (CEA- Saclay, F) without whose help the measurements would not have been possible.

6. References 1. 2.

3. 4. 5.

6. 7. 8.

Pluyette E., Sprauel J. M., Lodini A., Perrin M., Todeschini P.: Residual stresses evaluation near interfaces by means of neutron diffraction: modelling a spectrometer, ECRS4, Cluny: pp. 153-163, 1996. Sprauel J. M.: Stress evaluation by neutron diffraction: Modelling of a two axis spectrometer, Journal of Neutron research, in press, 2001. Webster P.J., Mills G., Wang W. P., Holden T. M.: Impediments to efficient through surface scanning, Journal of neutron research, 3, p. 223-240, 1996. McLaren EA.: All-Ceramic Alternatives to Conventional Metal/ceramic Restorations. Compendium 307-325, March 1998. Van Vlack L. H.: Materials for Engineering, Addison - Welsley AB 92 - chapter 11, 12, 13 and 15, 1982. Doremus RH, Review bioceramics, Journal of Materials Science Vol. 27, 285-297, 1992. Noyan I. C., Cohen J. B.: Residual Stress Measurement by Diffraction and Interpretation (Materials Research and Engineering), 1987. Carradó A., Sprauel J. M, Barrallier L., Lodini A., Neutron and Synchrotron evaluation of residual stresses in coatings, Journal of Neutron research, in press, 2001.

ELASTOPLASTIC DEFORMATION OF TWO PHASE STEELS STUDIED BY NEUTRON DIFFRACTION AND SELF-CONSISTENT MODELLING M. R. DAYMOND ISIS Facility, Rutherford Appleton Laboratory Chilton, Didcot, Oxon. OX11 0QX, UK H. G. PRIESMEYER Institut für Experimentelle und Angewandte Physik Christian-Albrechts-Universität, Kiel, Germany A. M. KORSUNSKY Department of Engineering Science University of Oxford, Parks Road Oxford, OX1 3PJ, UK

Abstract In situ neutron diffraction experiments allow the measurement of phase specific strain, and thus stress, in a multi-phase system. Further, monitoring of multiple hkl lattice planes provides information as to the response of differently oriented grains within the polycrystal. These experimental results can be compared directly with predictions from a self-consistent Hill-Hutchinson model modified to take into account the presence of the second phase. The techniques and ideas are demonstrated with recent tests on two practical engineering steels (1) ferritic steel, with a small volume fraction of carbon in the form of cementite and (2) duplex steel, consisting of approximately equal amounts of austenitic and ferritic phases. Good qualitative agreement was obtained between model and experimental data in each case. Issues affecting the use of such a model for two phase systems are discussed. The interpretation of residual stress measurements, for both single peak and Rietveld multi-peak measurements is considered.

1.

Introduction

It has been recognised for some time that internal and residual stresses in materials can have a considerable effect on material properties, including fatigue resistance, fracture toughness and strength. Such stresses can vary greatly as a function of position within engineering components, due to manufacturing processes during the production route. Their measurement and interpretation is thus of considerable interest to the engineer. Monitoring of phase specific internal strains can also provide a direct insight into the mechanical behaviour of multi-phase materials. The strength of a multi-phase material 495 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 495–506. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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depends on the level of in-situ matrix strengthening (e. g. due to increased dislocation density from the presence of a secondary phase), but also on the efficiency of load transfer between the two phases. The distribution of load between the phases is dependent on relative volume fraction, shape and orientation and on the relative elastoplastic properties of the phases [1]. The partitioning ratio of applied load between the two phases will remain constant with increasing applied load provided that both phases remain elastic. However, once the stress levels in a multi-phase material are high enough for relaxation or inelastic deformation processes to occur in one or more of the phases, the load partitioning ratio changes. The mechanics in the case of the classic “composite”, where a small volume fraction of hard reinforcement is dispersed uniformly in a more compliant matrix, has seen considerable discussion [1]. The distinction between reinforcing phase and matrix is more complicated in the case of a two phase system with approximately equal volume fractions of the two phases. In this case an interpenetrating microstructure may well exist. This paper addresses a number of issues relating to mechanisms of materials deformation, and to the practical measurement of stress in components by diffraction from two common two phase steels; the system, and an austenitic-ferritic duplex stainless steel. Ferritic steels are extremely common, having good mechanical and forming characteristics as well as being relatively cheap. Small volume fractions of dispersed secondary phases such as cementite are found in many practical ferritic steels, commonly found in the case of the pearlitic microstructure. Duplex stainless steels, consisting of approximately equal amounts of austenite and ferrite, often combine the best features of these constituent phases. They generally have good mechanical properties, including high strength and ductility, and are increasingly used as an alternative to conventional austenitic grades. Neutron diffraction is an important experimental technique, ideal both for profiling macro-strain variations as a function of position, and for measuring phase specific load transfer. It relies on the same physics as the analogous measurement of stress using xray diffraction [2], however with the principal benefit for the engineer or materials scientist of increased penetration depth compared to traditional x-rays. Since neutrons interact primarily with the nucleus, rather than the electron cloud as x-rays do, the penetration depth of neutrons is very large compared to x-rays. For example, the penetration depth (1/e) for steel is ~1cm for thermal neutrons, but less than for xrays of comparable wavelength [3]. This results in neutrons being a probe suitable for bulk average measurements of material properties. For many practical applications, it is bulk averaged properties that the engineer or material scientist is primarily interested in. Diffraction measurement of strain involves the monitoring of changes in separation of one or more suitably orientated crystallographic lattice planes. It is thus a direct measure of the elastic strain in the material. In a polycrystal a diffraction peak represents the average lattice separation over all the grains in the irradiated volume which are suitably oriented to diffract. Thus each diffraction peak is produced by a different family of grains within the polycrystal, giving a direct insight into the grain orientation dependence of elasto-plastic deformation. For instance in the presence of an applied stress during elastic loading, the strains observed for measurements on individual diffraction peaks will typically differ in magnitude with respect to each other, and to the continuum macrostrain (e. g. [4]), a phenomena commonly dealt with during

INTERGRANULAR STRAINS IN TWO PHASE STEELS

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engineering stress measurements through the use of a ‘diffraction elastic constant’. The importance of understanding the mechanisms of interaction between differently orientated crystallites is highlighted when one considers that not only do individual grains exhibit elastic stiffness anisotropy, but also that plastic relaxation occurs preferentially on certain slip systems (plastic anisotropy). Despite increasingly sophisticated models (e. g. [5]) the current state of the art is far from quantitatively predicting the evolution of hkl dependent strains or the implications of their spread on failure. The use of self-consistent models does however give a direct insight into plasticity in polycrystals; the onset of deformation in differently oriented crystallites and the importance of hardening mechanisms. Systems where an elastic phase interacts with a ductile phase have been treated extensively using continuum mechanics arguments [1], but are relatively unexplored using crystal plasticity arguments. This lack of data is even more clear for the more complicated case where both phases are ductile. This paper applies a self consistent model to the deformation of two phase steels and compares the results with lattice reflection data from neutron diffraction experiments.

2. Uniaxial tensile tests on steel A series of uniaxial tensile loads were applied sequentially to specimen in situ on the ENGIN instrument of the PEARL beamline at the ISIS pulsed neutron facility, Rutherford Appleton Laboratory. The load frame is purpose built for use within the neutron beam. The loading axis is horizontal and typically at 45° to the incident beam, allowing simultaneous measurement of lattice plane spacings both parallel and perpendicular to the loading direction in the opposing 90° detector banks (Fig. 1). Universal joints were used for these tensile tests in order to maintain uni-axiality of load during the test. The material used was a 38MnS6 steel consisting mainly of a pearlitic microstructure, with larger scale ferrite veins and cementite particles, indicating a

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hypereutectoid structure, The material was nominally untextured, with a volume fraction of cementite ~5%. A circular sample of 8mm diameter cross section was tested, with a nominal gauge volume (neutron sampling volume) 8mm high, 5mm wide defined by incident slits, and 1. 5mm wide outgoing defined by radial collimators. The duplex steel was cold rolled SAF2304, consisting of approximately equal volumes of austenite and ferrite phases. The duplex steel was received as 1. 5mm thick plate, with a heavily banded microstructure of elongated austenite islands in a ferrite matrix [6]. The material was strongly textured, with texture varying as a function of depth; from ~2.5x random at the surface of the plate to ~5x random at the centre of the plate for the austenite, and correspondingly from ~5x to ~25x random for the ferrite [7]. The sample was cut with tensile axis parallel to the transverse plate direction, and measured with the rolling direction vertical, thus strains were determined in the transverse and normal plate directions. A nominal gauge volume 3 x 15 x 1. 5mm was defined for this test. The austenite grains were approximately equiaxed with dimensions the ferrite grains around in the rolling and transverse directions, in the normal direction [6].

3.

Self-consistent modelling

The elastic-plastic properties of a polycrystalline aggregate have been described using the Hill self-consistent approach [8], which was first implemented by Hutchinson [9]. A population of grains is chosen with a distribution of orientations and volume fractions that match the measured texture. Each grain in the model is treated as an ellipsoidal inclusion, though spherical grains were used for all the results presented in this paper. Each grain is attributed anisotropic elastic constants and slip mechanisms characteristic of a single crystal of the material under study. Interactions between individual grains and the surrounding medium, (which has properties of the average of all the grains) are performed using an elasto-plastic Eshelby type self-consistent formulation. Since the properties of the medium derive from the average response of all the grains, it is initially undetermined and must be solved by iteration. Small total deformations are assumed (typically less than 4%), and no lattice rotation or texture development is incorporated in the model. The model used is described in more detail in [10]. The single crystal elastic constants used in the model are shown in Table 1. Plasticity in the austenite was assumed to take place on (111) systems. Since there is no close packed plane in bcc materials, slip is generally considered to take place on many slip systems, with the slip direction in common. In order to mimic this simply in the model, three slip systems were included, all with the same critical resolved shear stress and hardening behaviour, namely the {110}, {112}, and {123} [11]. The critical resolved shear stress and exponential hardening coefficients in each case were fitted to give optimum agreement with the macroscopic stress-strain curves, and are listed in Table 2. The hardening function used for each system is described by Equation 1,


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where is the accumulated shear strain in the grain. The threshold stress in Equation 1, describes (in an average way) the resistance to activation that the deformation modes experience; it usually increases with deformation due to strain-hardening. Equation 1 represents an extended Voce law [12] which, instead of stress saturation, exhibits an asymptotic hardening rate For the strains used in this work it is an adjustable hardening parameter. In addition, we allow for ‘self’ and ‘latent’ hardening by defining coupling coefficients which account for the obstacles that dislocations in system represent to the propagation of s dislocations. The increase in the threshold stress of a system due to shear activity in the grain systems is calculated as:

When 'self' and 'latent' hardening are indistinguishable, Equations 1 and 2 permit a description of the high hardening rate observed at the onset of plasticity, and its decrease towards saturation at large strains. Linear hardening is a limiting case of this law, and takes place when In this paper an isotropic hardening model was used with latent hardening equal to self hardening.

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In order to compare the model with the experiment a subset of the total population of grains used in the model is identified for each diffracting family defined by the condition of having an hkl plane lying within ±5 degrees of the loading axis, or perpendicular to it. These are the grains that would contribute to a diffraction measurement of strain in that direction, and an average over each subset of the strains across the diffracting plane is compared with the neutron results below. Each diffraction peak is produced by a different population of grains. Although the self-consistent model has been used extensively on single phase materials, some simple modifications are required for multiple phases. We here simply create a population of grains, with the appropriate volume fraction with properties of the individual phases; the model has already been applied to both fcc and bcc materials [5, 13]. For the ferritic steel / cementite system we take advantage of the fact that cemenite has approximately the same stiffness as ferrite [14], as has also been observed by diffraction measurements [15]. Therefore without introducing a new crystal structure into the model, we treat the cementite as having the elastic properties of iron crystallites but constraining these crystallites not to slip, and providing the appropriate volume fraction. Since the self-consistent model treats the homogenous matrix as having the average elastic properties of all the grains, provided we a) do not have texture in the cemenite, and b) require only an average phase strain (rather than lattice reflection strains) for the cementite, this is a good approximation within the self-consistent model. It is a valid approach in considering the reinforcing effect of the cementite particles on the iron lattice reflection strains in the plastic regime. It should be noted of course that the model therefore does not take into account the complexities of microstructure actually observed. The model was run with 1000 randomly oriented grains representing the iron with weight 0. 95, and 1000 randomly oriented grains representing the cementite with weight 0. 05. Both phases were therefore represented with sufficient grain populations for random textures, and with appropriate volume fractions. For the duplex steel, the measured texture was converted to a grain population (1152 grains per phase), with equal weighting, and again the appropriate elasto-plastic properties applied to each phase. The incident spectra at time-of-flight neutron sources are polychromatic, thus a wide range of possible lattice planes are recorded in each measurement. The scattering vectors for all reflections recorded in one detector lie in the same direction, and thus indicate the strain in that direction. Each reflection is produced from a different family of grains that are oriented such that a specific hkl plane diffracts to the detector. We have therefore analysed the experimental data in two methods, both carrying out fits to individual peaks within this spectrum, and fitting the whole spectrum simultaneously using a Rietveld refinement [16]. The use of this technique is discussed in detail in [17], but in brief has been shown to be a good representation of the mean phase elastic strain. Thus we can compare both the hkl specific and average phase strains obtained from the model directly with strains obtained from analysis of the diffraction data.


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4. Results The agreement that can be achieved between the model and experimental macroscopic curves is reasonable (Fig. 2); it should be remembered that this agreement has been achieved by choice of the parameters in Table 2. However important discrepancies exist, which further modifications to the hardening parameters do not remove. For instance, in the (Fig. 2a) a reverse yield can be seen at the base of the experimental unload, which is not captured in the model. Secondly, the measured

macroscopic response immediately after yield is somewhat softer than shown in the model; the plastic hardening curve is initially shallower in the experimental data. It is not possible to fully capture this in the model, even if a very low hardening rate is used. This fact can be partially understood from the mechanisms of the model, in which each grain is modelled as being a single, independent ellipsoid within a medium which has properties of the average of all the grains. When the first optimally oriented grain in the modelled population reaches yield and plasticity occurs, the relaxation caused has only a small effect on the stiffness of the medium due to the small volume fraction represented by each grain. This will initially have a relatively small effect in bringing other grains to their yield surfaces. This constrained transfer of relaxation effects between differently oriented grains via the interaction of the medium limits how steep the initial part of the macro plastic curve can be, given that the value of the asymptotic slip hardening parameter (Eqn. 1) is constrained by the requirement to match the observed macrostrain gradient at large plastic strains. It should be noted however that if a prior deformation or thermal treatment had created residual stresses a more rapid switch from elastic to long-term plastic behaviour is indeed possible. This is because the prior deformation biases the state of individual grains such that more reach their yield surfaces at a given macroscopic stress, most easily imagined in the case of an interrupted tensile test, though the effect is also possible in for example rolling followed by tension [18]. Correspondingly, in the duplex data (Fig. 2b), some slight discrepancies also exist. Initial departure from the elastic limit is seen experimentally perhaps at an applied stress of as low as 320MPa, but more significant yielding occurs

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at 450MPa; this two stage yielding behaviour is not fully captured. Further, the high plastic strain gradient is slightly too low in the model. The strains obtained from a Rietveld refinement on the phases, indicative of the ‘bulk’ or average strain in each phase are shown in Figure 3. These are compared with the mean phase elastic strain obtained from the model. Considering first the ferrite /

cementite system, the experimental cementite strains (Fig. 3a) show a large scatter due to the small volume fraction and broad peaks, but within experimental error there is good agreement. In both axial and transverse directions the cementite elastic response is the same as the iron, with larger strains generated in the cemenite once plasticity occurs and load transfer initiates from iron to cementite. The iron strains, with uncertainties of the same size as the symbols, are in excellent agreement with the model which captures the small but measurable deviation from linearity in both axial and transverse directions. This is a reassuring observation for stress measurement in bcc materials, since in common with fee steels it suggests that the Rietveld strain is indeed a good representation of the bulk macro strain, even in the plastic regime. However it does highlight the importance of considering even relatively minor volume fraction phases when making stress measurements. For the duplex system, again the initial elastic response in both axial and transverse directions are very similar for the two phases, indicating the similar ‘bulk’ moduli of the phases; as would be hoped this is correctly captured by the model. Experimentally there is then a gradual load transfer from the austenite phase to the ferrite, as the austenite initially yields (starting just above 300MPa). At an applied stress of around 500MPa the strain increment in the austenite with increasing applied stress changes again, presumably due to yield in the ferrite phase. This effect is seen in both the axial and transverse directions in the austenite, though the effect on the measured ferrite strains is small. The model qualitatively captures the load redistribution between the phases but does not correctly describe the magnitude of the effects. In particular the model has the austenite phase with nearly zero strain increment for increasing applied loads between


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300 and 450MPa, whereas experimentally this is still significant. Further the reverse transfer of load from the ferrite phase as it yields is much larger in the model than seen experimentally. The anisotropic response of individual lattice reflections during in situ loading continues to be a subject of considerable experimental interest, since it provides an excellent test of crystal plasticity models. Figure 4a shows the response of four (first

order) lattice reflections plotted against the macroscopic applied stress, parallel to the loading direction for the ferrite in the ferrite / cementite system. These strains represent the values determined by single peak fits. It should be remembered that each line thus represents the response of a different family of grains, oriented such that the given hkl lattice direction is parallel to the loading direction, respectively. Elastic moduli show generally good agreement between model and experimental results, though the differences are somewhat large to be due to experimental scatter alone. Deviations from an elastic linear response occur in the single peak strains close to the onset of macroscopic plasticity, but at different macro stresses for different grain populations. This is because once plastic deformation initiates, the yield of preferentially oriented grains relative to their neighbours causes strain redistribution, and a divergence from the hitherto linear response in other peaks. As would be expected the 110 and 211 peaks have the same elastic modulus [e. g. 17], but here can be seen to have different plastic anisotropy strains. The qualitative behaviour of the peaks in the plastic regime is well captured. There are some quantitative differences, notably the exact macro stress at which yield occurs in the different grain populations, though this is not perhaps overly surprising given the differences seen in the macroscopic response. Secondly, it appears that the initial yield is sharper in the experimental data than in the model; the departure from elastic slope to plasticity is experimentally nearly vertical in the axial direction, while the model shows a more gradual yielding. This suggests a very rapid initial transfer of load from the iron to the cementite. This observation is based on a small number of data points, but the

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effect seems consistent in the different individual peak strains, as well as the mean phase strains. It also ties in with the evidence from the macroscopic curve that we are not fully capturing the speed of onset of plasticity. It is interesting that the model 002 and 310 peak strains show a deviation first one way then the other from the elastic gradient, becoming softer than elastic, then harder. Correspondingly the 110 and 211 do not show this behaviour, deviating immediately to stiffer behaviour. This also appears to be the case in the experimental data, for instance in the 002, though these changes are at the level of experimental uncertainty. This is demonstrating that the 110 and 211 directions are preferentially oriented for slip (i. e. higher Schmid factor), and hence yield first whilst the 002 and 310 grain directions are poorly oriented for slip and hence some small degree of load redistribution is occurring onto these planes as slip initiates from the former grain sets to the latter. As larger macroscopic plastic strains are generated, the 002 and 310 oriented grains also start to slip, and the load transfer to the cementite dominates. The axial strains observed in the duplex austenite phase are shown in Figure 4b, and illustrate a number of points. The initial yield behaviour, shown as a reduction in the gradient of the hkl strains is captured. As with the mean phase strains (Fig. 3b), the model predicts a smaller strain increment with applied stress than is experimentally observed for the 311 and 200 peaks. Interestingly, the model appears to under predict the softening in the 111 peak; the experimental strain appears to be nearly vertical between 350 and 500MPa. While the effect of the initiation of ferritic yielding at ~500MPa on individual grain family strains can be seen in both model and experimental data, again the magnitudes are not correctly captured. The ferrite single peak strain data has not been shown due to lack of space, however we note that (a) the 200 peak is extremely weak in this direction due to the strong texture, and therefore useful strains could not be obtained, (b) the 211 and 110 peaks provide indistinguishable strain data in both the elastic and plastic regime (c. f. Fig. 4a), (c) as in Fig 3b, the qualitative behaviour of ferrite single peaks below 450MPa is well captured by the model, with strain magnitudes again also well captured. However there is no experimental evidence in the experimental single peak strains of the reverse load transfer which is seen in the modelled ferrite response as a reduction in gradient (in both phase and single peak strains)

5. Discussion From the results detailed above it is clear that, while the model of the duplex steel has captured some key features of the phase specific and grain orientation dependent elasto-plastic deformation, the model of the ferrite/cementite system has been considerably more successful, achieving excellent quantitative agreement between model and experiment. It is important to address the reasons for this discrepancy beyond the obvious issue of either one or both phases yielding, as this highlights the successes of this modelling approach, as well as present problems. Firstly, when modelling a two phase system where only one phase is elastic, as described above, the plasticity hardening parameters of the plastic phase are chosen to provide ‘best agreement’ with the macroscopic response. Comparisons are then made with the internal phase strains. The choice of ‘best agreement’ has some subjectivity,


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which could be removed for instance by incorporating the process in a least squares’ approach. However, the effect on the internal hkl strains of small changes in the hardening parameters to ‘tune’ agreement with the macroscopic response is small. However, when both phases can yield there is clearly opportunity for more flexibility in choice of yield stress and hardening parameters to obtain the macroscopic response. In this case, since it was known that the austenite would yield before the ferrite phase [6], and further this yield was identified with the slight initial change in macroscopic slope (at ~300MPa), sufficient extra constraints are introduced into the model to limit the range of feasible hardening parameters. While an appropriate through-thickness average texture was incorporated in the model, a number of important microstructural and processing issues were not included in the model. For example, the texture varied as a function of depth through the plate. This will result in a variation of mechanical properties with depth, and thus some level of load transfer between material at different depths, although the neutrons do produce a result averaged over the entire thickness of the plate. Further, the ferrite grains were treated as spheres, while they in fact have a distinct aspect ratio – this will affect the Eshelby stiffness of the grain in the model. While the austenite grains are indeed equiaxed, making a sphere approximation appropriate, they are found in elongated islands within a ferrite matrix. The self-consistent model treats each grain in a medium consisting of the average properties of the entire grain population, and therefore does not take into account the environment of grains. While this ‘dilute’ approximation has been successful with the ferrite/cementite system, it is possible that failure to capture these geometric effects is worsening agreement in the duplex system. Finally, and perhaps most importantly, some residual strains from the processing of the duplex steel will almost certainly exist. These strains will effect the yielding behaviour of the phases, as well as for given grain orientations. The generation of these strains is extremely complicated to model, since the process includes texture development as well as recrystallisation. Attempts have been made to model the development of internal strains due to processing in a single phase steel prior to such a uniaxial loading test [18] and this avenue needs to be explored with the duplex steel results described above.

6.

Conclusions

Good qualitative agreement has been obtained between experimental and model data for uniaxial loading of the duplex steel, where both phases were modelled with elastoplastic behaviour. Discrepancies between model and experiment were attributed to microstructural characteristics and processing induced residual strains. Excellent agreement has been achieved between experimental and model data for uniaxial loading of a simple two phase iron-iron carbide system, in which the elastic properties of the two phases are the same, while only one phase has been modelled as exhibiting plastic slip. The strains obtained from a Rietveld refinement have been shown to provide a good description of the mean behaviour of the iron. The importance of considering the influence of even a small volume fraction of secondary phase in the plastic regime has been demonstrated.

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7. Acknowledgements Diffraction experiments were carried out at the ISIS pulsed neutron facility at the Rutherford Appleton Laboratory, Oxon., U. K., under support from the Engineering and Physical Sciences Research Council. The authors would like to acknowledge DaimlerChrysler and Magnus Odén, Linköping University for supplying the and duplex samples respectively.

8. References 1. 2.

3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18.

Clyne, T. W. and P. J. Withers, An Introduction to Metal Matrix Composites. Cambridge Solid State Science Series, ed. E. A. Davis and I. M. Ward. 1993, Cambridge: Cambridge University Press. 509. Noyan, I. C. and J. B. Cohen, Residual Stress - Measurement by Diffraction and Interpretation. Materials Research and Engineering, ed. B. Ilschner and N. J. Grant. 1987, New York: SpringerVerlag. 272. Hutchings, M. T. and C. G. Windsor, Industrial Applications of Neutron Scattering, in Neutron Scattering (Ch. 25), K. Skold and D. L. Price, Editors. 1986, Academic Press: Orlando, Fla. p. 405482. MacEwen, S. R., J. Faber, and A. P. L. Turner, The Use of Time-of-Flight Neutron Diffraction to Study Grain Interaction Stresses. Acta metall., 1983. 31(5): p. 657-676. Clausen, B., T. Lorentzen, M. A. M. Bourke, and M. R. Daymond, Lattice Strain Evolution during Uniaxial Tensile Loading of Stainless Steel. Mat. Sci. Eng., 1999. 259(1). p 17-24. Johansson, J., M. Oden, and X. H. Zeng, Evolution of the Residual Stress State in a Duplex Stainless Steel during Loading. Acta Mater, 1999. 47(9): p. 2669-2684. Moverare, J. J. and M. Oden, Deformation Behaviour of a Prestrained Duplex Strainless Steel. Acta Mater., 2001 (submitted). Hill, R., Continuum Micro-Mechanics of Elastoplastic Polycrystals. J. Mech. Phys. Sol., 1965. 13: p. 89-101. Hutchinson, J. W., Elastic-Plastic Behaviour of Polycrystalline Metals and Composites. Proc. Roy. Soc. Lond. A, 1970. 319: p. 247-272. Turner, P. A., N. Christodoulou, and C. N. Tome, Modelling of the Mechanical Response of Rolled Zircaloy-2. Int. J. Plasticity, 1995. 11(3): p. 251-265. Chin, G. Y. and W. L. Mammel, Computer Solutions of the Taylor Analysis for Axisymmetric Flow. Trans. TMS-AIME, 1967. 239: p. 1400-1405. Kocks, U. F., C N. Tomé, and H. -R. Wenk, Texture and Anisotropy. 1998, Cambridge: CUP. Pang, J. W. L., T. M. Holden, and T. E. Mason, The Development of Intergranular Strains in a HighStrength Steel. J. Strain Analysis, 1998. 33(5): p. 373-383. Kagawa, A., T. Okamoto, and H. Matsumoto, Young's modulus and thermal expansion of pure iron-cementite alloy castings. Acta mater., 1987. 35(4). p. 797-803. Acker, K. V. and P. V. Houtte. Macro- and Microstresses in the Cementite and Ferrite Phases of Cold Drawn Steel Wires Measured with XRD and Neutron Diffraction, in Proc. of 4th Eurpoean Conf. on Residual Stresses. 1996. Cluny en Bourgogne, France. Rietveld, H. M., A Profile Refinement Method for Nuclear and Magnetic Structures. J. Appl. Cryst., 1969. 2: p. 65-71. Daymond, M. R., M. A. M. Bourke, R. B. Von Dreele, B. Clausen, and T. Lorentzen, Use of Rietveld Refinement for Residual Stress Measurements and the Evaluation of Macroscale Plastic Strain from Diffraction Spectra. J. Appl. Phys., 1997. 82(4): p. 1554-1562. Daymond, M. R., C. N. Tomé, and M. A. M. Bourke, Measured and Predicted Intergranular Strains in Textured Austenitic Steel. Acta Mat., 2000. 48: p. 553-564.

RESIDUAL STRESSES AND ELASTIC CONSTANTS IN THERMAL DEPOSITS

THOMAS GNÄUPEL-HEROLD National Institute of Standards and Technology Center for Neutron Research, Gaithersburg, MD 20899, U. S. A. University of Maryland, College Park, MD 20742-2115 HENRY J. PRASK National Institute of Standards and Technology Center for Neutron Research, Gaithersburg, MD 20899, U. S. A. FRANK S. BIANCANIELLO National Institute of Standards and Technology, Metallurgy Division MD 20899, U. S. A.,

Abstract

In this study we investigate experimentally and theoretically the influence of various aspects of the pore structure on residual stress, coating adhesion to the substrate and elastic properties of the coating. For that purpose, feedstock powders of Inconel 625 were prepared with four different particle size distributions. The coatings were prepared by air-plasma spraying under nearly identical spray conditions on grit blasted steel substrates. Neutron diffraction was used to determine the average in-plane residual stresses in the coatings. The in-plane Young’s modulus was determined using a fourpoint bending apparatus. The porosity and the pore distribution were characterized using small-angle neutron scattering as well as precision density measurements. It was found that both the residual stresses and the elastic modulus of the coatings sprayed with coarse powder were considerably lower than the stresses in the coatings sprayed with the smaller particles. As the particle size decreases, a rising oxide content in the coating as well as a change in the pore distribution elevate the elastic modulus and, as a consequence of that, the residual stresses. The most pronounced effect on the pore distribution is a lower fraction of connected porosity which effectively decreases the pore aspect ratio. With no ductility left due to the embrittlement effect of the oxide particles, these coatings exhibit a low strain tolerance and the residual stresses are close to their maximum level. 507 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 507–514. © 2002 All Rights Reserved. Printed in the Netherlands.

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1. Introduction Metallic coatings are used in a variety of applications ranging from protecting surfaces from corrosion, providing wear resistance, bond-coats for thermal barrier coatings or restorative layers for machine parts. Unlike with ceramic coatings, for metallic coatings there are comparatively little data available regarding the relationship between microstructure, residual stress and mechanical properties. Differences with respect to these properties between metallic and ceramic coatings can be expected due to the ductility of metals. For example, there is evidence that the elastic anisotropy between Young’s modulus in the plane of the surface, and normal to the surface, is different for metallic coatings where is found. For ceramic coatings it is usually found that The elastic anisotropy is rooted in the preferred orientation of elongated voids among which two main populations can be distinguished: horizontal (x, y), pennyshaped voids and vertically (z) elongated cracks. Depending on their concentration and their aspect ratios, pores can substantially reduce the elastic moduli and they can create large elastic anisotropy. It is easy to understand that, for example, if more flat voids are oriented horizontally (i. e. the two long axes are horizontal) than there are vertically oriented cracks then because the vertical cross sectional area is smaller. Spherical pores are also present but they simply decrease the effective elastic modulus without adding to anisotropy. The measurement of the elastic moduli is commonly done by four-point bending, indentation or sometimes by ultrasonic resonance. More recently, diffraction has also been used. Each of these methods comes with its own restrictions but all of them have in common that they provide results only for Young’s modulus. So far, there are very few data available about Poisson’s ratios or the shear modulus. With respect to the elastic modulus two different values have to distinguished: the “mechanical” value that relates macroscopic strain to macroscopic stress, and the diffraction modulus which relates lattice strain to macroscopic stress. The latter can be quite different from the first because in a diffraction measurement only lattice strain is recorded while a macroscopic load is applied. However, the straining of the lattice of the grains in a coating is still anisotropic and it depends on the reflection (hkl) used in that measurement. There are two main conditions that determine the level of residual stress in a coating: the coating elastic constants and thermal strains that originate from temperature differences and different coefficients of thermal expansion (CTE). Variations in certain spray parameters like particle velocity in the plasma jet can cause critical changes in the bonding of particles and in the pore structure of the coating. The coating elastic constants are very sensitive to the microstructure and can be up to a factor ten smaller than bulk (zero porosity material) values. Thermal strain develops first due to the rapid quenching from the splat solidification temperature to the substrate temperature. The second thermal strain contribution develops when the substrate-coating sandwich cools down to ambient temperature with different contraction strains due to different CTEs of substrate and coating. The most commonly used methods for stress determination in coatings are diffraction, deflection methods and material removal. For several reasons we choose neutron diffraction for the strain measurements. First, we deemed X-ray diffraction

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unsuitable because of the large surface roughness of the coatings and because of the possibility of stress gradients that would lead to misinterpretation by a surface technique like X-rays. Also, with neutrons it is easy to measure at different spots on the coating in order to check the uniformity of the bonding to the substrate. Second, neutron diffraction is especially suitable for strain measurements under applied load. In particular, the high penetration of neutrons becomes very useful for measuring the inplane strain in a four-point bending apparatus. For this application it is a valuable feature of diffraction methods that only elastic strain can be measured. 2.

Experimental Procedure

The coatings were prepared by atmospheric plasma spraying at the thermal spray facility of the NIST Metallurgy Division. The substrates were low carbon steel slabs of dimensions 2. 9(t) × 29. 5(w) × 120(l) mm with corund grit blasted surface. The particle size distributions of the Inconel 625 feedstock powders were set by sieving. The details are listed in Table 1.

The composition if the powder is listed in Table 2.

The crystal structure of the material is face centered cubic with a lattice parameter of approximately 3. 6 Å. The neutron diffraction measurements were done using the residual stress diffractometer (BT8) at the NIST Center for Neutron Research, Gaithersburg, MD, U. S. A. The technique is explained in detail by Allen et al and for coatings by Kesler et al so that we will restrict ourselves to the details of a stress measurement on medium thickness coatings. The coating thickness was found to be somewhat non-uniform because of the large surface roughness of the outer surface. The thickness measurements were done using a caliper on several points on the coating. The average values were from 0.4 mm up to 0.7 mm with an uncertainty of 0.1 mm whereas neutron beams usually have sizes of 0.5 mm or larger. Fig.1 shows how the neutron measurement is done.

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Neutron diffraction requires an unstressed d-spacing as a reference value to calculate strain and stress. An equally viable alternative is the use of mechanical boundary conditions which is a method of choice for coatings. Here, the condition is fulfilled through the thickness of coating and the unstressed d-spacing can be eliminated from the equations which is basically equivalent to the technique in Xray diffraction stress analysis (Hauk (1997)). The wavelength was chosen such that the (311) reflection could be measured at Once the strains are obtained, the specific elastic constants for each coating are needed to calculate the stress. These measurements were done both with freestanding coatings and with substrate-coating composites using a four-point bending apparatus. We also conducted density measurements by immersing small pieces of coating into ethanol under vacuum to achieve the best possible penetration. Subsequently the weight of the immersed and the dry coating was measured. The density measured that way includes the enclosed (not connected to the surface) pores.

3. Results and Discussion The effect of the feedstock powder particle size on the elastic moduli, residual stresses and porosity were investigated and the results are presented in the following. The results in Tab. 3 show that the particle size has a distinct effect on the density of the coatings. Also, preliminary results from small angle neutron scattering show that the surface area ratio of surface connected porosity to enclosed porosity is about four times bigger for the coarse coating than for the medium coating (Barker (2001), making this coarse coating more like a network than a porous solid. However, X-ray diffraction measurements also indicate that the lower densities of the fine, medium and mixed coating are mostly due to a high content of nickel oxide and chromium oxide that are


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formed in-flight especially on small particles during spraying. Clearly, formation is promoted by the higher surface to volume ratio of the smaller particles.

Four-point bending measurements on free standing coatings were performed with the outer surface in compression and in tension. The results in Fig. 2 indicate that the elastic modulus is consistently higher if the outer surface is in a tensile mode. This indicates that the pore structure through the thickness of the coatings is inhomogeneous. It is known that coatings are stiffer in compression than in tension because a compressive stress closes voids with high aspect ratios while tensile stresses open even more pores. It is likely that the inner layers of the coatings have an increased porosity compared to the outer layers, thus making it easier to strain the inner layers in compression with the outer layers in tension.

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The diffraction moduli are up to a factor of two bigger than the mechanical values. The reason for this behavior is the same for all coatings: the lattice of the grains is strained only to a fraction of the strain that deforms the coating as a whole. The “rest” of the strain goes into the deformation of the pores and cracks. That means that only some microscopic regions in the bulk of the coatings deform purely in tension or compression. Other region will be subjected to bending or twisting, both of which produce less strain than pure compression or tension. The combined effect of both reduces the average lattice strain as measured in the diffraction experiment so that approximately only 50 % of an applied strain are measured as lattice strain. Using these measured diffraction moduli assures that we obtain the correct stresses from the lattice strain measurements. The very low modulus of the coarse coating can be explained as follows. Preliminary results from small angle neutron scattering experiments show that the surface area of the medium coating is slightly higher than that of the coarse coating. The connected porosity, on the other hand, is approximately five times higher for the coarse coating. A connection between elongated voids of the same orientation effectively increases the aspect ratios of the average void which has a significant impact on the elastic constants as shown in Figure 3. The self-consistent model used to calculate the values in Figure 3 approximates the coating as a two-phase composite consisting of a crystalline phase and ellipsoidal voids with elastic constants close to zero (see Willis (1981), (Matejicek et al (2001)).


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The most significant impact of the pore aspect ratio on the elastic moduli is caused if the aspect ratios are in the range from 0.1 to 0.01 for horizontal voids and from 10 to 100 for the vertical cracks, respectively. The 50% drop in elastic constants can be well explained by assuming that the average pore aspect ratio for the coarse coating is five times larger than that of the medium coating due to increased connection between pores. The residual stresses in the coatings are presented in Fig. 4. These values were obtained from the measured strains using the diffraction modulus shown in Figure 2. Poisson’s ration was measured on the coarse coating to be 0.3. This value was also used for the other coatings.

There are only small differences among samples that were sprayed with the same particle size. The small variations in coatings thickness do not cause large changes in stress for the same particle size. However, the results for different particle size distributions are a clear indication that the actual stresses are mostly controlled by the elastic modulus because both the stress and the moduli follow the same behavior. The only source of the tensile coating strains are quenching strains and thermal strains which are approximately the same for all coatings. Other processes as sliding between splats due to imperfect bonding or plastic deformation can only decrease these strains. The contribution from plastic deformation is very small because especially the medium and the fine coating contain a substantial fraction of oxides. From X-ray measurements we estimate the oxide content to approximately 20 % for the medium and the fine coating. Such a high fraction of non-shearable particles effectively suppresses any ductility. The consequence is that plastic deformation can no longer contribute to stress relaxation. As a result, both stresses and elastic moduli are the highest for these coatings. Both effects can be detrimental for the adherence of the coating because there is very little strain tolerance left.

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4. Acknowledgement

The authors would like to thank Dr Jiri Dubsky from the Institute for Plasma Physics in Prague for allowing us to use his four-point bending apparatus, and Drs Steven Mates, Steven Ridder and Rodney Jiggets from the NIST Metallurgy Division for their help in spraying the coatings. 5. References 1. 2.

3.

4.

5. 6. 7.

Barker, J. (2001), NIST Center for Neutron Research, unpublished results Allen, A. J., Mulchings, M. T., Windsor, C. G. and C. Andreani (1985), Advances in Physics, 34 (4), 445 Gnäupel-Herold, T, Matejicek, J, Prask, H J (2000), Mechanical Properties of Plasma Sprayed Coatings - Measured by Diffraction, Proc. of the 9th International Metallurgical Conference Metal 2000, Ostrava, Czech Republic, paper no. 508 Hauk, V. (1997), Structural and Residual Stress Analysis by Nondestructive Methods, Elsevier Sciece Pub., Amsterdam Kesler, O., Matejicek, J., Sampath, S., Suresh, S., Gnaeupel-Herold, T., Brand, P. C., Prask, H. J. (1998): Measurement of Residual Stress in Plasma-Sprayed Metallic, Ceramic and Composite Coatings; Mater. Sci. Eng. A Vol. 257, No. 2, p. 215-224 Matejicek, J, J, Gnäupel-Herold, T (2001) Neutron scattering in studies of complex anisotropic microstructures, Int. Conf. on Materials Structure & Micromechanics of Fracture, MSMF-3, Brno, Czech Republic, in press Willis, J. R. (1981), Advances in Applied Mechanics 21, 1-78

NEUTRON DIFFRACTION ASSISTED RESIDUAL STRESS ANALYSIS IN WELDED STRUCTURES C. OHMS and A. G. YOUTSOS Joint Research Centre of the European Commission Westerduinweg 3 1755 LE Petten, NL Abstract Welding residual stresses in structural components can significantly compromise their performance and lifetime. Prediction of welding stresses based on numerical modeling has not yet proven to be reliable, while measurement of such stresses based on NDT remains a challenging task. It is shown in this paper that neutron diffraction is a reliable non-destructive method for residual stress analysis in structural weldments. The Large Component Neutron Diffraction Facility (LCNDF) at the High Flux Reactor (HFR), Petten has facilitated residual stress measurements in various weldments, including large steel piping welds. A key issue in applying neutron diffraction to welds is the reliable estimation of the stress-free lattice distance in the heat affected zone and weld pool and in all directions of interest. Results of numerous investigations at HFR show that this is achievable by testing small coupons, cut from a companion weld specimen, which are nearly free of macro-stresses and consequently it is reasonable to be used as reference specimens. In fact, the feasibility of this approach has been demonstrated in monolithic and bimetallic welds. In this paper residual strain/stress data in five welded specimens, based on neutron diffraction, are presented. The results presented in this paper consistently show that, by ignoring the spatial and directional variation of the reference lattice distance, which is exhibited throughout the weld pool and the heat affected zones, erroneous strain data can be derived leading to non self-equilibrating internal stress estimates. Keywords: Residual stress, welding, neutron diffraction, large components, coupons

1. Introduction Neutron diffraction is a powerful technique for non-destructive analysis of residual stresses deep within crystalline materials at reasonable spatial resolution. The HFR provides two facilities dedicated to this test method. One of these is the Large Component Neutron Diffraction Facility (LCNDF) capable of handling specimen weights up to 1000 kg. In testing thick structural weldments and their respective heat affected zones (HAZs) by neutron diffraction, one has to take into account variations of the stress free reference parameter throughout the weld and the HAZ. These are due to plastic 515 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 515–526. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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anisotropy, changes in chemical composition and texture formation caused by the welding process. In neutron diffraction this problem can be handled by using coupons carefully cut to the appropriate size as free of macroscopic stress representatives of the measurement locations that are going to be investigated. For the economy of the paper, due to its nature and also for the sake of the reader, we present together in Section 2 the experimental campaigns based on neutron diffraction and the respective results. Two brief Sections follow this on discussion of the results and conclusions respectively. In this paper residual strain/stress data, based on neutron diffraction, in five welded specimens are presented. The first specimen is a section cut from an austenitic steel piping butt weld of 68-mm thickness. A large campaign of reference measurements was conducted, which was based on a large number of coupons cut from a thin companion specimen. This has shown the extent of potential errors, which are introduced in the strain analysis, if reference variations are neglected. The second specimen is a 25-mm thick post-weld heat-treated austenitic/ferritic steel fusion welded piping section. In this case "comb" type reference specimens were used, and the derived stress data were found to be in good agreement with computational results based on numerical modeling techniques. The third sample is a 65-mm thick section girth weld joining two ex-service AISI type 316H austenitic stainless steel forgings. Due to the high wall thickness, residual strains were measured adequately only in the radial direction in this case. This is confirmed by comparison of measured strains with computational data based on FEM techniques. Measurements were made before and after PWHT in order to evaluate the efficiency of the applied stress relief process. The fourth specimen is a ferritic steel alloy fusion welded plate of 12. 5-mm thickness. Residual stresses in this specimen have been investigated at various neutron diffraction facilities worldwide to establish feasibility of neutron diffraction. For the HFR investigation reported in this paper, reference measurements based on “comb” type reference specimens, which have revealed negligible reference variations. Finally, the fifth sample is an aluminum alloy friction welded plate of 6-mm thickness, representative of an aircraft wing welded section.

2. Experimental Campaigns and Results Based on Neutron Diffraction 2. 1. AUSTENITIC STEEL BUTT WELD SECTION Measurements have been performed on a thick nuclear piping austenitic steel butt weld. Fig. 1 shows the location and size of the coupons, which were cut from a companion specimen for the investigation of the macro-stress free lattice parameters [1]. All coupons are 5-mm cubes with their centres at the strain measurement locations. Measurements were made at all locations of the weld transverse direction - line G, which coincides with the piping axial direction, and all locations of the weld axial direction - column 6, which coincides with the piping radial direction. For each of the investigated coupons the lattice parameter was measured using the 113- and 002-refleciton planes in the normal, axial and weld directions. The sampling volume chosen for all tests was 3x3x3 and at a fixed wavelength of 1. 528 Å using the Ge-113 neutrons of the monochromator at the E3 neutron beam at former CRNL, CAN. This wavelength gives

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rise to a value near for the 113-reflection, which renders a cubical sampling volume, but only for the 002-reflection.

Figs. 2 – 3 and 4-5 show the measured diffraction angles in all the coupons examined for the weld axial and normal directions respectively. It can be readily observed that in all cases reported in Figs. 2-5 the values within the weld are much higher than in the parent material. In fact for the 113-reflections varies by as much as and for 002 by about resulting into compressive pseudo strain levels in the range of 2200 and 2800 respectively. Strain measurements were made in the weld specimen and in its normal and axial directions using both 002 and 113.

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Fig. 6 shows the resulting residual strains in the axial direction along the transverse axis of the weld based on the 002-reflection and taking correctly into account the


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macro-stress free lattice parameter variation as shown in Fig. 3. Fig. 6 also shows the pseudo strain data, which one would obtain if this variation were neglected. The plotted results in this Figure suggest that the parent material is undergoing large compressive strains while the strains in the weld are tensile and relatively low. Fig. 7 shows strains measured in the weld axial direction and along its axis of symmetry. These results show, as those plotted in Fig. 6, that the weld material is exhibiting low tensile strains along the weld axis. Position M6 appears as an exception due to its position with respect to the geometry of the weld (see Fig. 1).

2.2. BIMETALLIC WELD SPECIMEN The specimen investigated is a tube of 393 mm length, 168 mm outer diameter and 25 mm wall thickness. The weld has been applied circumferentially at mid-length. On the outer and inner surfaces the transition from the austenitic to the ferritic material is clearly visible. Measurements were performed at various locations within the weld, the buttering layer and the HAZs in three ortho-normal directions, i. e., tube hoop, axial and radial directions. The measurement locations are shown in Fig. 8. In order to facilitate

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these measurements windows had to be cut into the component for providing access for the neutron guides. From the thus removed material a slice was then cut in order to provide for the reference coupons; these were wire eroded in form of 6 x 6 columns. Thus two reference specimens were made, one representing the locations within the buttering layer and the ferritic steel HAZ and the other representing the locations within the weld and the austenitic steel HAZ [2]. Based on the assumption of macro-stress relief in the columns through cutting, these were used for reference measurements at each location in each direction. The most appropriate test procedure was found to be operating at a fixed scattering angle 29, while varying the neutron wavelength, using an adjustable double-monochromator, for the various material phases.

First tests revealed a strong (200)-texture within the weld and buttering layer, while in the austenitic and ferritic HAZs the (111)- and (110)-reflections could be used, respectively. In order to illustrate the necessity of measuring location dependent references, Fig. 9 shows apparent strains derived based on a common reference


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value for each of the four measurement regions (ferritic HAZ, buttering, weld and austenitic HAZ). It can be seen that apparent strains of more than 1000 can be introduced by ignoring the reference parameter variation. This effect was found to be most significant within the weld. Nevertheless it is also present in the other measurement regions. The measurements were performed using sampling volume of 4x4x5 to 4x4x10 scattering angle of 76.150° and the following crystallographic reflections: ferrite (110), austenite (200) in weld and buttering, and austenite (111) in HAZ austenite. From the thus derived strains, residual stresses were calculated. The elastic constants related to the various material reflections were taken from the literature [3]. Figures 10-12 show the residual stresses that were found in the hoop, axial and radial directions, respectively. These show compression in the ferritic steel, and tension in the austenitic phase in the hoop direction, while in the axial direction high stresses are present closer to the outer and inner specimen surfaces and low stresses near the middle of the piping wall. Radial stresses remain within ±100 MPa.

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2. 3. AUSTENITIC STEEL GIRTH WELD The spatial distribution of residual stress in a thick section girth weld (432 mm outside diameter, 65 mm thick) joining two ex-service AISI type 316H austenitic stainless steel forgings was characterized by both neutron diffraction testing and numerical modeling [4]. A neutron wavelength of 2. 8 Å was employed rendering a scattering angle of about 84. 5° with the austenitic (111) reflection plane. A large sampling volume was selected for measurements in the specimen radial, longitudinal and hoops directions. Seven locations through the piping wall in the HAZ were measured (nearly 20 mm from the weld centerline). Further measurements were made 6 and 20 mm below the outer specimen surface at various distances from the weld centerline. All hoop, longitudinal and radial direction strain measurements at mid-thickness of the specimen wall necessitated a total neutron path length within the steel of about 90 - 95 mm. This proved to be beyond the capabilities of the diffractometer because of the resulting severe neutron beam intensity attenuation. The above experimental campaigns were repeated following completion of the PWHT of 2. 5 hours at 750°C. The strains presented in Fig. 13 are based on a unique reference value derived from measurements in the parent material far from the weld. This Figure shows the apparent radial strains before and after PWHT measured through thickness in the HAZ 20 mm from the weld centre line. The strain relief is positive for the test locations closer to the outer surface whereas is becomes negative at the inner surface. The difference in measured radial strains, that is the strain relief, is in good agreement with equivalent results based on FE simulation performed by British Energy, as Fig. 14 clearly shows. Additional tests indicate that measurements, based on the selected sampling volume and wavelength, would have been feasible for up to 25-28 mm below the specimen surface locations, while beyond this depth the deteriorating quality and reliability of data seems to be prohibitive for neutron diffraction stress testing. This means that, and in agreement with previous experience, reliable neutron diffraction data could be obtained for a piping wall thickness of up to 50 - 55 mm.


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2. 4. FERRITIC STEEL WELD Residual strains and stresses were measured at mid-length of a ferritic steel plate of 200x150x12.5 mm. A 12-pass TIG-weld was applied to the plate at mid-width. Various neutron diffraction laboratories worldwide tested the specimen and a comprehensive report on the obtained results is due to appear shortly. At JRC tests were performed using a gauge cross section and a neutron wavelength of 2. 57 Å. Reference coupons in this case did not reveal significant variations in the stress free lattice parameter thus an average was used as reference value for all tests. Figs. 15-16 show the strains and stresses measured at the HFR-Petten at the plate mid-thickness in the weld longitudinal (X), weld transverse (Y) and plate normal (Z) directions.

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2. 5. ALUMINUM ALLOY FRICTION STIR WELD

Two Al-7010 plates of 6 mm thickness and 120 mm width were joined through friction stir welding. ILL, Grenoble, and HFR-Petten performed strain measurements based on neutron diffraction. The aluminium (111) reflection was chosen and an incident beam wavelength of about 2. 99 and 2. 57 Å, respectively, giving rise to Bragg angles of about 79. 4° and 67°. Both facilities employed gauge volumes of cross section. An eventual variation in the reference parameter was not taken into account in this case. Whereas the agreement between ILL and JRC results in the weld transverse direction is remarkably good (see Fig. 17), in the plate normal direction only a reasonable qualitative agreement could be established (Fig. 18) due to the unfavourable texture of the material. The alignment discrepancy between ILL and JRC results is 0. 5 to 0. 75 mm. Fig. 19 shows the weld transverse strains obtained by JRC at the midlength of the specimen compared to test results from other measurement locations. The


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good agreement in the data suggests uniformity of the stress distribution over a large range of the specimen.

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3. Discussion of Results The austenitic steel butt weld section analysis reveals significant spatial and directional variations in the reference lattice spacing. The bimetallic weld exhibits, as expected, an asymmetric stress pattern in the hoop direction, whereby the ferritic part is predominantly under compression, while the austenitic HAZ is mainly under tension. The transition from the compression-dominated to the tension-dominated zone takes place at the ferritic steel buttering interface, which exhibits a very significant stress gradient. In radial direction only minor stresses were found varying between +100 and -100 MPa. Axial stresses are tensile at the outer specimen surface and compressive at the inside. In both cases the stresses tend to be higher in the ferritic than in the austenitic HAZ. Slightly lower stresses were found in the weld itself. It seems that the stresses found in the axial direction do not balance everywhere. Both the bimetallic weld stress tensor data and the austenitic steel girth weld radial strain relief data are in good agreement with similar results based on numerical modelling. In the ferritic weld case the test data, for strain and stress alike, clearly show symmetry where expected. Finally, strain/stress data for the ferritic steel weld and strain data for the aluminium alloy weld are in very good agreement with similar results based on neutron diffraction testing carried out at other than the HFR facilities.

4. Conclusions Neutron diffraction proves to be a very suitable tool for non-destructive determination of residual stress in structural welds. It was shown that the HFR LCNDF could accommodate efficient strain/stress testing of steel welded structures with up to 50-mm wall thickness. Measurement of the reference scattering angle at each location and in all measurement directions proved to be necessary in austenitic and austenitic/ferritic welds for accurate strain estimates. The statistical error of uncertainty associated with all neutron strain/stress analyses is, generally, quite low, i. e., 75-150 and 5-20 MPa.

5. Acknowledgements This research was carried out within the European Commissions’ Research and Development Programme. The authors wish to thank Messrs. Th. Timke and P. van den Idsert for their valuable contributions to the above work. The contributions of Messrs. Th. Holden (ex-CRNL) and Th. Pirling (ILL) are greatly acknowledged.

6.

References

1. 2. 3. 4.

C. Ohms and A. G. Youtsos, J. of Textures and Microstructures, 33, (1999) 243-262. C. Ohms, A. G. Youtsos, Materials Science Forum, 347-349 (2000) 658-663. B. Eigenmann, E. Macherauch, Mat. -wiss. U. Werkstofftech., 27 (1996) 426-437. P. J. Bouchard, S. K. Bate, D. George, R. H. Leggatt and A. G. Youtsos, Proceedings of the Conference on residual stress, Vol. 2, IOM Communications Ltd., London, 2000, 972-979.

Int.

SYNCHROTRON RADIATION IN-SITU ANALYSES OF AA 6061 + DURING TENSILE DEFORMATION AT AMBIENT AND ELEVATED TEMPERATURE A. PYZALLA TU Berlin, Institute for Materials Science and Technology Sekr. BH 18, Ernst-Reuter-Platz 1, D-10587 Berlin, Germany B. REETZ TU Berlin, Institute for Materials Science and Technology Sekr. BH 18, Ernst-Reuter-Platz 1, D-10587 Berlin, Germany A. JACQUES L.P.M. (UMR CNRS 7556), Ecole des Mines de Nancy Parc de Saurupt, F-54042, Nancy, France J.-P, FEIEREISEN L.P.M. (UMR CNRS 7556), Ecole des Mines de Nancy Pare de Saurupt, F-54042, Nancy, France O. FERRY L.P.M. (UMR CNRS 7556), Ecole des Mines de Nancy Parc de Saurupt, F-54042, Nancy, France T. BUSLAPS European Synchrotron Radiation Facility (ESRF) BP 220, F-36043 Grenoble, France Abstract High energy synchrotron radiation enables the in-situ determination of the elastic strain developing during deformation of metals and especially of metal matrix composites. By using a white beam the strain response can be characterised simultaneous with texture development. The high photon flux of the synchrotron beam allows short data acquisition times and thus an in-situ combined stress and texture determination is possible at high temperature. Here results of investigations of the tensile load stress – elastic strain response of aluminium metal matrix composites are presented. Keywords: synchrotron radiation, high energy, metal matrix composite, aluminium oxide, tensile deformation, elevated temperature.

1. Introduction Metal matrix composites (MMCs) combine the beneficial properties of their constituents. The embedding of suitable ceramic hard phases in an aluminium base alloy produces light weight wear resistant materials. These materials are used in 527 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 527–534. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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aerospace and automotive applications, e. g. in automobile motors. In the automobile motors the aluminium metal matrix composites are subjected to elevated temperatures. Thus, the combination of the aluminium alloy matrix with alumina particles is beneficial with respect to the weight as well as with respect to the overall thermal expansion of the component. But, the differences in the physical properties and the mechanical behaviour of the constituents result in a non-uniform load uptake of the thermal and the mechanical loading of the component and also in the formation of residual stresses after cooling. In order to determine the elastic strain and the load sharing between the aluminium matrix and the particles at elevated temperature, in-situ tensile tests of AA6061 + - samples in a furnace with a loading device were performed using white synchrotron radiation [1, 2]. At ambient temperature in-situ stress analysis in composites can well be done using neutron diffraction [e. g. 3, 4]. Typical data acquisition times are in the range of several minutes. At elevated temperature during this time interval severe stress relaxation or creep effects may occur. Thus, at high temperatures the use of synchrotron radiation with its high flux and, thus, a short acquisition time is preferable. The high photon flux at the sample position enables the recording of a spectrum with sufficient counting statistics within a few ten to a few hundred seconds. By using an energy dispersive set-up simultaneous to the information about the elastic strain in both phases of the composites also information about the texture development in the composites is available.

2. Experimental Details

Material The AA 6061 + – composites were manufactured by hot extrusion. From the extrusion products samples with a diameter of 4, 4 mm and an overall length of 45 mm were manufactured. The active length amounts to 25 mm. The longitudinal axis of the samples corresponds to the longitudinal direction of the extrusion profiles. The microstructure of AA6161 + in the as - manufactured condition is characterised by a homogeneous distribution of the alumina particles in the aluminium matrix. (Fig. 1).

The ceramic – particles have a rectangular shape and a size of The grain size of the AA6061 matrix is 40 At the interface between the

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AA6061 – matrix and the particles a fringe containing an aluminiummagnesium - spinell - phase is present (Fig. 2).

Experiment Set-Up The experiments were performed at the High Energy Beamline ID 15A of the European Synchrotron radiation facility (ESRF) in Grenoble, France. For the experiments an energy dispersive arrangement using the white beam was chosen. This set-up was described in detail in [1, 5]. For the experiments a diffraction angle of was chosen as a compromise between maximising the resolution respectively the photon flux available. In order to define the gauge volume a slit with variable horizontal and vertical gap as well as 2 slits with 60 horizontal gap and 10 mm vertical gap were inserted in front of the sample. In the diffracted beam two further slits with gaps of the same size 60 x 10 mm were placed. The first of these slits was directly attached to the surface of the furnace. The second slit was mounted closed to the detector. The size and the shape of the gauge volume was determined experimentally using a thin aluminium foil. It was found to be 70 wide and 1200 long, approximately. The gauge volume was centred in the sample using maximum integrated intensity values of translation scans. In order to increase the number of scattering grains the sample was continuously oscillated by ± 1° in during the measurements. An energy– dispersive Ge-detector was calibrated in an energy range of 25 keV to 250 keV using the characteristic lines of a source.

Furnace with Integrated Tensile Test Device The furnace allowing in-situ tensile tests (Fig.. 3) with a load up to 5 KN at elevated temperatures up to 1200° was developed at L. P. M. (Ecole des Mines Nancy). The sample is heated-up by two pairs of graphite resistors. The combined tensile test – heating device was set-up on the sample stage. The sample was installed horizontally, so that the longitudinal axis of the sample pointed in the direction of the scattering vector, that is nearly perpendicular to the incoming beam. The samples were heated-up in vacuum. Their temperature during the tensile deformation was measured using three thermocouples. Using the local temperature recorded at the different sample positions by the thermocouples the temperature of

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the samples could be controlled. For experiments scheduled at a temperature of 200°C the temperature of the sample could be kept constant within a temperature range of approximately ± 20K

Data Evaluation Diffraction spectra were taken from the unloaded samples and after defined loading steps up to a maximum load stress of 150 MPa. For all spectra an acquisition time of 300s was used. From the energy dispersive spectra the line position, the integrated intensity and the full width at half maximum of different reflections were obtained by fitting the reflection profiles using a Gaussian function. The energy value representing the line position corresponds to the lattice spacing including the lattice strain. Thus can be calculated according to Bragg’s law. Then the stresses are obtained by Hooke’s law taking account of the phase and reflection specific XECs. For the determination of the strain-free lattice distance of the particles, powder was chemically extracted from the samples. A spectrum of the powder was recorded at ambient and at elevated temperature.

3. Results A typical high energy synchrotron radiation spectrum of AA6061 + 15vol. -% is shown in Fig. 4.


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Obviously in the energy range between 50 keV and 150 keV a number of reflections of the aluminium matrix is accessible. Due to the low – content of the metal matrix composite and due to the hexagonal structure of the there is a multitude of reflections with significantly lower intensity. The ratio between the intensity of the reflections and the background of the spectrum obtained justifies the short acquisition time of 300s chosen for both the aluminium matrix and the particles.

Tensile deformation at ambient temperature The elastic strain – load stress - curve obtained for the particles (Fig. 5) shows initially compressive residual strains which result from the cooling of the AA6061 – alloy after the extrusion processes and the heat treatment due to the differences in the thermal expansion of the particles and the aluminium matrix.

During the tensile loading the elastic strain in the particles changes into tensile strain values which increases steadily with increasing tensile external load. This indicates that part of the load stress during the deformation is taken up by the particles. The spectra reveal further, that the escalation of the external load does not only result in increasing strain of the particles but also in changes of the reflection intensities. As an example the development of the intensity of the 0 2 10 reflection of the – particles due to increasing loading is shown in Fig. 6. This particle behaviour is quite different from our earlier observations on an Al-MMC with globular silicon particles [6]. The values presented are peak maxima normalised by the total photon count rate and based on the peak intensity obtained in the initial state, before loading. The change of the reflection intensity monitored reveals that the particle orientation changes slowly at external stress below 150 MPa approximately and increase strongly if the sample is subjected to higher external tensile stresses. The change of

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the particle orientation is accompanied by an increase in the full width at half maximum (FWHM) of the reflections of the – particles (Fig. 7).

This indicates that strong micro strains are present in the particles due to inhomogeneities in their shape and orientation as well as their individual strength. Microscopical investigations of the composites even reveal that some of the particles show damage effects (Fig. 8).


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The aluminium matrix contains tensile residual strains before loading, which balance the compressive residual strains in the particles. With increasing tensile external load also the tensile strains in the aluminium matrix increase. This increase is linear up to an external strain of MPa which corresponds to the start of the formation of strong micro stresses in the – particles. The matrix reflections also show that the aluminium matrix tends to enhance its / double fibre texture, especially the - fibre component. This texture formation during the in-situ tensile deformation, however, is weak, which is due to the / - fibre texture being present even before loading due to the manufacturing of the profiles by hot extrusion. Tensile deformation at 200°C The lattice distances of powder and the particles in the composite reveal that at 200°C, before loading, due to stress relaxation effects during the heating of the sample the particles and thus also the aluminium matrix are strain-free within the limits of experimental accuracy. The load stress – elastic strain curve of the – particles obtained during tensile loading at 200°C is shown in Fig. 9.

A comparison of this curve with the curve obtained at room temperature (fig. 5) reveals that at 200°C the strains of the – particles are higher at equivalent load stresses than at room temperature. Further on the elastic strain in the aluminium metal matrix in case of the deformation at 200°C is higher than it is at equivalent load stresses in case of deformation at room temperature. This is due to the decrease of the modulus of elasticity at increasing temperature. As was observed during room temperature deformation also during deformation at 200°C the - fibre component of the aluminium metal matrix increases in strength.

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4. Conclusions and Outlook Tensile deformation of AA6061 + 15 vol. -% was carried out at ambient and at elevated temperature using white high energy synchrotron radiation. The results of the experiments show the development of the lattice strain in the aluminium matrix and the particles with increasing tensile load stress. At high external stresses the – particles tend to assume a preferred orientation and this orientation change is accompanied with the formation of micro stresses in the particles. Further work will concentrate on the investigation on the load transfer between the phases and its relation to particle damage when the sample is subjected to high external strains and stresses.

5. Acknowledgements The authors gratefully acknowledge the allocation of beamtime and the use of the beamline ID 15A at the ESRF Grenoble, France. A. Pyzalla and B. Reetz would like to thank the Deutsche Forschungsgemeinschaft for the grant Az. Py 9/1-1.

6. References 1. 2. 3. 4.

5. 6.

Reimers, W., Pyzalla, A., Broda, M., Brusch, G., Dantz, D., Liss, K.. -D., Schmackers, T., Tschentscher, T.: J. Mat. Sci. Letters 19 (1999) 581 – 583. Pyzalla, A., Reimers, W.: Mat. Sci. Forum Vols. 347 – 349 (2000) 34 – 39. Seol, K., Krawitz, A. D.. Mat. Sci. Eng. A127 (1990), 1-5. Korsunsky, A., Daymond, M. R., Wells, K. E.: Mat. Sci. Forum Vols. 347– 349 (2000) 492-497 Pyzalla, A . : J. Nondestructive Evaluation 19 (2000) 21-31. Pyzalla, A.; Jacques, A.; Feiereisen, J. P.; Buslaps, T.; D’Almeida, T.; Liss, K. In - situ Analyses of the Microstrains during Tensile Deformation of an AlSi – MMC at Room Temperature and Elevated Temperature, J. Neutron Diffraction, in print.

7. Hybrid Methods


REFLECTIONS ON THE IMPORTANCE OF EXPERIMENTAL RESULTS TO ALL MECHANICISTS, ESPECIALLY THEORETICIANS ROBERT M. JONES Emeritus Professor of Engineering Science and Mechanics Virginia Tech Blacksburg, Virginia 24061-0219 Abstract All too often, mechanicists divide themselves into two camps, theoreticians and experimentalists. This unfortunate state of affairs denies us the essential two-sided, realistic view of important physical problems. We must realize that theory cannot exist without suitable experiments, and that experiments cannot be properly designed or used in engineering practice without some form of theory. Thoughtful, penetrating interaction between theoretician and experimentalist is crucial. The truly effective experimentalist must have a strong theoretical background. The truly effective theoretician must have at least a rudimentary, if not a strong, experimental background. Of course, the ideal mechanicist would be neither a theoretician nor an experimentalist, but both simultaneously! The important factor is cross-talk between the two points of view with the objective of achieving a unified view of a phenomenon and its model that can be used to treat behavior beyond that which was observed in experiments. This paper is about the elements that go into making up the unified viewpoint. Those elements include appreciation of the actual characteristics of the phenomenon studied as determined by experimental techniques and how those charcteristics help develop and refine every theory. The point is that experiment is the driving force behind theory.

1. Introduction We will consider examples of agreement between theoretical predictions and experimental measurements of behavior of two complicated mechanical systems. They are the illustrations of the usual comparison between theory and experiment. But, what more do we want? Can we perceive a stronger useful purpose for experimental measurements than verification of theory? Of course we can! To perform any theoretical prediction well, we must have a lot of information about the behavior of the widget we are investigating. Without that information, we are totally in the dark about how to model the widget. Or else we’re kidding ourselves that we can construct a useful model without extensive fundamental (experimental) information. However, some people attempt to do just that! They seem to drastically oversimplify the problem so they can use a convenient overly simplistic mathematical model. This paper is addressed to those of us who are more rational and realize, if we only thought about it, that theoreticians need experimentalists much more that we might admit. This paper has four major parts: (1) examples of good agreement and interaction between theory and experiment, (2) how experiments are used more than ‘just’ to verify theory, (3) how experiments are the very basis of the presumptions we make in every theory to effectively capture the essence of the observed behavior, and (4) why the term ‘presumption’ should be used in place of assumption because of a clear relation to experimental evidence. 537 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 537–550. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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2. Examples of Good Agreement between Theory and Experiment Two examples that are representative of the complexities involved in contemporary behavioral modeling will be discussed. One is the interlaminar stress phenomenon found in laminated fiber-reinforced composite materials, and the other is the high-temperature behavior of particulate composite materials under extreme thermal gradients. The fundamental modeling of complex material properties is described along with how the final behavior is predicted and measured.

2.1 INTERLAMINAR STRESSES IN FIBER-REINFORCED COMPOSITE LAMINATES In classical lamination theory for fiber-reinforced composite laminates, no account is taken of stresses such as and in the coordinates of the laminate in Figure 1. Those stresses are called interlaminar stresses, and they exist on surfaces between adjacent layers, although they also exist within the layers but are usually largest at the layer interfaces. Thus, only the stresses in the plane of the laminate, and are considered; that is, a plane-stress state is presumed to exist. Unfortunately, classical lamination theory often implies values of and where they cannot possibly exist, namely on the unloaded edge or free edge of a laminate. Accordingly, classical lamination theory includes some stresses that cannot exist and does not include some of the stresses that do exist and actually cause failure of a composite laminate. High interlaminar stresses are the basis for one of the failure mechanisms uniquely characteristic of composite laminates, namely, free-edge delamination and subsequent delamination growth as shown in Figure 1. There, the laminae could come apart in the as shown, or they could also merely experience a crack between them and then slide along the crack in either the or

The contribution of Pipes and Daniel [1] was to firmly establish that the theoretical stress distribution predicted by Pipes and Pagano [2] was indeed correct. Their approach was the Moiré fringe technique to examine the surface displacements of a symmetric angle-ply laminate under axial extension. At various load levels on long, flat graphiteepoxy specimens, Moiré fringes were photographed on the top surface of the upper angleply lamina. That lamina is one lamina thickness away from the interlaminar plane where both laminae must be a rectangle at their interface. The farther away from that interlaminar plane, the more the top lamina tends to deform into a parallelogram. The S-shaped Moiré fringes associated with the axial displacements are shown along with the elasticity

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solution of Pipes and Pagano in Figure 2. If an orthotropic lamina is loaded off-axis with a tensile stress, then shear-extension coupling exists, leading to an originally rectangular shape both elongating and shearing into a parallelogram. That shape is the natural shape toward which even a lamina in a laminate is tending as we observe the behavior as we go away from the interface between the two top layers. Thus, a line drawn across the specimen perpendicular to the axial load before loading tends to deform into a diagonal line indicating shear deformation. However, the influence in the top layer of the shear stress which is high at the free edge and quickly decreases in the direction away from the edge and in the direction toward the top surface, is to deform the diagonal line more at the free edge than as the middle of the laminate is approached. Thus, the predicted surface deformation is the somewhat S-shaped divergence from a straight line. That predicted divergence is plotted with the measured deformation in Figure 2 where we see excellent agreement. Thus, the physical existence of interlaminar stresses has been clearly demonstrated. One observation is that the ’weird’ shape of the theoretical displacement results was confirmed by the identically ’weird’ shape of the measured displacements. Thus, both theory and experiment must be correct!

2.2 THERMALLY STRESSED ATJ-S GRAPHITE DISKS An annular disk, similar in shape to a lifesaver as seen in Figure 3, is exposed to very high transient radially inward induction heating on its outer perimeter as a test of how well some materials behave in thermal shock conditions. In this Southern Research Institute thermal stress disk test, a high outside-to-inside temperature gradient results. The outside portion of the disk tends to expand more than the inside, so very severe thermal stresses are generated. The stresses in the circumferential direction are compressive near the outer diameter and tensile near the inner diameter. The slanted wedge disk has an inclined inner

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diameter surface to provide a well-defined target for a laser diameter-measuring device. All the many references are contained in the summary paper for this problem by Jones and Starrett [3].

2.2.1 ATJ-S MATERIAL CHARACTERISTICS ATJ-S graphite is a transversely isotropic granular or particulate composite material made in cylindrical billet form, as shown in Figure 4. The flake-like graphite particles are aligned in planes of isotropy because of billet compaction in the The resulting material stress-strain behavior is highly nonlinear, bimodular (quite different in tension than in compression), and strongly temperature-dependent, as displayed for the plane in Figures 5 and 6. There, the boxes are actual experimental data reported by Starrett and Pears, and the curves are the Jones-Nelson nonlinear material model fits to the data. In Figure 5, the tension behavior becomes stiffer as the temperature approaches 2000°F (1100°C), and even stiffer at 3000°F (1650°C). However, the stress-strain curve at 3000°F (1650°C) is slightly lower for high strains than that at 2000°F (1100°C). The compression behavior monotonically becomes more flexible as the temperature increases from 3000°F (1650°C) to 5000°F (2800°C) in Figure 6. Moreover, the stress-strain behavior is different under tension loading than under compression loading at every temperature, although the two behaviors are shown here only for3000°F (1650°C).


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2.2.2 INNER DIAMETER CHANGE PREDICTIONS Times late in the test run are selected for correlation of predicted and measured diameter change results. By then, the disk should be deforming nonlinearly, i.e., the stresses should be inelastic. The measured inner diameter changes are shown in Figure 7 for two orthogonal directions as a function of time. The change of diameter in the two directions is measured with two electronically equivalent circuits. The measurement in the y-direction is much less noisy than in the However, both measurements are sufficiently accurate for the present correlation effort without calibration. The difference in measured deformations in the and also can be attributed to the fact that the disk hole

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does not remain perfectly circular. Of course, the hole should remain nominally circular because the material is nominally isotropic in the plane of the hole, and the temperature distribution is nominally axisymmetric about the perpendicular to the plane of the disk. However, the material does not have perfect transverse isotropy, nor is the temperature distribution perfectly axisymmetric. The disk inner diameter change is predicted with the nonlinear material model embedded in a version of the SAAS III finite element computer program. The predicted diameter change is remarkably close to the measured diameter change (within 3%) in Figure 7. The nonlinear predictions are consistently from 2.2 to 3.3% below the average measured values. However, the elastic predictions are 5% lower than the nonlinear predictions at sec, 3.6% lower at sec, but 3.4% higher at Thus, the nonlinear predictions for this body with a strong nonhomogeneous character (because of the temperature gradient and temperature-dependent mechanical properties) are more consistent (qualitative) and more accurate (quantitative) relative to the measured deformations than are the elastic predictions. However, the percentage differences between the various approaches are not high enough to warrant strong claims of accuracy for the present approach. The reported agreement between predictions and measurements is a necessary but not sufficient condition for testing the accuracy of the nonlinear model.

2.2.3 STRESS, STRAIN, AND TEMPERATURE PREDICTIONS The predicted stresses and are shown, along with the corresponding temperature distribution at in Figure 8. Stresses of course cannot be measured; however, their predicted values are examined to determine the degree of stress-strain curve nonlinearity at various times in the thermal stress disk test. Although we expect to predominate, substantial values of tensile radial stress, do exist in Figure 8. Despite the fact that the surfaces have zero radial stress, can have substantial values elsewhere because of the small disk inner diameter and the high circumferential stresses to which is inversely proportional and proportional, respectively. Thus, the disk stress state consists of areas of biaxial tension near the inner diameter. Accordingly, the bimodulus capability of the Jones-Nelson model is essential to accurate stress analysis of the disk.


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2.2.4 SUMMARY The Jones-Nelson nonlinear material model is verified for rapid thermal loading problems involving irregularly shaped nonhomogeneous bodies. The nonhomogeneity results from a temperature gradient over a body with strongly temperature-dependent mechanical properties. Moreover, the model is shown to be valid for materials with highly nonlinear stress-strain behavior that is different under tension loading than under compression loading. The vehicle for the verification of the model is the Southern Research Institute thermal stress disk test. The inner diameter changes of this annular wedge-shaped disk made of ATJ-S graphite are predicted with the model to within about 3% over a relatively wide range of temperatures and hence material behavior. This close agreement is a vivid demonstration of the good performance of the nonlinear material model in an environment with temperatures that change rapidly in space and time, a complex geometry, and a material with properties that are a strong function of temperature level, stress level, and stress sign. The irregularly shaped body is treated with a finite element model in which the material model is embedded. Because of the agreement between nonlinear predictions and measurements (a necessary but not sufficient condition for validity of the model), the present results are an important step in the qualification of the model for general use in nonlinear material deformation problems. How were we able to develop such an accurate model of material and structural behavior? Answer: by integrating our knowledge of many behavioral characteristics obtained solely by experimental observation! The credit goes to the superb high-temperature and high-temperature-gradient property measuring skills, equipment, and understanding of the people at Southern Research Institute.

2.3 EFFECT OF GOOD AGREEMENTS BETWEEN THEORY AND EXPERIMENT When we get good agreement between theory and experiment, we tend to feel we understand the phenomena involved, and we congratulate each other. However, getting to the point where we agree on how to model a phenomenon as well as how to measure that phenomenon might take years of effort by both experimentalists and theoreticians.

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3. Experimental Results are Used More than ”Just” to Verify Theory There is an absolute need for experimental results to go beyond mere verification of theory and for theory to extend the domain of usefulness of experimental results that obviously cannot be obtained for every conceivable circumstance. We considered two examples of good correlation between theory and experiment in the previous section. The point was that both theory and experiment had become refined enough that clearly superior results were obtained, and they were in agreement. Here, we address further uses of experiment in the all-important theory-experiment interaction that must exist for mechanicists to make progress. Josef Singer describes eight motives for experiments in the area of shell buckling, and their interpretation in light of more general mechanics topics should be fairly obvious:

1. “Better understanding of buckling and postbuckling behavior and the primary factors affecting it. 2. To find new phenomena. 3. To obtain better inputs for computations. 4. To obtain correlation factors between analysis and test and for materials effects. 5. To build confidence in multipurpose computer programs. 6. To test novel ideas of construction or very complicated elements of a structure.

7. For buckling under dynamic loading and in fluid-structure interaction problems. 8. For certification tests of full-scale structures.”

Singer originated those points in 1982, and he and his coauthors, Arbocz and Weller, go on to explain each of them in their remarkable book Buckling Experiments: Experimental Methods in Buckling of Thin-Walled Structures [4]. Drucker hit the alarm bell in 1962 when he lamented that too many students were choosing to study theoretical approaches to mechanics problems rather than taking up the experimental reins. In fact, he was so upset that he said “Unless appreciable numbers of the most qualified students aim at combined experimental and theoretical research, the storehouse of physical information will be depleted by the tremendous emphasis on analysis and theory, and the theorist will be reduced to playing useless games. Experiment is essential, it is vital, and it is creative. Over the years, experiment provides the basis for the refinement of existing theory and the development of new theory” [5]. That these two luminaries of mechanics put so much emphasis on the importance of experimental work ‘speaks volumes’. Dan Drucker spoke those words in 1962, and his laments and cautions are probably more applicable today than when he first spoke them! Joe Singer spoke twenty years later, and addressed the still-needed work in experimental research, especially in shell buckling. The combination of those two points of view is a powerful commentary on the status of mechanics today. Our focus on theory at the expense of experiment has ‘led us down a garden path’ to overreliance on the computer to solve our problems to the extent that we hardly appreciate experimental work, and that state of affairs is a real professional shame.


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4. Experimental Results are the Very Basis of all Theoretical Models That experimental results are the fundamental basis of all theoretical models is a truism known to all (“Whatever you do, try to have a reason to do it”, said Forrest Gump [6]). Experimental results should be the driving force for theory! Experiments should be revealing new behavioral characteristics and motivating ever-advanced theory until the phenomenon is well understood. In the theoreticians’ world of curly overbars, underbars, various squiggles, bold-faced symbols, and other manifestations of what we often (questionably) regard as elegant, theoreticians must recognize that experimental work provides the soul behind all theoretical work. ‘Theory is important, yet theory without practice is worthless’. Sounds correct, but you wouldn’t ordinarily guess who said it! Nikita Khrushchev was actually discussing Marxism-Leninism at the time [7], but his remark holds true for us, too! Experimental investigation is necessary to provide the observations of specific behavioral features that must be modeled with a theory. Without the observations, there is absolutely no basis for a model! That is, there must be a firm foundation of multiple observations of how a material or structure behaves under various loading conditions to serve as the evidence to be modeled. Sometimes, it is very difficult to perform or conduct a proper experiment because of inadequate or severely challenged instrumentation, difficult to simulate events, or even impossibility or improbability of causing certain events to happen. Witness the severe trials of the astrophysicist who must rely on bits and pieces of information to construct or verify a theory. Similarly, a structural mechanicist involved in analyzing and designing structures to resist earthquakes has ground-motion spectra from past earthquakes to guide the loading definition but little except a few earthquake simulator “tables” to determine or confirm what happens to actual structures. Moreover, heating and support of plates for thermal buckling experiments are two challenging topics in structural mechanics. Fortunately, many structural mechanics simulations are performable. The reality of “how things work” is quite hard to determine in some cases. For example, shell buckling under axial compression loading occurs far too rapidly for the human eye to perceive all that is happening. It took the amazing experiments and high-speed motion pictures of Almroth, Holmes, and Brush [8] to bring those facts to our attention and to help us understand how shells really buckle. Note that Almroth and Brush were primarily theorists, although Almroth would have considered himself a practical engineer. An engineer should seek “clues” in the form of revealing information from critical experiments to develop an appreciation for the behavior of a particular type of mechanical system. With that appreciation, an engineer is prepared to model the behavior with the appropriate physical laws and principles. If the model is good and the analysis correct, then the behavior might be predicted well. If the model is lacking in essential features, then the behavior cannot be adequately predicted. The point is that the actual behavior provides modeling clues that must be addressed. Thus, we must observe the behavior before we can model it! I am personally disappointed when I see some try to develop a model with an obviously poor understanding of the phenomenon or phenomena addressed. In some cases, the purported elegance of the mathematical approach is used to “cover up” the fundamental fact that the theory has little basis in fact, namely the fact of the actual behavior. Some (fortunately few) theoreticians quite apparently choose to totally ignore all experimental

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data related to the phenomenon addressed with their theories! Seemingly, they do not use experimental results to justify their assumptions (presumptions, as we will see in the next section) or even to use as a comparison for their theoretical results! How do they ever know if what they did has any meaning? Or, was the work done to satisfy the insatiable proclivities of a dean or department head with minimal ability to think about the significance of scholarly work, but seemingly maximal ability to count papers? That is the min-max problem of academia! That unfortunate problem is enormously compounded when a book is regarded by those who have lost their sense of scholarship as ‘worth’ only a paper or two. Recall Drucker’s prediction that ‘the theorist will be reduced to playing useless games’ [5].

5. Clarification of the Terms Assumption, Presumption, and Restriction and Their Relation to Experiments When theoreticians ‘assume’, ‘presume’, or ‘restrict’ at various stages in their analyses, two areas of concern are apparent. Just what do they mean by, one, presumptions as opposed to assumptions, and, two, restrictions as opposed to presumptions. The distinctions are related to the reasons why we perform certain acts in analysis. Again, “Whatever you do, try to have a reason to do it”, said Forrest Gump [6].

5.1 PRESUMPTION VERSUS ASSUMPTION Too often, a theoretician ’assumes’ that a certain class of behavior exists and proceeds to develop a model and theory based on that assumption. However, the fundamental concept in analyses of mechanical behavior of materials and structures is not as simple as merely making an assumption because to assume is ‘to take for granted; suppose (something) to be a fact’ [9]. Thus, an assumption is arbitrary, not fact! Accordingly, the word assumption leaves out entirely whether the (something) has any reason whatsoever to be considered. We in mechanics and more generally in engineering cannot be that illogical! We are not interested in arbitrarily taking something for granted! Our concern is with finding why we should consider some line of reasoning or approach. The far more appropriate word for the mechanics analysis is presume because it ‘implies taking something for granted or accepting it as true, usually on the basis of probable evidence in its favor and the absence of proof to the contrary:ehp 1.’ [9]. Thus, presume (presumption) is a far stronger and, indeed, far more appropriate word for use in analysis than assume (assumption). For example, we could assume something that violates equilibrium, but certainly we cannot presume something that violates equilibrium! Assuming is poor practice, and violating equilibrium is unacceptable. An engineer should never make an arbitrary decision, i.e., one without a reason based in fact. Instead, an engineer makes ‘presumptions’. Clearly, the physical reason that is an essential part of every presumption must come from personal observation of the real behavior of the system addressed. In fact, some engineers use the term assumption in place of the far more appropriate term presumption, i.e., they say assumption, but really mean presumption as described in the foregoing discussion. A crucial part of almost any mechanics analysis is the identification of the probable physical evidence that leads us to the correct presumption. That is, the steps often called assumptions in a mechanics analysis are actually presumptions because they are not arbitrary assumptions (a redundant term), but are presumptions based on evidence that we


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can examine, and we often critically evaluate them, although sometimes not until after the analysis is completed. Our concern in mechanics analyses is with the probable evidence of material and structural behavior, so we deal with presumptions, not assumptions, and we must identify the probable behavior. That is, we must have a reason to justify our presumption, i.e., to make it plausible enough to consider. Thus, every presumption has two elements: (1) what? and (2) why? Or, (1) what is the presumption? and (2) why is the presumption believed to be true? What is the evidence to support the presumption? Without some form of evidence, we have no justification to make the presumption! The obvious basis for every presumption is experimental fact! The classical Euler-Bernoulli beam-bending hypothesis in elementary mechanics of materials is a good illustration of the two elements of a presumption. In accordance with that theory, we presume that (1) planes originally perpendicular to the neutral axis1 of an undeformed beam remain perpendicular to the deformed neutral axis after bending because (2) we can see from the deformation of a rubber beam (a material that is easily and visibly deformable) that vertical lines drawn on the side of the beam of rectangular cross section in Figure 9a remain essentially straight during pure bending as in Figure 9b. Those vertical lines are traces in the plane of Figure 9 of planes that are perpendicular to the beam neutral axis. However, those originally vertical lines do rotate to a position such that, if extended, they all pass through a common point, namely the center of curvature. Thus, plane sections remain plane after bending. During bending, the horizontal lines in Figure 9b below the neutral axis stretch, and the horizontal lines above the neutral axis compress, describing arcs of circles with centers at the center of curvature through which the originally vertical lines pass after bending. Note that we are relying on the resolution capability of the naked eye in our observations. That is, at the level of our vision, we see that plane sections remain plane. However, if we were to bend a much deeper, shorter beam, we would observe that plane sections do not remain quite plane because shear deformation of the original planes is evidenced by some curvature of the deformed ‘planes’. Alternatively, if we used a refined measurement technique such as

1 The neutral axis of a beam is the originally straight horizontal line that does not change length (strain) when pure bending moments are applied at each end of the beam. The neutral axis is at the mid-height of a rectangular cross-section beam.

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Moiré interferometry, we would detect non-planeness of the deformed ‘planes’. Most engineers do not distinguish between assumptions and presumptions. They typically use the word assume (or assumption), but generally mean presume (or presumption) because good engineers always have a reason for doing anything (just like Forrest Gump). In contrast, a presumption (assumption) might be quite different from an approximation. An approximation is often more related to the mathematics of treating the model of a physical situation than to the actual physical situation itself. In mathematics, we might consider only the (apparently) dominant terms in a series as an approximation (and we ‘assume’ that the rest of the terms are small enough to ignore, although often we are not certain). This approach is a definitive and well-demonstrated mathematical approach to analysis. A presumption (or assumption) is often more related to the physical side of a problem than to the mathematics. We attempt to simplify the process of solving a physical problem, or the model thereof, by making certain simplifying presumptions about the modeling of the physical behavior based upon physical observations. Those simplifying presumptions might involve the perceived nature of the physical response that occurs, e.g., a deformation.

5.2 PRESUMPTION VERSUS RESTRICTION The seemingly dual terminology of restrictions and presumptions is used because the terms have fundamentally different meanings that unfortunately are not always acknowledged. Restrictions are limitations on the use of the theory that are obviously either satisfied or they are not. Thus, restrictions are concerned with the known. For example, a theory for square plates does not apply to round plates. Presumptions are limitations on the theory that have a nature of uncertainty to them. That is, presumptions are concerned with the unknown. For example, stresses perpendicular to the surface of a plate, but within the plate, are commonly presumed to be small enough to be regarded as zero, or assumed to be zero; however, we do not know for certain just how small the stresses are unless we appeal to a more accurate theory. Also, displacements might be presumed to be small to enable certain approximations. However, whether the displacements actually are small can be determined only when the final results are known. In summary, the difference between restrictions and presumptions is that restrictions involve the known and presumptions involve the unknown (about which we wish to speculate). The following restrictions and presumptions provide further opportunity to clarify the difference between the two classifications for the common example of a plate as shown in Figure 10. Recall that presumptions in engineering must be justified, i.e., we must know why we believe the presumption to be true from a physical standpoint.


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Restrictions The plate is isotropic, linear elastic, and of constant thickness. The plate thickness is very small compared to its length and width (such a configuration is commonly called a thin plate, although the name plate itself implies such a geometry). No body forces exist.

Presumptions Stresses acting in the plane (the plane of the plate) dominate the plate behavior. Then, and are presumed to be zero such that an approximate state of plane stress is said to exist (wherein only and are considered). The Kirchhoff hypothesis of negligible transverse shear strains, and and negligible transverse normal strain, constitutes a statement of nondeformable normals to the middle surface although there is an inherent, but commonly ignored, conflict with the presumption of zero transverse normal stress, Displacements and are small compared to the plate thickness (generally, although not necessarily, indicative of small-deflection theory). Strains,

and

are small compared to unity (small-strain theory).

Rotatory inertia terms are negligible. Some engineers assume something that can be clearly determined as fact or fallacy if they would just look at the phenomenon. For example, some engineers ‘assume a rectangular plate’ when all you have to do is look at the plate to determine whether it is rectangular or in fact some other shape. Those engineers really mean to say they restrict their attention to a rectangular plate. That is, they confuse restriction with assumption. Clearly, there is no need or place for such confusion. Engineers are responsible for communicating their opinions, results, designs, etc. in an absolutely unambiguous manner. Otherwise, the possibility certainly exists that serious misinterpretation will occur. As an example of misinterpretation, in the 1920s, a building was designed to be built just off Red Square in Moscow. The architect expressed the possibility of two different (alternative) exterior designs in his plans by drawing a single end view of the building with one design on the left half of the building and the other design on the right half of the building, all on one sheet. No note was placed on the drawing to explain that there was a choice of exterior designs to be made, so the constructor built the building just as he saw it on the plans. After all those were Stalinist times, so asking too many questions could get a person into serious trouble. That unusually nonsymmetric building stands to this day! Obviously, engineers cannot afford to be ambiguous when we present our results!

6. Concluding Remarks We have seen examples of good agreement between theoretical predictions and experimental measurements of behavior for two complicated mechanical systems, and other

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examples abound. But, what do they mean? They mean that we have been able to successfully perform two challenging tasks: (1) develop a material model and a structural model that are capable of high-fidelity prediction of behavior and (2) measure behavior in a complex environment in which the many involved behavioral aspects of the widget are excited. But is our sole interest obtaining verification of theoretical predictions by performing useful measurements? No! Absolutely not! We must have experiments to help us understand the behavior of our widget so that we can determine the very basis for any model that we might develop. To properly do so, we must first establish the behavior and its characteristics, and then we’ll see if we can develop useful approximations to capture the essence of the behavior of the widget. Those approximations must be both realistic and useful. The approximations are decidedly not ‘assumptions’ because they are not arbitrary as is inherent in the definition of the word assumption. The approximations must have a solid, reasonable basis in fact, so they are ‘presumptions’. The distinction between assumptions and presumptions (and restrictions) is extremely important, yet tends to be glossed over by all too many. The most important message to theoreticians is: Seek and rely on experimental evidence to motivate, develop, verify, and extend your theories. Work closely with an experimentalist to maximize your opportunity to capture and to understand the phenonmenon. Mere information is not knowledge. It takes much thoughtful work to mold information into a body of knowledge – it’s called research! For some complex phenonomena, there can’t be enough information. “Too much ain’t enough” said Willie Nelson.

7. References 1. R. Byron Pipes and I. M. Daniel, Moire Analysis of the Interlaminar Shear Edge Effect in Laminated Composites, Journal of Composite Materials, Volume 5, April 1971, pp. 255–259.

2. R. Byron Pipes and N.J. Pagano, Interlaminar Stresses in Composite Laminates Under Uniform Axial Extension, Journal of Composite Materials, Volume 4, October 1970, pp. 538–548.

3. Robert M. Jones and H. Stuart Starrett, Nonlinear Deformation of a Thermally Stressed Graphite Annular Disk, AIAA Journal, Volume 15, Number 8, August 1977, pp. 1116–1122.

4. J. Singer, J. Arbocz, and T. Weller, Buckling Experiments: Experimental Methods in Buckling of Thin5.

6. 7. 8. 9.

Walled Structures, Volume 1, Basic Concepts, Columns, Beams and Plates, 1998, Wiley, Chichester, United Kingdom. D. C. Drucker, On the Role of Experiment in the Development of Theory, Proceedings of the 4th U. S. National Congress of Applied Mechanics, 1962, pp. 15–33. Forest Gump ref. personal communication daily. Nikita Khrushchev, Khrushchev Remembers – The Glasnost Tapes, Translated and edited by Jerrold L. Schecter with Vyacheslav V. Luchkov, Little, Brown and Company, Boston, 1990, p. 3. B. O. Almroth, A. M. C. Holmes, and D. O. Brush, An Experimental Study of the Buckling of Cylinders Under Axial Compression, Experimental Mechanics, Volume 4, Number 9, September 1964, pp. 263– 270. Webster’s New World Dictionary, Second College Edition, David B. Guralnik, Editor in Chief, Prentice Hall Press, 1984, pp. 84 and 1126.

MIXED NUMERICAL-EXPERIMENTAL TECHNIQUES : PAST, PRESENT AND FUTURE A.H. CARDON, H. SOL, W.P. DE WILDE, J. DE VISSCHER, K. HOES, D. DINESCU Dept. Mechanics of Materials and Constructions (MEMC) Faculty of Applied Sciences and Engineering (FTW) Free University Brussels (V.U.B.) Pleinlaan 2, B1050 Brussels, Belgium, EU Abstract Direct methods in continuum mechanics start from the thermomechanical characteristics and by solving the equilibrium or motion equations, satisfying boundary and critical conditions, result in stress, strain and/or displacement fields. Inverse methods are starting from the obtained solutions in order to compute a priori unknown characteristics or parameters. Direct experimental measurements of some characteristics or parameters are sometimes very difficult, if not impossible, to perform. Special applications of inverse methods combining some experimental global results with computed results are known as mixed numerical-experimental techniques or hybrid methods. Those MNET can be applied in order to obtain thermomechanical characteristics of material systems with elastic, viscoelastic, elastoplastic or viscoplastic behaviour. They can also be applied for damage, interphase, damping and processing characterisation. The MNET-approach can also be used in many other fields out of mechanics if a good theoretical model exist and a sensitive “experimental” method is available.

1. Introduction Classical definitions of stress and strain fields in continuum mechanics are based on some homogenisation procedure over a representative volume element (RVE). That RVE has to be small enough to avoid smoothing of high gradients but large enough to represent an average of the microbehaviour, J.Lemaitre, [1], Direct experimental measurements of stresses are only possible in some very special situations due to the definition of the stress vector. For the direct experimental measure of strains, strong “uniformity” conditions are necessary. For classical materials those conditions can be satisfied but with composite or multimaterial systems such direct experimental measurements of stress and strain fields are often difficult, if not impossible. In many other domains also the theoretical basic elements are not accessible for a direct “experimental” measure. In such a case it is possible, after definition of a set of control parameters, to start with estimated values of those control parameters, to compute some general effect obtained by application of the 551 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 551–560. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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theoretical model and compare that computed result (CR) with a direct experimental measure (EM) of the same global effect. If the “distance” between CR and EM is small enough, norm to be defined, the estimated set can be considered as acceptable, even if it cannot be proven that this set is unique. If the “distance” between CR and EM is too large an iteration procedure has to be developed till an acceptable set of control parameters is obtained. Perturbations of the so obtained acceptable control parameters have to be performed in order to check if the set can be considered as unique and to validate the results. This is the general basis of mixed numerical-experimental techniques (MNET) closely related to the inverse methods in model theory. In solid mechanics those methods were developed for the characterisation of material systems, for the characterisation of the interaction regions between basic materials in a multimaterial system, for the “measure” of some damage parameters and for the “measure” of residual stresses. In many other domains such as processing, general production and manufacturing methods and fluid-solid interactions those MNET can be applied. Those methods are not limited to mechanics and biomechanics, but can be used in any domain where some models describe a global evolution or behaviour starting from some locally defined parameters. For the application of MNET it is necessary to have a good model in order to obtain an accurate computed value of some global result and a sensitive experimental technique for the direct measure of that same global result.

2. Short history of the basic aspects and developments of MNET In the years following world war II, with the development of photoelasticity, it was necessary to arrive at the separation of the principal stresses. A first generation of hybrid experimental-numerical stress analysis methods were proposed such as the shear difference method and the Filon’s method, as described by A.J. Durelli and W.F. Riley, [2]. This approach was later developed and formalised by K.H. Laermann, [3], and G.V. Rao, [4], An overview of this first period, 1950-1985, is given by A.S. Kobayashi, [5]. In fact inverse methods are applied since a long time in order to obtain the value of some material characteristics directly on construction components under loading instead of using separate test coupons. If we consider the 4-point bending of an elastic isotropic beam we have as approximation for small displacements w such that equation :

With the boundary conditions w = 0 for

we obtain

the equilibrium

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and in the centre of the beam, x = 0, As consequence we obtain an estimation of the elastic modulus of the material of the beam by measuring w; M, and I given, as :

In this example we use for the computation an “analytical” solution and for the experimental measure an explicit measure of the displacement w for different values of the bending moment M. The relation between CR and EM is linear and a direct computation is possible. In a more general case the relation between CR and EM is nonlinear and an iteration cycle has to be applied. The number of analytical available solutions and approximations is limited, especially for composite materials and non-elastic behaviour, but we can use also the results of asymptotic solutions obtained for thin walled elastic and viscoelastic bodies as shown by A.H. Cardon et al., [6], It is not absolutely necessary to start from the estimation of the complete set of control parameters of a given phenomenon. It is perfectly possible to use a direct experimental measurement of the values of a subset of control parameters, the MNET being only applied in order to obtain the other control parameters. If we consider the characterisation of long fiber reinforced polymer matrix lamina assumed as transversely isotropic we are able to measure directly 4 of the material characteristics and we only need to apply a MNET for the fifth caracteristic as shown by A.H. Cardon et al., [11]. In the seventies with the development of modern computers and computational facilities based on finite and/or boundary element methods, see e.g. K.T. Kavanagh, [7], we saw a new start of MNET-applications as developed by W.P. De Wilde et al., [8], L.R. Deobald et al.,[9] and H. Sol, [10].

3. Characterisation of material systems In the period 1982-1990 a central problem was the measure of the material constants of fibre reinforced composites and especially long fibre reinforced lamina. Those lamina are strongly anistropic and, function of the relation between the fibre diameter and the thickness of the laminated plate, we can assume the material system as being orthotropic or transversely isotropic. Considering the complete set of material constants we have in general the possibility to obtain some of those constants by a direct experimental measure as e.g. the longitudinal modulus, and the transverse modulus In an elastic situation those two moduli are real. In case of a viscoelastic situation we have complex moduli and for a fibre reinforced polymer matrix will be real and complex. We start from the direct measured values of and we introduce estimations for

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Based on this completed set a global characteristic is computed by an appropriate numerical method, or from some analytical or asymptotic solution, if available, CR. The same global characteristic is measured directly, EM. Finite element or boundary element methods are very convenient for the computation and different programs are available especially for thin and moderate thick plates. The global characteristic can be a static one, but in that case iterations are not easy and the experimental technique is not very sensitive. Following different results as obtained by H. Sol, [10], a dynamic analysis based on resonant frequencies or a more general modal analysis is much more sensitive and efficient. After definition of a norm for the measure of the distance between CR and EM, we consider If this distance is smaller than the defined norm, we have an acceptable set of characteristics If for a first computation the distance is too large we have to change the values of the estimated subset and compute CR(1). The general MNET-scheme can be presented as follows:

to


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Different research groups developed this method in the period 1985-1990 and presented their results at the International Conference of Experimental Mechanics in August 1990. The group of the TUDenmark by P. Pedersen et al., [12]; the group of TUEindhoven by M.A.N. Hendriks et al., [13], and the group of Brussels by H.Sol, [14]. At the same conference some other presentations related to MNET were presented, e.g. by K.H. Laerman, [15], B.L. Agee, [16] and J.L. Jensen et al., [17]. The same year of the International Conference of Experimental Mechanics a European Mechanics colloquium (Euromech 269) was organised in St. Etienne by A. Vautrin and H. Sol on “Mechanical Identification of Composites”, [18]. Besides a number of papers on specific experimental methods, impact behaviour, damage characterisation and interfaces a large number of papers are applications and developments of MNET. On the basis of dynamic characteristics Hua Hongxing, [19], showed the possibilities of this scheme. The composite properties to be measured are assumed to be macroscopically homogeneous, globally averaged and elastic. The assumption of an elastic behaviour is convenient but seems not to be a limitation for the application of MNET. The plates used as test specimen are thin in order to be able to use the Love-Kirchhoff assumptions and from the stiffness coefficients the material engineering constants can be obtained. The dynamic behaviour, characterised by resonance frequencies and/or mode shapes, if the damping is very small, is sensitive to the plate rigidities. The measurement and the calculation of those dynamic characteristics are easy to be obtained by one test on the same specimen. A detailed discussion of the finite element method, the correlation criteria between numerical and experimental models and the updating methods are given in [19]. Applied to a bi-directional glass fibre reinforced 10-layers plate following MNET, results were obtained for the engineering constants :

Those results have to be compared with the direct measured values by static experiments on beams in tension, bending and torsion loadings :

A satisfactory agreement appears clearly, but a complete error analysis is necessary from the numerical modelling errors, over the classical measurement errors to the external induced errors and those coming from the numerical computations.

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4. More recent developments The material identification does not have to be limited to an elastic behaviour and it became possible to obtain the entire complex stiffness matrix of orthotropic materials by some generalised mixed numerical experimental technique, as shown by J. De Visscher, [20]. As expected an important number of papers presented at the ICEM, Lisbon, 1994, [21], were related to one of the aspects of the application of the mixed numerical-experimental methodology. The MNET was applied not only to problems of characterisation of material systems, but progressively to general nondestructive techniques, damage analysis, residual stresses, processing methods, interphase characterisation and was also applied to non-mechanical engineering problems and even to more general non- engineering fields. A large number of examples were presented during the European Mechanics Colloquium (Euromech 357), organised by H. Sol and C.W.J. Oomens in April 1997, [22], We also have to menition some papers presented at the International Conference on Experimental Mechanics, Oxford, 1998, [23], such as the contribution e.g. by K.H. Laermann, [24], that by T. Comlekci et al., [25] and by Inoue et al., [26]. As illustrated by those two last papers, inverse methods, or mixed numericalexperimental techniques, are the declared or hidden basis of many non-destructive testing methods. Any signal coming out of the volume of a construction component through the free surface contains some integrated information on the internal state. The reconstruction of the acceptable interior elements from the measured signal in combination with a model of the material system behaviour may be considered as a possible definition of an inverse problem and an application of some MNET. This was also illustrated recently by J.D. Achenbach, [27], as part of the general report of the US National Committee on Theoretical and Applied Mechanics “Research Trends in Solids Mechanics”, [28]. Many examples of the MNET applications in non-destructive testing can be found in the proceedings of the First Joint Belgian-Hellenic Conference on Nondestructive Testing, Patras, [29] and later on a lot of papers presented at NDT-NDE conferences are showing, explicitely or not, the application of the MNE concept. An immediate extension of the method out of the strict elastic behaviour is the measure of damping as shown by J. Vantomme, [30], and J. De Visscher et al., [31]. The MNET is a global method. The results are based on the comparison of integrated or averaged quantities. This is an advantage in some situations but if some localised information is needed a large number of specific iterations are necessary.

5. General applications of MNET 5.1. INTERPHASE In a multimaterial system such as a composite or an adhesively bonded joint one of the essential elements of the mechanical performance is the interaction level between the different basic materials. This is known as the problem of interfaces or interphases.


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This interaction region is very small and it is impossible to obtain any direct experimental measure of the characteristics of this interphase. The only possibility is to include this region in a multilayered plate or shell system and to apply a mixed numerical experimental technique. In order to work out a MNET, we have to model the interphase. This can be done by the introduction of imperfect interface conditions as proposed by J.D. Achenbach et al., [32], or by a continuum model and a plate type approximation as proposed by A.H. Cardon, [33]. The application of MNET for the characterisation of the interphase was discussed e.g. by A.H. Cardon et al., [34].

5.2. RESIDUAL STRESSES If

is the Cauchy stress tensor, we have for an elastic equilibrium problem : in the volume V, along S around V,

and Hooke’s law For residual stresses

we have :

in V, along S. In order to obtain information on the stresses we relax them by some destructive technique (e.g. drilling hole method or cutting techniques), measuring the resulting strain field and computing the related stress state. Instead of relaxing the existing residual stress state it is also possible to introduce a supplementary loading and applying a numerical-experimental scheme starting from some estimated values of the residual stress field and using some global characteristics, displacements, resonant frequencies or mode shapes, we can obtain acceptable values of the residual stresses.

5.3. DAMAGE Damage can be defined as the irreversible rupture of internal bonds in a continuum from micro- to macroscopic level. The characterisation of damage in a direct way is not easy and it is in general the influence of damage on the thermomechanical characteristics of the continuum that gives us some measure of the damage level. MNET can be applied for damage characterisation based on an estimation of the damage parameters in the stiffness characteristics and the comparison of computed results with experimentally obtained values.

5.4. PRODUCTION PROCESSES Many production processes of polymer matrix composites are the result of some flow over, or through, a reinforcement element. A good example is the resin

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transfer moulding (RTM) process. In all those flow related production processes the efficiency of the process is related to the velocity of the flow propagation around the reinforcement. That velocity can be calculated from some diffusion or permeability coefficients. Those coefficients are difficult to be measured experimentally or to be computed on the basis of non Newtonian fluid mechanics in interaction with some solid reinforcements and different surface conditions. It is possible to apply a MNET in order to obtain the permeability coefficients necessary to optimise the resin transfer moulding process as shown by the research group of H. Sol, [35],[36], in collaboration with the group of I. Verpoest, (University Leuven, KULeuven).

5.5. BIOLOGICAL MATERIALS The knowledge of the thermomechanical characteristics of living tissues is an essential element not only for healing procedures but also for any type of artificial elements to be introduced in the human body. For such prosthesis biological compatibility is essential but thermomechanical compatibility is also necessary in order to avoid any type of stress concentrations leading to singular material productions in reactive or living materials. In the characterisation of living tissues the most difficult problem is the fact that we are not able to obtain normalised test specimen and to use normalised test methods. We have to take the living material in the geometrical form and shape as it appears in nature. As consequence a direct experimental measure of the thermomechanical characteristics of living tissues, such as. e.g. tendons, muscles, bones, arterial wall and skin is very difficult if not impossible. Only the application of MNET can give us results obtained in vivo or on material elements with their initial shape. The research group of Eindhoven initiated such applications e.g. M. Van Ratingen, [37], and W.K.L. Van der Voorden et al., [38]. We have also to mention for bones the work of B. Van Rietbergen et al., [39] and many examples are discussed in the IUTAM Symposium on Biosolid Mechanics, [40].

6. Conclusions The mixed numerical-experimental techniques based on the adjusted comparison of computed results with direct measured experimental values of some global effect or behaviour have proven their efficiency in many applications where no direct experimental measure of the details of the basic elements are possible. MNET is a global method where integrated and averaged quantities are used. The “localisation” of some sources can be obtained by adjustments of the basic set of parameters. MNET need a good model to obtain the computational results and a sensitive experimental technique for the experimental results. Taking into account the important development of our computational facilities and the new possibilities of our experimental methods MNET is probably the most promising method for the future of experimental mechanics with applications in all domains where models are used in order to describe evolutions of phenomena and systems. MNET is not a miracle method and a carefully analysis is necessary on computational and experimental level, but the potentials are high.


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7. Acknowledgments Large parts of this research were made possible by financial support from the Research Fund of the Flemish Region of Belgium (FWO-Vlaanderen), from the Research Council of the Free University Brussels (OZR-VUB), from the Belgian Office of the Prime Minister for Scientific, Technical and Cultural Affairs (UIAPproject), and from the International Association for Promotion of the Cooperation with Scientists from the New Independent States of the former Soviet Union. (Intas-contract 96-2113).

8. References 1. 2.

Lemaitre J. (1966): A course on damage mechanics Edition), Springer. Durelli A.J. and Riley W.F. (1965): Introduction to Photomechanics, Prentice Hall, pp. 185186. 3. Laermann K.H. (1981): Recent developments and farther aspects of experimental stress analysis in the Federal Republic of Germany and Western Europe”, Exp. Mech., 21, pp. 4957. 4. Rao G.V. (1982): Experimental-numerical hybrid techniques for body-force and thermal stress problems – Applications in power industry, SESA Conference on Experimental Mechanics, pp. 398-404. 5. Kobayashi A.S. (1987): Hybrid experimental-numerical stress analysis, Handbook on Experimental Mechanics (Ed. A. Kobayashi), Prentice Hall, chapter 17, pp. 739-767. 6. Cardon A.H., Kossovich L., Kaplunov J., Emri I. and Gallegos C. (1999): Interactive theoretical-numerical-experimental methods for the characterisation of composite systems, Proc. SEM Annual conference 1999, SEM, pp. 825-827. 7. Kavanagh K.T. (1971): Finite element analysis in the characterisation of elastic solids, Int. Journal of Solids and Structures, vol. 7, pp. 11-23. 8. Hermans Ph., De Wilde W.P. and Hiel C.C. (1982): Boundary integral equations applied in the characterisation of elastic materials, Computational Methods and Experimental Measurements (Eds. Karamidas G.A. – Brebbia C.A.), Springer pp. 189-199. 9. Deobald L.R. and Gibson R.F. (1986): Determination of elastic constants of orthotropic plates by a modal analysis/Raleigh Ritz technique, Proc. Int. Modal Analysis Conference, pp. 682-690, Union College, Shenectady, NY, U.S.A. 10. Sol H. (1986): Identification of anisotropic plate rigidities using free vibration data”, PhDdissertation, VUB, Brussels. 11. Cardon A., De Muynck E., Hiel C.C. and Boulpaep F. (1979): Utilisation d’une méthode mixte, numérique et expérimentale pour la détermination des caractéristiques mécaniques de matériaux composites”, Proc. of the Canadian Congress of Applied Mechanics, pp. 105106. 12. Pedersen P., Frederiksen P.S. (1990): Sensitivity analysis for identification of material parameters, Proc. ICEM, Copenhagen, pp. 545-551. 13. Hendriks M.A.N., Oomens C.W.J., Jans H.W.J., Janssen J.D. and Kok J.J. (1990): A numerical-experimental approach for the mechanical characterisation of composites”, Proc. 9th ICEM, Copenhagen, pp. 552-561. 14. Sol H.: Identification of the complex moduli of composite materials by a mixed numericalexperimental method, Proc. ICEM, Copenhagen, pp. 562-571. 15. Laerman K.H. (1990): On a computer oriented numerical-experimental method in photoviscoelasticity, Proc. ICEM, Copenhagen, pp. 2059-2068. 16. Agee B.L. and Mitchell L.D. (1990): Frequency-dependent viscoelastic property measurement via modal analysis techniques, Proc. ICEM, Copenhagen, pp. 1978-1988. 17. Jensen J.L., Brincker R. and Rytter A. (1990): Uncertainty of modal parameters estimated by ARMA-models, Proc. ICEM, Copenhagen, pp. 2095-2104. 18. Vautrin A., Sol H. (Eds), (1991): Mechanical identification of composites", Elsevier Appl. Science.

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19. Hongxing Hua (1993): Identification of plate rigidities of anisotropic rectangular plates, sandwich panels and circular orthotropic disks using vibration data, PhD-thesis, VUB, Brussels. 20. De Visscher J. (1995): Identification of the complex stiffness matrix of orthotropic materials by a mixed numerical experimental method, PhD-thesis, VUB, Brussels. 21. Silva Gomes J.F. et al (Eds), (1994): Recent advances in experimental mechanics, A.A. Balkema Publishers. 22. Sol H. and Oomens C.W.J. (Eds), (1997): Material identification using mixed numerical experimental methods, Kluwer Academic Publishers. 23. Allison I.M. (Ed), (1998): Experimental mechanics – Advances in design, testing and analysis, A.A. Balkema Publishers. 24. Laermann K.H. (1998): Contribution to solving inverse problems in NDT of structures, Proc. ICEM 11 (Ed. I. Allison), A. A. Balkema Publishers, pp. 755-760. 25. Comlekci T. and Boyle J.T. (1998): A general hybrid numerical-experimental technique for thermoelastic stress separation”, Proc. ICEM 11 (Ed. I Allison), A.A. Balkema Publishers, pp. 463-468. 26. Inoue K., Kishimoto K., Hayakusa K. and Shibuya T. (1998): Stress separation in thermoelastic by means of boundary element inverse analysis, Proc ICEM 11 (Ed. I. Allison), A.A. Balkema Publishers. 27. Achenbach J.D. (2000): Quantitative non-destructive evaluation, Int. Journal Solids and Structures 37, pp. 13-27. 28. Dvorak G.J. (Ed), (2000): Research trends in solid mechanics, Pergamon Press. 29. Van Hemelrijck D., Anastassopoulos A. (Eds) (1996): Non-destructive testing, A.A. Balkema Publishers. 30. Vantomme J. (1992): A parametric study of material damping in glass fibre reinforced thermosetting plastics, PhD-thesis, VUB, Brussels. 31. De Visscher J., Sol H., De Wilde W.P., Vantomme J. (1997): Identification of the damping properties of orthotropic composite materials using a mixed numerical experimental method, Applied Composite Materials, vol. 4, n°1, pp. 13-33. 32. Achenbach J., Zhu H (1989): Effect of interfacial zone on mechanical behaviour and fracture of fibre reinforced composites, J. Mech. & Phys. Of Solids, 37, pp. 381-393. 33. Cardon A.H. (1993): From micro- to macroproperties of polymer based composite systems by integration of the characteristics of the interphase regions, Composite Structures, 24, pp. 213-217. 34. Cardon A.H., De Wilde W.P., Van Hemelrijck D., Van Vinckenroy G., Sol H., De Visscher J. (1994): Experimental characterisation of the interface-interphase properties in composite systems by mixed numerical-experimental techniques, Recent Advances in Experimental Mechanics, J.F. Silva Gomes (Ed), Proc. ICEM, A.A. Balkema Publishers, pp. 35-38. 35. Dinescu D., Sol H., Hoes K. (2001): A fast mathematical model for permeability identification in resin transfer moulding using a mixed numerical-experimental method, Proc. Of the 3rd Canadian Composite Conference – Cancom 2001, Montreal, pp. 109-117. 36. Hoes K., Dinescu D., Sol H., Luo Y., Verpoest I., Parnas R.S. (2001):New sensor based setup for permeability identification, Proc. Cancom 2001, Montreal, pp. 101-108. 37. Van Ratingen M. (1994): Mechanical identification of inhomogeneous solids – A mixed numerical-experimental approach, PhD-thesis, TUEindhoven. 38. Van der Voorden W.K.L., Douvers L.F.A. (1997): Characterisation of the in-plane mechanical behaviour of the human skin in vivo, Proc. Euromech 357, Material Identification using Mixed Numerical Experimental Methods (H. Sol, C.W.J. Oomens, Eds), Kluwer, pp. 173-182. 39. Van Rietbergen B., Kabel J., Odgaard A., Huiskes R. (1997): Determination of trabecular bone tissue elastic properties by comparison of experimental and finite element results, Proc. Euromech 357, Material Identification using Mixed Numerical Experimental Methods (H. Sol, C.W.J. Oomens, Eds), Kluwer Academic Publishers, pp. 183-192. 40. Pedersen P., Bendsoe M.P. (Eds), (1999): Iutam Symposium on Synthesis in Bio Solid Mechanics, Kluwer Academic Publishers.

A MOIRÉ-FE METHOD FOR INTERNAL CTOA DETERMINATION

J. H. JACKSON Idaho National Engineering and Environmental Laboratories P.O. Box 1625, Idaho Falls, ID 83415-2218 A. S. KOBAYASHI University of Washington Box 352600, Seattle, WA 98195-2600

Abstract A hybrid moiré-finite element (FE) analysis was used to determine the crack tip opening angle (CTOA) along a tunneling crack front in a three-point bend (SENB) specimen. The specimen was machined from a ductile 2024-T351 aluminum plate of 8.1 mm thickness and was pre-fatigued to an initial crack length to width ratio of The specimen was then subjected to stable crack growth of varying after which the specimen was post-fatigued to mark the final crack front and loaded to failure. A quarter segment of the SENB specimen was modeled with a truncated 3-D elastic-plastic FE model. Measured surface displacements, which were obtained by moiré analysis, and stable crack growth with increasing load were prescribed on the FE model. The CTOA was obtained from the computed crack opening displacement, approximately 1 mm behind, and normal to the crack front. The good agreement between the measured and computed surface CTOA are indicative of the accuracy of the procedure. 1. Introduction A ductile fracture criterion, which relates to the plastic strain field in the crack tip plastic zone, is the crack tip bluntness which was quantified by the crack opening displacement (COD) criterion advanced by Wells in 1963 [1]. This COD criterion, which was initially related to Irwin’s plastic zone estimate [2], and later to the Dugdale plastic zone [3], was promoted mainly for fracture assessment of thick-walled pressure vessels by The Welding Institute in the 60’s and 70’s [4]. Since the COD along a surface crack front in a thick plate cannot be readily measured and must be computed, its acceptance was hindered by the lack of easily accessible threedimensional elastic-plastic computer codes with the sensitivity to accurately map the crack tip bluntness. The CTOA, which uniquely characterizes the crack tip strain field [5], is another local field parameter which is measured as close to the crack tip as is feasible within the limits of experimental accuracy on the specimen surface. Shih et al [5] as well as Kaninnen et al [6] in the 70’s concluded that the CTOA was a computationally attractive operational parameter and an alternative to the J-integral 561 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 561–570. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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criterion. Unlike the J-integral criterion, which subsequently dominated the ductile fracture research of the 80’s and 90’s, the CTOD/CTOA criterion is also valid in the presence of unloading and crack extension. The CTOA criterion was used in the 80’s to analyze axial crack propagation in a subscale gas transmission line [7]. The rupturing profile of a propagating crack in a pressurized 2-in-diameter, schedule 10, carbon steel pipe was recorded with a highspeed framing camera and was processed through a special software [8] which yielded the CTOA and the flap motion. The CTOA criterion was also used in a ring model of the rupturing pipe and a thermal hydraulic depressurization code to simulate the largescale burst tests of 48-inch diameter x 0.720-inch thick, X70 line pipes [9]. The excellent agreement between the measured and the computed CTOA versus crack velocity relations demonstrated that the CTOA was an effective static and dynamic fracture parameter for pipe rupture [10]. A renewed research thrust in the CTOA criterion was undertaken in the early 90’s in response to the NASA Aircraft Integrity Program. The thin aluminum specimens considered for aircraft structure were ideally suited for the CTOA criterion. Newman et. al. showed that the CTOA criterion will predict the residual strength in laboratory specimens [11, 12] as well as in the FAA/NASA wide panels with multiple site damages [13]. Dawicke et. al. [14] and Gullerud et. al. [15] modeled the tunneling behavior in 2024-T3 specimens through a series of elastic-plastic FE analysis and found that during the initial, transient stage of crack propagation, the CTOA is on the order of a few degrees smaller than its steady state value in the center of the specimen and is a few degrees higher on the surface. When steady state (stable tearing) crack growth is reached, the two values become nearly equal and coincide with the typical steady state value. This study used a unique finite element model that incorporated crack tunneling during the initial stable tearing phase which was measured experimentally by fatigue marking the extent of tunneling after subjecting the specimen to various load levels and crack extensions. These crack front shapes were digitized from photographic images and fit with polynomial curves to describe the crack front shape as a function of through-thickness position. The elastic-plastic portion of the FE analysis relied on the incremental theory of plasticity, and was composed of several layers of elements to model the curved crack front. Gullerud et. al. [15] used steady state CTOA as a crack tip node release parameter for 3-D FE analysis. That is, nodes are released to allow crack extension when all values of CTOA are equivalent across the crack front. The FEA and experimental load versus displacement trends of 2024-T3 aluminum CT and MT specimens compared well. In the following, the authors’ finding on the utility of CTOA in a SENB aluminum specimen is presented. 2.

Method of Approach

A 3-D FE model of the tunneling crack was driven in the generation mode by the experimentally determined crack profile and applied load on 2024-T351 aluminum SENB specimen. The CTOA computed from the surface displacements determined by moiré interferometry was compared with FE computed CTOA for partial validation of the analysis.

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2.1 EXPERIMENTAL PROCEDURE Six 2024-T351 SENB specimens, shown in Figure 1, were monotonically loaded under displacement control, at a rate of 0.254 mm/min until stable crack growth initiated. Here the tests were stopped and the first displacement fields were captured via a CCD camera, connected to a frame grabber in a Windows NT computer. The load rate was then reduced to 0.127 mm/min to maintain stable crack propagation and the ensuing monotonic loading was stopped at crack extension intervals of 0.25 mm to record the and fields. Each test was stopped after a unique level of crack extension had been reached. Since the 3-D flaw analysis will rely on the tunneling present in the interior of the specimen, the experimental approach involved marking the crack front at various crack extensions by post-fatigue marking. Each of the specimens tested in this experimental work were loaded to a different crack extension length to provide a view of the tunneling behavior as the crack extends. Post-fatigue cycling was then used to mark the new crack front in a similar manner compared with pre-fatigue cycling. However, since the extent of tunneling was unknown prior to fatigue marking, an estimated average crack length had to be used for calculation of the values to grow the post-fatigue crack at an appropriate rate. This tunneling analysis is similar to the approach used by Dawicke [1], and Gullerud et. al. [15] in their 3-D CTOA analyses. The post-fatigue crack front profiles were digitized into a computer from photographic images and curve fit to allow it to be mapped in the FE model.

The moiré interferometery technique used for capturing surface displacements was developed by Wang et. al. [16] for measuring relatively large deformations. This method utilizes a low spatial frequency, steep grating of about 40 lines/mm on a highly polished surface to achieve very high contrast fringes. It combines the advantages of the geometric and traditional moiré interferometry methods to allow measurement of relatively large deformations. Here, two collimated beams are directed onto the surface of the specimen at a very shallow angle of approximately 6 degrees. A moiré pattern is visible to the naked eye and can be photographed without the need to capture the 1st order diffraction, which is the typical way to extract moiré data.

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2.2 NUMERICAL MODELING Only one-quarter of the SENB specimen was modeled, due to symmetry considerations. In addition, this model was further truncated at a specific distance from the crack plane for computational efficiency since a full 3-D elastic-plastic model can be very computationally expensive. The truncation distance was determined by an elasticplastic FE analysis of a full quarter model of relatively coarse mesh, i.e. 0.5 mm elements in the vicinity of the crack tip and transitioning to 1.0, and 2.0 mm elements further away. The coarse model, which contained only four elements through the half thickness of the specimen, was loaded to 3.6 KN at which crack extension was expected. The truncation distance for prescribing experimentally obtained surface displacements through the thickness of the FE model was set as the distance from the crack face where the von Mises stress was approximately constant through the thickness, implying a 2-D approximation. For this analysis, the truncation distance was set to y = 15 mm. The application of boundary conditions at a truncation point in the case of bending is a complex issue. The v-displacement (normal to the plane of the crack) varies linearly from top to bottom to ensure a bending load. In contrast, the u-displacement (parallel to the plane of crack) boundary condition is applied at only one point on the 15 mm truncation boundary as it is nearly constant in the vertical direction. To ensure that the prescribed u-displacement was prescribed to reflect reality, several different scenarios were explored in a simple 2-D model of one quarter of the specimen with actual bending employed. Numerical experiment snowed that u-displacement prescribed at the mid height of specimen offered the best comparison with the true shear stress distribution near the truncation boundary in the full model. The final model used in this analysis is shown in Figure 2.


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The extent of tunneling seen in the experimental analysis required a sufficient resolution of element layers through the specimen thickness. To accommodate the extreme slope of the tunneled crack front, thin (0.25 mm) layers were utilized in the areas of the steep crack front gradient with a transition to thicker, 0.5 mm element layers near the center of the specimen where the crack front gradient is not as large. The final model (Figures 2 and 3) consisted of ten, 0.25 mm thick layers near the outer surface followed by three, 0.5 mm thick layers to total the specimen half-thickness of 4.0 mm for a total of 13 element layers. This arrangement of layering also helps to accommodate the natural surface singularity that is expected near the free surface. Element sizes were 0.25 mm in the x and y directions near the crack tip area and transitioning to 0.5 mm and 1.0 mm further away. Figure 2 also shows the model ydirection truncation distance of 15 mm, with experimentally obtained displacement boundary conditions prescribed. The measured tunneling profiles were prescribed as boundary conditions defining the crack face in the final model as seen in Figure 3.

3.

Results

3.1 TUNNELING CRACK PROFILE Since the crack extension behavior is somewhat difficult to control, only four of the six tests yielded usable results. Images of these four tunneling profiles are shown in Figure 4. Each of these images was calibrated to known dimensions, and the pre-crack and tunneling crack front profiles were digitized using Sigma Scan software. The digitized crack fronts were then fit to polynomials of varying orders from 4th order to 6th order depending on the level of complexity. The polynomial curve fit served to fill in sparse data at regular (0.25 mm) increments through the thickness (z) direction and to provide a means for calculating local crack front tangents via differentiation for use with the numerical analysis. Since the amount of stable crack propagation is difficult to control in an experimental environment, tunneling profiles at very sparse points were obtained. To compensate for the sparseness of data without performing what could amount hundreds of different tests, a linear interpolation scheme was written using Matlab to obtain the intermediate tunneling profiles and increase the data resolution by two times. Figure 5 shows the interpolated crack fronts after two interpolations, and Figure 6 is a comparison of the raw, digitized data to the polynomial fit data.

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Although a fairly insignificant specimen shear lip was expected, and observed in this thick specimen configuration, it is worth noting its value. For this specimen configuration, a shear lip of approximately 0.5 mm per side was measured. This amounts to a total shear lip of 1.0 mm, or 13% of the specimen thickness. After approximately 2.0 mm of surface crack extension, the crack is observed to propagate in a relatively self-similar manner. After several levels of crack extension however, the near mid-plane crack front flattens as it approaches the externally applied mid-span load. This will affect the tunneling as well as the numerical analysis of these crack fronts.

3.2 MOIRÉ FRINGE PATTERN Figure 7 shows typical moiré fringe patterns corresponding to the and v-displacement fields observed in the SENB tests. A program was written in Matlab to compute the and v-displacements from digitized moiré fringe patterns. This allowed easy extraction of relevant displacement quantities at chosen nodal points along the truncated boundary of the FE model as well as along crack faces for CTOA measurement. 3.3 EXPERIMENTAL CTOA The CTOA was calculated on the surface of three specimens labeled T3-3PB, T4-3PB, and T5-3PB. The CTOA calculation consisted of extracting the v-displacement at approximately 1.0 mm behind the current crack tip, obtaining the inverse tangent, and

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multiplying by 2.0 (symmetry). Figure 8 is a plot of the resulting CTOA for these three experiments.

3.4 CALCULATED CTOA Figure 9 shows the CTOA calculated at three locations through the thickness at approximately 1.0 mm behind and normal to the crack tip for the surface, the quarter point (midpoint of numerical model), and the specimen mid-point. The surface CTOA trend shows the typical sharp increase at the beginning of crack growth, followed by a decline to a fairly steady state value of approximately 7-8 degrees. The quarter point and mid-point trends exhibit an interesting slow rise as the crack extends. This behavior is due to the rapid crack propagation in the center (tunneling) near the beginning of the test, which will produce a small amount of crack tip blunting and hence low CTOA. As


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the tunneling slows, the CTOA on the inner layers should increase as seen here due to increasing amounts of plastic deformation and crack tip blunting. It is observed that the CTOA roughly follows the same trend as the near-field calculated on an mm contour. Both the CTOA and the near-field show a decreasing trend followed by a fairly sharp increase at local crack extensions longer than approximately 6.0 mm.

4. Conclusions

A hybrid experimental-numerical procedure was used to determine the CTOA of a tunneling crack in a plane strain, SENB specimen. Moiré interferometry was used to obtain prescribed displacement boundary conditions for application in the numerical model as well as crack opening displacement values for obtaining the surface CTOA. The interior CTOA was lower than the surface CTOA at the initial phase of crack extension due to the difference in plane stress state on the specimen surface and plane strain state in the interior. As the crack extension approaches the specimen thickness, the interior CTOA approaches the surface CTOA thus indicating that a pseudo plane stress state is achieved.

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Acknowledgement

This work was sponsored by U.S. Department of Energy Grant #03-97ER14770. 6. References 1. 2. 3. 4.

5.

6.

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8.

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10. 11.

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14. 15.

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Wells, A.A. (1963) Application of Fracture Mechanics at and Beyond General Yielding, British Welding Journal, pp. 563-570. Irwin, G.R., Kies, J.A. and Smith, H.L. (1958) Fracture Strengths relative to Onset and Arrest of Crack Propagation, Proc. of ASTM, pp. 640-657. Goodier, J.N. and Field, F.A. (1963) Plastic Energy Dissipation in Crack Propagation, Fracture of Solids, eds. D.C. Drucker and J.J. Gilman. Interscience, pp. 103-118. Harrison, J.D., Dawes, M.G., Archer, G.L., Kamath, MS. (1979) The COD Approach and Its Application to Welded Structures, Elastic-Plastic Fracture, eds. J.D. Landes, J.A. Begley and G.A. Clarke. ASTM STP 668, pp. 606-631. Shih, C.F., deLorenzi, H.G. and Andrews, W.R. (1979) Studies on Crack Initiation and Stable Crack Growth, Elastic-Plastic Fracture, eds. J.D. Landes, J.A. Begley and G.A. Clarke. ASTM STP 668, pp. 65-120. Kanninen, M.F., Rybicki, E.F., Stonesifer, R.B., Broek, D., Rosenfield, A.R., Marshall, W.C. and Hahn, G.T. (1979) Elastic-Plastic Fracture Mechanics for Two-Dimensional Stable Crack Growth and Instability Problems, Elastic-Plastic Fracture, eds. J.D. Landes, J.A. Begley and G.A. Clarke. ASTM STP 668, pp. 121-150. Kobayashi, A.S., Emery, A.F., Love, W.J. and Chao, Y.H. (1988) Subsize Experiments and Numerical Modeling of Axial Rupture of Gas Transmission Lines, ASME Journal of Pressure Vessel Technology, 110, pp. 155-160. Emery, A.F., Kobayashi, A.S., Love, W.J., Place, B.W., Lee, C.H. and Chao, Y.H. (1986) An Experimental and Analytical Investigation of Axial Crack Propagation in Long Pipes, Engineering Fracture Mechanics, 23, pp. 215-226. Sugie, E., Matsuoka, M., Akiyama, T., Tanaka, K. and Kawaguchi, Y. (1985) Notch Ductility Requirement of Line Pipes for Arresting Propagating Shear Fracture, Proceedings of the 1985 Pressure Vessels and Piping Conference, ASME PVP, 98(8), pp. 53-61. Emery, A.F., Chao, Y.H., Kobayashi, A.S. and Love, W.J. (1992) Numerical Modeling of FullScale Pipe Rupture Tests, ASME Journal of Pressure Vessel Technology, 114, pp. 265-270. Newman, J.C., Dawicke, D.S. and Bigelow, C.A. (1992) Finite-Element Analyses and Fracture Simulation in Thin-Sheet Aluminum Alloy, Durability of Metal Aircraft Strucutres, eds. S.N. Atluri, C.E. Harris, A. Hoggard N. Miller and S.G. Sampath, Atlanta Technology Publication, pp. 167-186. Dawicke, D.S., Plascik, R.S. and Newman, J.C. (1997) Prediction of Stable Tearing and Fracture of a 2000-Series Aluminum Alloy Plate Using a CTOA Criterion, Fatigue and Fracture Mechanics: 27th Volume, eds. R.S. Plascik, J.C. Newman and N.E. Dowling. ASTM STP 1296, pp. 90-104. Newman, J.C. (2000) Advances in Fatigue and Fracture Mechanics Analysis for Aircraft Structures, Structural Integrity for the Next Millennium, eds. J.L. Rudd and R.M. Bader, 1, pp. 342. Dawicke, D.S., Newman, J.C. Jr., and Bigelow, C.A. (1995) Three Dimensional CTOA and Constraint Effects During Stable Tearing In A Thin-Sheet Material, NASA TM-109183. Gullerud, A. S., Dodds, R. H. Jr., Hampton, R. W., and Dawicke, D. S. (1998) 3-D Finite Element Modeling of Ductile Crack Growth in Thin Aluminum Materials, Fatigue and Fracture Mechanics: 30th Volume, Jerina, K. L., and Paris, P. C., Eds., ASTM. Wang, F.X., May, G.B. and Kobayashi, A.S. (1994) Low-spatial Frequency Steep Geometric Grating for Use in Moiré Interferometry, Optical Engineering, 33 (4), pp. 1125-1131.

PATTERNS OF MODERN EXPERIMENTAL MECHANICS Principles of Modeling of Actual Responses of Materials. Machines and Structures

JERZY T. PINDERA Department of Civil Engineering University of Waterloo. Waterloo. Ontario. Canada. N2L 3G1 Abstract The paper presents – in a very condensed form – origin, consequences, and a suggested resolution of the existing dichotomy in applied mechanics, which eventually causes unnecessary failures of machines and structures. The major contributing factor is the fuzzy distinction between the rigorously defined properties of hypothetical abstract mathematical constructs used in applied mechanics, and experimentally determined responses of simplified, approximate mathematical models of real engineering objects. Four figures illustrate this dichotomy. Only selected references are given. A reader is referred to the original publications. 1. Introduction - Frame of Reference 1.1. GENERAL FRAME OF REFERENCE Experimental mechanics is a component of applied mechanics, within the general framework of engineering mechanics. Engineering mechanics is understood here as synergy of applied mechanics and applied mathematics, numerical mechanics and mathematics, experimental mechanics and physics, materials science with physics and chemistry, and computational materials science. All activities occur within a framework provided by society. It is impossible to discuss in a short paper all major aspects of modern experimental mechanics. Thus, out of necessity, the theme of the paper is limited to the issue that – in the author’s judgment – is of a paramount theoretical and practical importance: How to assess reliability and applicability of particular engineering theories, particular design methodologies, and evaluated results of engineering experiments, which could be safely used in real design, including the requirement of a controlled failure. References contain data on major original contributions. In accordance with the author’s academic and professional background in aeronautical engineering, the following criteria are accepted as axioms: Theory and practice may be compatible. Theory must be tested experimentally. No engineering theory must violate any accepted scientific theory. Safety first, the stress is put on rational strength optimization and on controlled failure (in engineering parlance “an airplane is build around a pilot”). 571 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 571–584. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Dichotomy in Engineering Applied Mechanics

2.1. BRIEF CHARACTERIZATION

For the purpose of this paper engineering applied mechanics (EAM) is understood in the narrow sense, as a discipline encompassing analytical mechanics (AM) intertwined with experimental mechanics (EM) and materials science (MS). EAM is based on mathematics and utilizes natural sciences to deal with responses of hypothetical bodies that are assumed to reliably represent real physical bodies. However, the methodology of approach accepted in contemporary EAM often deviates from that accepted in science and in mathematics. This results in dichotomy between very technological methodology that does not require any comprehensive theoretical background, and methodology based on the present state of knowledge in natural sciences that does not accept convenient mathematical simplifications. In science a theory is accepted only when it is tested by experiment or observation - no discrepancy between a theory and an experiment is accepted. This rule is accepted, as an axiom, in aeronautical-aerospace engineering, but is not accepted in general engineering mechanics. Thus, in natural sciences a series of experiments are needed to confirm or to refute the assertions. The purpose of scientific experiments is to determine various responses of a tested body to various energy inputs, in accordance with the obvious truism that no energy, no information. To describe correctly an information-producing energy flow, it is necessary to introduce the concept of a dynamic system response that can degenerate to a static system response. A testing system consists of the tested object, of a loading and measurement subsystem, of a theory of experiment, and an experimenter. The role of experimenter is essential. The notion of system response encompasses mathematical tools, such as transfer functions, and physical parameters, such as impedances and parameters of the flow of various forms of energy through the system [4, 12]. In mathematics only arguments and rigorous computation are needed. In mathematics theories pertain to properties of mathematical objects that exist in a hypothetical, abstract space, and are unequivocally defined by a set of axioms. Mathematics deals with abstract concepts and their interrelations [1]. Computations in mathematics are rigorous – no convenient simplifications are permitted. No energy flow is needed to investigate properties of mathematical constructs, and no analysis of system response is needed. The only unresolved intellectual problem is the surprising effectiveness of mathematics when applied to describe the real, perceived world [35]. Engineering applied mechanics uses the concepts of abstract mathematical objects, or constructs, which can be rigorously presented analytically, and uses the concepts of physical bodies, which are described by physical responses to energy flow. (Unlike natural sciences and mathematics, EAM often uses physical terms to denote abstract bodies, usually without satisfactory experimental evidence that an assumed abstract mathematical object really represents a real physical body, and describes responses of abstract bodies and systems to various energy inputs by relations that contain arbitrary mathematical simplifications, without any proof that the influence of the eliminated mathematical terms is negligible). It is commonly disregarded that mathematics is only a language of a book. Such a situation leads to a dichotomy in EAM – dichotomy between a simple phenomenological approach to the real engineering tasks that accept

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unproven assertions, and a physical approach to the real engineering tasks that complies with scientific requirements [16, 17, 20, 25, 26]. This dichotomy exists at both the analytical and the experimental levels, both in teaching and in engineering practice. Quite often analytical relations for stress/strain/temperature/time states are used that may be considered rigorous in the abstract mathematical space, but are crude when used to represent real physical processes. Some examples are given below. The problem is rooted in curricula. The engineering curricula are usually constructed in such a manner that the students believe the abstract mathematical constructs denoted as physical objects – beam, plate, or column – correctly and completely represent the behavior of real beams, plates or columns. As a result, a number of engineering students request Do not teach me formulas based on assumptions – give me the correct formula, or state As an engineer I do not care where my formulas came from. All I am interested in is how to use them. Such requests are inconceivable in sciences and in aeronautical-aerospace engineering. The notion of idealization, a common term in EAM, does not comply with the Platonic notion of idea – it usually represents a crude simplification that is often not justified [8, 15, 18]. At the experimental level it is often ignored that the reliability of experimental results depend on whether or not the experimenter is satisfactorily conversant with all involved physical theories related to the experiment. There was a time when measurement problems limited reliability of engineering measurements. This problem does not exist any longer in theories and techniques of engineering measurements – (the sky is the limit). However, this incredible progress exposed major weaknesses in applied mechanics: the theoretical framework of experiments, which is commonly and uncritically based on relations of abstract analytical mechanics, and the very simplified relation of physics, including materials science that are used as components of the theoretical foundation , are often unnecessarily deficient as illustrated below. Consequences of this dichotomy are not only intellectual or esthetical but also most practical –many costly failures of engineering structures were caused not by fabrication errors, but by theoretical mistakes made by the designers or by the experimental consultants. Only a few classical examples on unnecessary failures are mentioned here: the collapse of the Tacoma Narrows bridge caused by the well known but neglected aeroelastic interaction (flutter); the failures of the first passenger jet Comet caused by the known but neglected fatigue mode loading; the damages to the John Hancock Tower in Boston that was caused by well known but neglected torsional aeroelastic interaction; or the massive collapse of the power network in Quebec that was caused by neglecting the well known principle of a controlled failure, which was accepted in the aeronautical engineering almost a century ago. Interestingly, the extensive experimental model studies of the Tacoma Narrows Bridge were performed before construction started, but the investigator was not aware of existence of the wellknown flutter, designed a wrong theory of experiments, and could not predict the failure caused by flutter. Also, it is common to neglect the influence of thermoelastic coupling at static and dynamic loadings, which results is misinterpretation of test results: a small increase of the value of apparent elasticity modulus, but a significant increase of value of the dynamic strain wave released by a fracture front [35]. The neglected heat production at plastic deformations in steel often results in apparently mysterious fatigue tensile fractures in compressively loaded machine components.

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This discussed dichotomy is intellectually unpleasant and leads to unnecessary engineering failures, which are often costly. It is possible to eliminate this dichotomy by stressing the importance of scientific methodology in engineering theories, that is, by accepting the same patterns in the development of a theory as accepted in sciences and mathematics. This would necessitate changes in engineering curricula. 2.2. A FEW EXAMPLES OF DICHOTOMY IN EAM 2.2.1. PROBLEMS IN ENGINEERING APPLICATIONS OF SOLUTIONS OF THE THEORY OF ELASTICITY The singular solutions in elasticity theory are very popular, but they must be applied very cautiously as mentioned in [31] and illustrated by Figure 1, [20]. In a large region of a circular disk loaded diametrically the pertinent elegant singular solution yields wrong results, both regarding values and signs of stresses. An analogous situation exists in the isothermal mathematical theory of plasticity [36] Practically all solutions of the mathematical theory of elasticity, which are accepted in engineering design, pertain to a plane stress state, including the notions and analytical relations of fracture mechanics. However, it has been well known for more that seventy years that the stress states in the region of notches in plates are noticeably three-

dimensional, [33]. Extensive quantitative data on the actual three-dimensional stresses (3D) in notches and cracks are given in [15-17, 19-26]. Compatible experimental


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evidence shows that the actual maximal stress in notches and cracks are up to 30% higher that those evaluated analytically or by means of averaging photoelasticity. Local loads also produce 3D stresses in prismatic beams, as shown in [18]. The problem is not only limited do the stress magnitude, but directly pertains to our definition of the stress concentration factor and of the stress intensity factor. Local effects are more significant than it is commonly believed [19]. As a result, two incompatible models of t stress state in a plate with symmetric notches, concurrently used in engineering mechanics and in design procedures: a plane model and a 3D model. Both models lead to different notions of the stress concentration factors and stress intensity factors and yield different vales of maximal boundary stresses [20]. The 3D stress distribution along the crack tip in a beam is quite interesting. The maximal stress occurs at the beam middle plane, and the maximal thickness stress is not negligible, see Figure 2, [20].

The actual distribution of the stress intensity, presented as a function of the distance r from the crack tip, shows that this function is not monotonic and it does significantly depend on the thickness ordinate, Figure 3, [26]. 2.2.2. DICHOTOMY IN MATERIALS CHARACTERIZATION AND IN SYSTEM RESPONSE ANALYSIS Engineering materials can be roughly divided into two classes: time-independent and time-dependent materials. The time-independent materials, such as steel, usually are

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tested at arbitrary strain rates, with disregard of thermoelastic coupling that influences indicated strain values and slightly alters the evaluated modulus of elasticity [9, 14]. It is common to neglect the existence of the huge thermal strain that is produced by heat

generated during plastic deformation of metallic and plastic material, and its influence on the actual stress state in machines and structures. Figure 4, [26], presents the essential difference between two interpretations of the same tensile test for steel: a common phenomenological interpretation, and an advanced engineering physical interpretation. Obviously, the test diagram does not represent material response to a tensile load; it represents response of the system consisting of tested specimen-testing machine, as it is indicated in the testing standards [2] and in bibliography [30]. The evaluation depends on the theoretical level of the accepted models of material and system responses. The typical evaluation of the test results in introduction of artifacts that do not exist, such as notions of the upper and lower yield stresses, or the notion of isothermal plastic deformation processes. Time-dependent materials are of major engineering interest [6, 29, 32, 34]. A very interesting situation exists in the testing of mechanical and optical responses of high polymers. Time-dependent responses to loads of plastic are commonly described by linear ordinary differential equations leading to hereditary integrals, what often is quite satisfactory. The coupling between the mechanical and optical responses is commonly assumed to be proportional, what is often wrong, and the optical responses at different spectral frequencies are assumed to be proportional to ratios of corresponding wavelengths, what violates basic theory and results in large errors [13, 20, 32]. The accepted mathematical models of viscoelastic responses require that the test results be presented as isochronal diagrams. However, it is a common practice to present a three-


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parameter stress-strain-time relations by strain-stress relation at a constant strain rate, and on that basis to decide whether or not the tested material is linearly or nonlinearly viscoelastic. Figure 5, [26], illustrates the issue, and shows that only an isochronal presentation allows for determination of the range of linear response and its change with time [13, 20, 26]. That figure presents graphs of the typical isochronous relations in isothermal conditions at uniaxial loads, which characterize mechanical and birefringence responses of various polymeric materials. Birefringence response is spectral frequency dependent. The isochronous relations show the existence of two linear response ranges, limited by the linear limit stress.

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A very confusing situation exists in applications of photonic radiant energy to the determination of mechanical quantities such as stress, strain or displacement. It is common to apply the most elementary relations of optics and of photonics, disregarding classical textbooks [3, 28]. This eliminates possibility of optimally utilizing actual patterns of interaction between radiation and the deformed body for determination of mechanical quantities of interest. Figure 6 presents, briefly, the existing dichotomy between the traditional engineering understanding of optics and the needed understanding [20]. It should be noted that the theoretically correct understanding of the patterns of propagation of photons inside stressed body resulted in the development of theory and techniques of the strain-gradient stress analysis [7, 26], and of the theory and techniques of the three-dimensional, tomographic isodyne stress analysis – the only nondestructive 3D stress analysis. 3. Basic Patterns of Modern Experimental Mechanics 3.1. VARIOUS NOTION OF A THEORY It is recognized that in engineering mechanics a balance should be maintained between reality and complexity, but this requirement does not justify unnecessary and destructive simplifications and unnecessary vagueness. In engineering mechanics the term theory quite often denotes intellectual structures formulated in language of mathematics, which are supposed to represent quite accurately responses to real loads of real materials and real bodies of various shapes.


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Very often the term rigorous solution is used, which implies that behavior of a the body in questions is presented completely and accurately. Usually no information is given whether the system in question, consisting of a body and of acting loads, actually exists in the physical space, or whether it only exists as a hypothetical system in an abstract space. It is uncommon to consider the influence of the thermo-elastic and thermo-plastic coupling what results in a drastic violation of the conservation of energy principle [9, 34]. This approach contrasts with approaches accepted in natural sciences and in mathematics. In natural sciences it is common to distinguish between a physical theory and a phenomenological theory [10, 15, 27]. A physical theory gives a deep insight into the mechanism of a phenomenon. A phenomenological theory in sciences in based on a set of assumed relations that must be compatible with all known pertinent facts. When a theory is phenomenological, the author states this fact. A basic theory of stress-induced birefringence presented in [28] is a scientific phenomenological theory. Patterns of development and testing of scientific theories are elegantly and succinctly presented by Feynman: guess – compute consequences – compare with experiment. If it disagrees with experiment, it is wrong [5]. Interestingly, theorists in theoretical physics accept this approach, but theorists in applied mechanics, both analytical and experimental, mostly reject it. Evidently, mathematics is a tool, the language of the book. Mathematics deals with mathematical constructs, or object, which exist in a hypothetical, abstract space – elliptic curves, matrices, or the mathematical theory of elasticity [1]. One can denote parameters of those mathematical constructs using terms taken from a real world, but this changes nothing – a beam defined as a mathematical construct still only exists in an abstract space and it must be analytically proven that such a beam partially simulates some patters of behavior of a real beam. Computations in mathematics are rigorous, without exceptions. Thus, mathematics is about mathematical objects, and mathematical theories are satisfied in mathematical models as it is said. There is no uniform approach to engineering mechanics in engineering sciences [11]. Different approaches in various branches of engineering are highly visible. This results in drastically different levels of engineering mechanics, so the levels of corresponding theories in EAM are different. For example, some publications discuss experimental impact problems as obviously isothermal solid body responses to an applied delta-function force, when some other researchers present analogous problems within a framework of a comprehensive system response, involving processes in solid and liquid bodies, patterns of energy transfer and flow, heat generation resulting in major temperature increase, and characterized by varying velocities of elastic waves. The common publications in EAM are usually written within the framework of a particular kind of phenomenological theory, which often violates one, or more, basic physical principles. It is common to use terms that are only commensurate with abstract mathematical constructs, but are misleading when understood as responses of real bodies. The comment presented in [29] about the philosophical flavor of the term “property” is instructive. The influence of the prevailing paradigm [10] in common EAM is strong, despite the incredible progress in technology and sciences.


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3.2. MODELING OF REALITY IN ADVANCED EXPERIMENTAL MECHANICS Reality is understood here as a perceived reality that is a reflection of a real, objective world [27]. It is realized that this is a strong philosophical statement, but it is unavoidable. Reality is perceived by means of energy flow. No energy flow, no perception. Thus we deal with a system, with its transfer functions and impedances of various kinds, and with all related constrains. The levels of perception of an observer and of his knowledge are major components of the perception system. The perceived reality must be simplified to basic models that can be presented in an analytical form, preferably as systems of linear differential equations. Presently, there are two approaches in modeling reality: a simple phenomenological approach based on utilization of the concept of “black boxes”, and a physical approach based on a deeper insight into the mechanism of modeled body responses, or processes. Figure 7 presents a simplified diagram of both modeling approaches. Both approaches result in development of basic physical models and of representative mathematical models, but the scientific levels of both kinds of models are drastically different [4, 8, 15,]. Advanced experimental mechanics is based on the physical methodology that excludes internal and external incompatibilities. Practical experience shows that the physical approach in development of physical and mathematical models is superior, and opens new research horizons [15, 17, 20, 25, 26]. Perhaps it should be noted that the intellectual level and elegance of analytical solutions in applied mechanics that deal with abstract mathematical constructs, and the physical reliability of the developed relations, are not causally connected. 3.3. ADVANCED EXPERIMENTAL MECHANICS: DEVELOPMENT PATTERNS Particular measurement procedures and their application to determine responses of materials, bodies, and systems no longer represent the main tasks of experimental mechanics. New measurement theories and techniques allow performing any measurement with accuracy surpassing engineering requirements. Experience shows that physicists are better equipped than engineers to develop advanced measurement systems and procedures. Theories and techniques of measurement have already developed as separate disciplines. Consequently, the major problem of applied mechanics became the reliability of the analytical and experimental procedures. One major task is to produce physical data on the actual behavior of objects, materials, and systems to be used as the basis for constructing more advanced and more reliable pertinent analytical relations, in particular for constructing reliable constitutive relations. The basic underlying issue is a rational designing of pertinent physical models starting from first principles, such as mechanical, thermodynamic, and photonic interactions at the atomic, molecular, and crystalline levels during the processes of deformation and fracture. Models must be physically admissible, thus energy and power must be finite. The other major task is to produce reliable experimental data. Thus, experiments in AEM must be based on correctly developed physical models that are used as a basis for a rational theory of experiments.


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These features allow characterizing the developing AEM by two principles: No theory – no experiment, No experiment – no theory. The major features of Advanced Experimental Mechanics (AEM) are the theoretical compatibility of applied theories and techniques, and the compatibility of theories, observation and experiments. These features allow for eliminating the unnecessary and destructive dichotomy in contemporary applied mechanics. 4. Concluding Remarks.

The major problems in contemporary experimental mechanics are of a theoretical nature. Disregarding approaches accepted in scientific theories and in mathematical theories mostly causes them. It would be useful if authors of papers presenting results of research in applied mechanics – including analytical and experimental mechanics, together with computational fracture mechanics and computational materials science would clearly state when they deal with the hypothetical mathematical constructs that may, or may not, bear some resemblance with responses of real physical objects, and when they attempt to assure a reliable correspondence between the developed system of analytical expressions and the selected responses of real bodies, supported by a discussion of experimental evidence. In other words, it would be useful to have a clear distinction between the responses of constructs existing within an applied mathematics realm, and the responses of models of bodies, existing within a physical realm. Evidently, the meaning of the same differential equations and their parameters depend on the general frame of reference. Modern engineering is an interdisciplinary field of intellectual and technological activities, involving physics, chemistry, biology, mathematic, economic, societal issues, political issues, and ethical issues. Thus, to satisfy societal expectations, education patterns of leading engineers should be commensurate with the notion of a modern Renaissance man. 5. 1.

References

Aleksandrov, A. D., Kolgomorov, A. N., Lavrent’ev (eds.): Mathematics. Its concepts, Mthods, and Meaning, The M.I.T. Press, Cambridge, Massachusetts, 1963. 2. ASTM,: 1969 Book of ASTM Standards, The American Society for Testing and Materials, Philadelphia, Pennsylvania, 1969. 3. Born, M. and Wolf, E.: Pricciples of Optics, Pergamon Press, Oxford – New York, 1975 4. Doeblin, E. O.: Measurement Systems: Application and Design, McGrow-Hill Book Co., New York, 1983. 5. Feynman, R.: The Character of Physical Law, MIT Press, Cambridge, Massachusetts,, 1993. 6. Haddad, Y. M.: Viscoelasticity of Engineering Materials, Chapman& Hall, Lomdon- New York, 1995. 7. Hecker, W. F. and Pindera: General relations of strain-gradient stress analysis, Theoretical and Applied Fracture Mechanics 25 (1996), 233-261 8. Kac, M.:Some Mathematical Models in Science, Science 166 (1969), 695-699. 9. Kestin, J.: A Course in Thermodynamics, McGraw-Hill Book Company, New York, 1979. 10. Kuhn, T. S.: The Structure of Scientific Revolution, University of Chicago Press, Chicago, 1970. 11. Leipholz, H. H. E.: On the Role of Analysis in Mechanics, Transactions of the CSME 7 (1983), 3-7. 12. Natke, H. G.: About the Role of Mathematics in Engineering – Thoughts of a Scientist between both Branches, GAMM Mitteilungen 1966, Akademie Verlag, Berlin 19 (1996), 121-131.

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13. Pindera, J. T. and Straka, P.:On physical measures of rheological responses of some engineering materials in wide ranges of temperature and spectral frequencies, Rheological Acta 13 (1974), 338-351. 14. Pindera, J, T., Straka, P. and Tschinke, M. F.: Actual Thermoelastic Responses of some Engineering Materials and its Applicability in Investigations of Dynamic Responses of Structures, VDI-Bechichte 313 1976), 579-584 15. Pindera, J. T.: Foundations of Experimental Mechanics: Principles of Modelling, Observation and Experimentation, in J. T. Pindera (ed.), New Physical Trends in Experimental Mechanics (CISM Courses and Lectures No. 64), Springer-Verlag, New York (1981) 199-327. 16. Pindera, J. T.: Advanced Experimental Mechanics in Modern Engineering Science and Technology, Transactions of the CSME 11 (1987) 125-138. 17. Pindera, J. T.: Advanced Experimental Mechanics and its Components: Theoretical, Physical, Analytical and Societal Aspects, in A. P. S. Selvadurai (ed.), Development in Engineering Mechanics, Elsevier, Amsterdam – New York, (1987) 347-414. 18. Pindera, M.-J., Pindera, J. T. and Ji, X.: Three-dimensional Effects in Beams – Isodyne Assessment of a Plane Solution, Experimental Mechanics 29 (1989), 23-31. 19. Pindera, J. T.: Local Effects and Defect Criticality in Homogenous and Laminated Structures, Trans. ASME, J. Pressure Vessel Technology 111 (1989), 136-150. 20. Pindera J. T. and Pindera M.-J.: Isodyne Stress Analysis, Kluwer Academic Publishers, Dordrecht, 1989. 21. Pindera, J. T., Hecker, F. W. and Wen, B.: Testing theoretical bases of caustic method in fracture mechanics and stress analysis, Theoretical and Applied Fracture Mechanics 15 (1991), 11-33. 22. Pindera, J. T. and Liu, X.; On the Actual Three-dimensional Stresses in Notches and Cracks, Composites Engineering 1 (1992), 281-301. 23. Pindera, J. T. and Wang, G.: Isodyne Stress Analysis of Adhesively Bonded Symmetric Joints, Experimental Mechanics 32 (1992), 348-356. 24. Pindera, J. T., Josepson, J. and D. B.: Electronioc Techniques in Isodyne Stress Analysis. Part 1: Basis Relations. Part 2: Illustrating Studies and Discussion, Experimental Mechanics 37 (1097), 33-38, 110-114. 25. Pindera, J. T.: Principles and Approaches of Advanced Experimental Mechanics in Service of Modern Technology, in Y. M. Haddad (ed.), Advanced Multilayered and Fibre Reinforced Composites, Kluwer Academic Publishers, Dordrecht, 1998. 26. Pindera, J. T.: Techniques of Tomographic Isodyne Stress Analysis, Kluwer Academic Publishers, Dordrecht, 2000. 27. Popper, K. R.: The Logic of Scientific Discovery, Harper and Row, London, 1977. 28. Ramachanidran, G. N. and Ramaseshan, S.: Crystal Optics, in S. Flügge (ed.), Encyclopedia of Physics, volume XXV/1 Crystal Optics. Diffraction, Springer-Verlag, Berlin, (1961) 1-592. 29. Reiner M.: Rheology, in S. Flügge (ed.), Encyclopedia of Physics VI, Springer-Verlag, Berlin, (1958), 434-520. 30. Siebel, E. and Ludwig, N.: Handbuch der Werkstoffprufung (Material Testing Manual), SpringerVerlag, Berlin, 1958. 31. Sokolnikoff, I. S.: Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956. 32. Stuart, H. A. (ed.): Die Physik der Hochpolymeren, Dritter Band (Physics of Highpolymers, Third Volume), Springer-Verlag, Berlin, 1955, 33. Thum, A. et all.: Verformung, Spannung und Kerbwirkung (Deformation, Stress and Notch Action), VDI-Verlag, Düsseldorf, 1960. 34 Valanis, K. C. (ed.):Constitutive Equations in Viscoplasticity: Phenomenological and Physical Aspects, The American Society of Mechanical Engineers, United Engineering Center, New York, N. Y., 1976 35. Wigner, E. P.: The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications on PURE AND APPLIED MATHEMATICS XIII (1960) 1-14. 35. Zehnder, A. T. On the temperature distribution at the vicinity of dynamically propagating crack in 4340 Steel, J. Mech. Phys. Solids 39 (1991), 385-415. 36. M.: Combined Loadings in the Theory of Plasticity, PWN, Warszawa, 1981.

INVERSE METHODS IN EXPERIMENTAL MECHANICS J. F. DOYLE School of Aeronautics and Astronautics Purdue University West Lafayette, IN 47907 Abstract This paper provides a framework for the active use of experimental mechanics in solving partially specified (or inverse) problems. It is based on a wavelet (or unit load) representation of the unknown applied loading that allows the FEM analyses to be performed as an external process and thus utilize the power and versatility of commercial codes. As a demonstration of the method and its attributes, different structures are studied experimentally using strain gage and whole-field data. 1.

Introduction

Experimental analyses are regularly combined with finite element analyses to give a deeper understanding of a problem. More often than not, however, these analyses are used passively in the sense that the separately generated results are simply compared on some common basis; a simple example is comparing measured strain gage results with those predicted using FEM. What this paper addresses is the active use of experimental mechanics as an integral part of an FEM analysis and to explore some of the new analysis opportunities made available by this merger. A fully specified problem in FEM analyses is one where all the material properties, geometries, boundary conditions, loads, and so on are known and the displacements, stresses and so on are required. A partially specified problem is one which deals with an existing structure or prototype where some information is missing. An example is modeling a structure with a loose bolt or one with wind loading. A common practice is to try to make all problems fully specified by invoking reasonable modeling assumptions. For example, it would be reasonable to assume an elastic joint for the bolt and report the results for joint stiffness ranging from very stiff to somewhat flexible. Because assumptions are used, the results must be reported over the full possible latitude of the assumptions. This adds considerably to the total cost of the analysis. Additionally, and ultimately more important, the use of assumptions make the results uncertain and even unreliable. On the face of it, the most direct way of narrowing the uncertainty in the results is to simply measure the unknown. For a dimension this is probably straightforward but what about the force due to the impact of hale on an aircraft wing? This is not something that can be measured directly. The primary concern of research reported in this paper is the establishment of formal methods for including measurements as part of the analysis of partially specified problems. 585 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 585–594. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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One key to our approach is to take the applied loading as the fundamental unknown even if we are not specifically interested in it. To see this, consider a linear complex structure subjected to an excitation, the response is governed by the equations

Within a structural context, therefore, system changes such as fracture, erosion, bolt loosening, and so on, can only appear as changes of stiffness and/or mass (damping is usually an unknown anyway). The changed condition is represented as a perturbation of the stiffness and mass matrices from the original state as and the applied loading as equations can be rearranged as

the governing

That is, the changed structure can be thought of as the original structure (and hence assumed completely known) plus a set of additional loads. The vector clearly has information about the changed (unknown) state of the structure; therefore, if somehow, we can determine then it is a direct (but sometimes subtle) process to extract the unknown structural and/or loading information. Thus, the essential problem is to determine an applied load from some appropriate measurements. This is a problem in force identification and References [1-8] can be used as an entry to some of this literature. The example problems discussed here are chosen to illustrate just some of the possibilities of new analyses and questions that can be explored although they are all restricted to being static and linear. 2.

Computer Implementation for Static Problems

The formulation of the problem is detailed in Reference [9] and will not be repeated here; instead we just summarize the approach as implemented in the computer code Wavelet.

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The governing equations for a static linear system can be discretized using the finite element method as

In this, is the vector of all the free degrees of freedom and is of size [K] is the banded symmetric stiffness matrix, is the vector of all applied loads (some of which are zero), is the vector subset of that contains the non-zero applied load scales, and is the matrix that associates these load scales with the degrees of freedom. In the inverse problem of interest here, both the applied loads and displacements are unknown but we know some information about the responses. In particular, assume we have a vector of measurements of size which are related to the structural DoF according to where the logic matrix [Q] is of size We want to find the forces that make the system best match the measurements. The unknown forces are represented as the collection

where is a distribution of forces and the corresponding scale. There are unit distributions. The key to understanding the computer implementation is this series of load vectors in – each load case can be performed externally by a commercial FEM code. The basic relationship is shown in Figure 1. One of the roles of the inverse program is to prepare the scripts to run the external programs. Each causes a response such that the total response (because of the linearity of the system) is

Because the responses are obtained by the FEM, they will have imbedded in them the effects of the complexity of the structure. Furthermore, it is important to note that these responses are not necessarily just displacement, they could be strain or principal stress difference as required by the experimental method, but in each case it would be precisely the same force scales Consequently, the inverse program does not need to perform any operation such as differentiation of displacements in order to get strains. In other words, anything particular to the modeling or mechanics of the structure is handled by the FEM program. The measured data at a collection of locations are related to the computed responses as where [Q] only symbolically plays the role of selecting the subset of participating in forming the data. That is, the arrays are established directly from the participating components of and The minimizing principle is

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where [W] is a diagonal data weighting array, and [H] is one of the regularization forms discussed in Reference [9, 8]. A readable discussion of regularization is given in Reference [10] and a more extensive mathematical treatment can be found in Reference [11]. This latter reference has an extensive bibliography. The least squares minimization leads to

If supplementary information is available, it is incorporated simply by appending to the matrix [A] before the least squares procedure is used. This system of equations is symmetric of size and can be solved by the standard techniques such as LU decomposition [10]. The purpose of and the regularization term is to overcome the ill-conditioning usually associated with least squares solutions. A reasonable beginning value of is to try where Tr is the trace of the matrix computed as the sum of diagonal components. Three cases are implemented in Wavelet: zero, first, and second order. 3.

Experiment I: Displacement Data

Figure 2 shows a double exposure holographic fringe pattern for the out-of-plane displacement of a pressure loaded plate. The displacements are related to the fringe order by

where N are the black fringes, is the wavelength of light, and illumination and viewing, respectively.

and

are the angle of

The fringe pattern is manually digitized along the center of some of the black fringes and the data stored as pixel location and fringe order. This is then scaled to the physical coordinates of the mesh. Figure 3 shows the limited number of discretized data points taken from the image. The empty circles shown were used to scale the data from pixel coordinates to physical coordinates. Figure 3 also shows the FEM mesh used.


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The data were relocated onto the nearest nodes of this mesh by the scheme discussed in Reference [9].

In formulating the inverse solution for this problem, there is only one unknown and that is the pressure. That is, is comprised of a single vector representing a uniform pressure of unit magnitude. (Because of the different element sizes, does not necessarily contain constant values.) This is a case where the script is for a uniform pressure and the specific contents of is determined by the Mesh/FEM program. The nominal measured pressure is 7.63 kPa and computed pressure is 7.77 kPa. The comparison is very good. Figure 2 shows a comparison of the reconstructed fringe pattern with that measured. Since the pressure comparison is good, it is not surprising that this comparison is also good. A theme running through the inverse solutions of this paper is that once the unknown loads have been determined, then the apparatus is already available for the complete stress analysis. Figure 4 shows, for example, the and stress contours. 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. Note that for plate problems this would require double differentiation. Furthermore, it is because of this attribute, that only a limited number of data points needed to be recorded.

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Experiment II: Strain Gage Data

Figure 5 shows a photograph of a 3-D thin-walled reinforced box-beam structure. The vertical loading is applied through a turn-buckle which has a force transducer. The dial gauges can be used to locate the shear center (the load position which causes no rotation). Figure 6 shows the geometry and the mesh used in the modeling. The reinforcing steel bars are attached to a thick end-plate; the thin aluminum skin is riveted to these bars at every 25 mm (1 in.). The box-beam structure is modeled as a 3-D shell.

Figure 6 shows the locations of three delta rosettes placed mid-way along the beam; the shear strain is easily obtained from a delta rosette as [12]

where the gage is oriented along the x-axis. Figure 7 shows the recorded shear strains for six load increments when the nominal load position is at x/W = 0.71 (this location is actually very close to the shear center); the data trends for the other load locations are similar. These data were fitted with straight lines and then the values adjusted to the origin. The maximum shear strains at the nominal maximum load of P = 60 were used in the inverse solutions.


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In the inverse problem, both the load and its position are assumed unknown. Rather than treat the unknown position as a variable (which would require an iterative solution) the problem instead is converted to a load problem by considering two unknown loads at the lower corners. These two forces are equipollent to the single force at position x and by taking moments about the load point, gives or Figure 7 shows a plot of the computed load position against its nominal value. The comparison is quite good. This experiment shows that while an instrumented structure can be its own load transducer, it is also capable of determining where the load is applied. As a consequence, this provides more flexibility in designing the load scheme.

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Experiment III: Photoelastic Data

The previous sections dealt with experimental methods where there is a linear relation between the data and the mechanical quantity. Photoelasticity is different because it has a nonlinear relationship between the observed fringes and applied loads. While it is true that a doubling of the load causes a doubling of the fringe orders, this is not true for non-proportional loading cases. 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. Since our inverse method is based on superposition, we must modify it to account for this nonlinear relation. The measured fringe data are related to the stresses by [12]

where N are the black fringes, is the material fringe constant, and h is the model thickness. As is usual in nonlinear problems, we begin with a linearization about a known state. That is, suppose we have a good estimate of the stresses as then the true stresses are only a small increment away. Do a Taylor series expansion about this known state to get

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Let the stress increments be computed from the scaled unit loads so that for a particular component of force it is given by

This must be computed for each component of force. In anticipation of using regularization, it is best to deal directly with the force distributions. Noting that we write the approximate relation as

with and

The minimum principle can now be established leading to

This is solved repeatedly using the newly computed to replace In terms of the computer implementation of Figure 1, the unit solution is determined only once (for each load), but the updating of the stress field requires an FEM call on each iteration. Figure 8 shows the loading arrangement for a photoelastic model with a hole. The specimen is essentially under three-point loading. The inset shows the light field fringe pattern. The fringes were recorded using a diffuse light polariscope with white light and the grey scale image shown is that recorded directly by the camera. The material fringe value was calibrated with this arrangement. Although there is some wash-out of the fringes near the concentration points (load and support point plus edge of hole), the fringe patterns are deemed adequate because we will mostly use data away from these locations.

In this experiment, the contact load was distributed by placing a soft layer between the model and the applied load; consequently, there was no possibility of knowing a priori the distribution. It is not the intention here to determine the precise distribution of this load – to do that would require collecting data close to the distribution. Rather, we are concentrating on doing a stress analysis in the region close to the hole. The fringe data of Figure 8 were discretized in a manner similar to that of the holographic fringes. That is, the center of the dark fringe was manually digitized at a number of points and this data relocated to the nodes in its vicinity. Only fringes that are clearly visible were discretized.


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In all iterative schemes, it is important to have a method of obtaining good initial guesses. Figure 9 shows the results of finding force pairs at neighboring nodes at many node locations. No regularization was used and convergence occurred in about five iterations for each of these two-force problems. Shown is the sum and difference of the forces. The sum gives an idea of the resultant force of the actual distribution while the force difference gives an idea of the location of the resultant. The separation between the force pairs depends on the ability of the data to resolve the forces without regularization. In the present test, the separation used was four node spacings. This information can now be used to make a reasonable initial guess.

The initial guess for the iterative scheme was taken as the resultant load with all other fourteen being zero. Figure 9 shows a typical reconstructed distribution, where about ten iterations was required for convergence. The precise distribution depends on the amount of regularization used. A key point, however, is that the solution in the vicinity of the hole does not depend sensitively on the precise nature of the distribution of applied stress. That is, all reconstructed distributions gave nearly the same solutions in the vicinity of the hole. Figure 8 shows a comparison of the measured and the reconstructed fringe patterns. The patterns close to the hole are almost identical. The patterns near the distribution show a difference which might be due to the presence of a shearing traction. Figure 10 shows the reconstructions for the and stress contours.

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6. Discussion This paper provides a framework for the active use of experimental mechanics in solving partially specified problems. That the FEM analyses can be arranged as an external process is a significant attribute because it means the power and versatility of commercial codes can be leveraged and this greatly extends the range and type of problems that can be solved. A point worth reiterating is the significant side benefits of using an FEM-based modeling. First, there is a consistent underlying framework for handling the experimental problem. That is, the same mesh is used for the manipulation of the experimental data (e.g., smoothing, relocating, cropping). Second, the postprocessing apparatus is already in place for obtaining information such as stress, as well as simply presenting the results. The example problems discussed were for linear, static problems – the emphasis was on structural complexity as represented by the box-beam structure. The underlying formulation, however, is also applicable to dynamic problems as illustrated in References [13,8]. It is also applicable to nonlinear problems. 7. Acknowledgements This work was supported in part by a U.S. Army Multi-Disciplinary University Research Initiative (United States Army Grant No. DAAH04-96-10331) awarded to Purdue University. 8. References 1. 2. 3. 4.

5. 6.

7. 8. 9. 10. 11. 12. 13.

K. K. Stevens, “Force Identification Problems: An Overview,” Proceedings of SEM Spring Meeting, Houston, 1987, pp. 838-844. M. T. Martin and J. F. Doyle, “Impact Force Identification from Wave Propagation Responses,” International Journal of Impact Engineering, 18, 1996, pp. 65-77. A. S. Kobayashi, “Hybrid Experimental-Numerical Stress Analysis,” Experimental Mechanics, 23, 1983, pp. 338-347. C.-H. Haung and W.-Y. Shih, “An Inverse Problem in Estimating Interfacial Cracks in Bimaterials by Boundary Element Technique,” International Journal for Numerical Methods in Engineering, 45, 1999, pp. 1547-1567. T. H. Baek and R. E. Rowlands, "Hybrid Stress Analysis of Perforated Composites using Strain Gages,” Experimental Mechanics, 41, 2001, pp. 195-203. T. Nishioka, “An Intelligent Hybrid Method to Automatically Detect and Eliminate Experimental Measurement Errors for Linear Elastic Deformation Fields,” Experimental Mechanics, 40(2), 2000, pp. 170-179. S.-M. Cho, A Sub-Domain Inverse Method for Dynamic Crack Propagation Problems, M.S. Thesis, Purdue University, 2000. J. F. Doyle, “Reconstructing Dynamic Events from Time-limited Spatially Distributed Data,” International Journal for Numerical Methods in Engineering, to appear, 2001. U. Kang and J. F. Doyle, “An Inverse Method for Static Problems using Whole-field Data,” Experimental Mechanics, submitted, 2001. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes, ed., Cambridge University Press, Cambridge, 1992. A. Neumaier, “Solving Ill-conditioned and Singular Linear Systems: A Tutorial on Regularization,” Society for Industrial and Applied Mathematics, 40(3) 1998, pp. 636-666. J. W. Dally and W. F. Riley, Experimental Stress Analysis, Ed., McGraw-Hill, New York, 1991. J. F. Doyle, “A Wavelet Deconsolution Method for Impact Force Identification,” Experimental Mechanics, 37, 1997, pp. 404-408.

COMPLEX STIFFNESS IDENTIFICATION BY INVERSE METHODS H. SOL and W.P. DE WILDE Vrije Universiteit Brussel (V.U.B.) Department Mechanics of Materials and Constructions Pleinlaan, 2 1050 Brussels, Belgium Abstract The basic principle of an inverse method for material stiffness identification is to compare measured values of a test specimen with computed values from a numerical model of the test specimen. The parameters in the numerical model are the unknown material stiffness properties and they are iteratively updated starting from an initial set of parameters until the computed output matches the measured values. This paper first outlines the general procedure used for material stiffness identification with inverse methods. The procedure is next illustrated with an example on the identification of complex engineering constants of composite materials. In this example the vibration behaviour of a freely suspended test plate is used as the information source. The measured quantities are the resonance frequencies of the test plates and the damping ratios of their associated mode shapes. The resonance frequencies are used to tune the real part of the complex moduli, while the damping ratios are used to identify the imaginary part.

1.

Introduction

Making models is very common in engineering science. With a model one tries to simulate the behaviour of a physical reality, but it is clear that even the best model will always have its limitations [13], [10]. The appearance of computers has increased the possibility of making accurate numerical models to simulate the static and dynamic behaviour of engineering constructions. The most powerful and popular numerical method for simulation of the behaviour of constructions is the finite element method. Since its appearance in 1960 [6], many good textbooks have been written about finite element modelling e.g. [19]. A finite element model of a construction contains geometrical and constitutive parameters. For an assumed known geometry and constitutive parameter values, the construction model allows the computation of the response ('Output') for given excitation forces ('input'). It must be however clear that one can only expect a physically reliable output if the model is sufficiently accurate and if the inputted parameter values are sufficiently correct (if not: garbage in, garbage out!). The values of most constitutive or material parameters involved in a model can be measured experimentally by appropriate standard testing. The ASTM society [2] has developed state of the art procedure for the identification of many material properties. Unfortunately, in some cases it is not possible to apply standard testing procedures. Sometimes, the material shows such 595 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 595–608. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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complex constitutive behaviour that standard testing can not give satisfying solutions. In such cases it can be considered to use an inverse method. In other cases, inverse methods sometimes just offer an advantageous alternative for standard testing. The principle of an Inverse Method for Material Identification (IMMI) is to compare a measured output from an experiment on a test specimen with the computed output of a numerical model of the same specimen loaded with the same input values (Figure 1). The difference between the measured and computed output is called "the residual". The residual can be incorporated in a user defined cost function that describes the goodness of fit of the numerical model with the test specimen. A simple example of a cost function is the summation of the squared differences between the corresponding output components.

The - a priori unknown - material properties in a IMMI are the parameters in the numerical model. These material properties are (usually iteratively) tuned in such a way that the computed output matches the measured output. The parameter modifications are computed by minimisation of the cost function. The final result of the output matching procedure is an estimate of the parameters values and eventually an estimate of their error bounds.

2.

Working principles of material identification by inverse methods

All a priori unknown material parameters of the numerical model can be stored in a parameter column {p}. The computed output column {y} contains the corresponding components of the measured output column {m}. The relation between the output column {y} and the parameter column {p} can be linear or non linear. The two cases will be considered in the next two paragraphs.

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2.1. LINEAR RELATION BETWEEN THE OUTPUT COLUMN AND THE PARAMETER COLUMN The relation (2.1) gives a general expression for a cost function with the parameter column {p} as an independent variable (a repeated index indicates a summation) and thus also as a function of the residual {m} - {y}:

In (2.1) is a Cost function yielding a scalar value, (or alternatively written as or ) is a column containing the NP unknown material parameters, contains initial user estimates for the material parameters, {m} contains the (NM x 1) measured output values, {y} is a (NM x 1) column containing the NM computed output values using {p} in the numerical model, (or or is a (NM x NM) weighting matrix applied on the difference between the measured column and the output column, and is a (NP x NP) weighting matrix for the difference between the initial parameter column and the parameter column {p} . Because the output column must contain enough information to evaluate properly the parameter column, NM > NP in a IMMI. The Cost function C(p) has a minimal value for the optimal parameter values column . It can be seen easily that (2.1) reduces to a weighted least squares cost function of the residual if is set to zero. Optimisation algorithms for minimising the cost function can be categorised in two classes: (i) algorithms that use function evaluations only (direct search algorithms) and (ii) algorithms that use output gradient information. An overview of algorithms from both classes can be found in [1], [14], [12]. The algorithms of the second class are more efficient for both IMMI and will be illustrated in the following text. Let be an initial estimate of the parameter values. The value of the Cost function in an arbitrary position {p} close to the position can be found with a Taylor expansion:

or where

is the (symmetrical) Hessian matrix evaluated at

The partial derivative of (2.3) with respect to the parameter component

is:

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For

the above expression becomes zero, hence:

thus: The partial derivative in expression (2.6) can be evaluated by taking directly the partial derivative of (2.1) with respect to component and evaluating the result at

with: For

the (NM x NP) sensitivity Matrix. (2.7) becomes:

The sensitivity matrix [S] is constant because of the assumed linear relation between {y} and {p}. The Hessian matrix can be obtained by taking the partial derivative of (2.7) with respect to component and evaluating the result at

Thus with (2.9) and (2.8) in (2.6):

The formula (2.10) is called the estimator'. In the linear case, (2.10) gives directly a value for starting from an arbitrary initial value (2.10) contains an expression that must be inverted. can be used to obtain the necessary numerical stability in case the sensitivity of the output {y} for parameter variations is poor.

2.2.

NON-LINEAR RELATION BETWEEN THE OUTPUT COLUMN AND THE PARAMETER COLUMN

Unfortunately, the difficulty with most IMMI is that the relation between the output vector {y} and the parameter vector {p} is nearly always non-linear. This means that updating the parameter values {p} from an initial value to a final optimal value has to be done in an iterative way. Expressions (2.2) - (2.9) hence need to be evaluated (linearised) in intermediate positions between and


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(2.6) becomes:

with:

and: or, if the non-linear part of the sensitivity matrix is neglected:

(2.12) and (2.14) in (2.11) yields:

The estimator (2.15) is now a recurrent formula that updates {p} from an intermediate value towards a better value with j = 0 , 1, 2, ....till convergence is reached at the value The linearised value of the sensitivity matrix must be re-computed in every iteration step. The consecutive values of {p} iterate towards if convergence occurs and if the convergence is not blocked at a local minimum of the Cost function (instead of in the desired global minimum). In (2.15) the weighting matrices and and the column with the initial parameter estimates remain constant during the consecutive iterations. The 1 second term in the general cost function (2.1) hence tends to act as a kind of 'spring' that prevents the parameter column of evolving too far from the initial estimate The 'stiffness' of this 'spring' is dependent on the relative values of the weighting matrices. It is also possible to re-evaluate the cost function (2.1) in each iteration by replacing the initial estimate by

In that case, the last term in (2.15) vanishes and the estimator becomes:

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Illustration: the resonalyser procedure

3.1. GOAL OF THE RESONALYSER PROCEDURE The resonalyser procedure is a IMMI that aims the identification of complex orthotropic engineering constants by the measurement of the vibration behaviour of rectangular test plates. The procedure is based on the above described inverse method and results in averaged material stiffness values over the whole plate area. The obtained material stifnesses are therefore ideally suited as input values for finite element models. The visco-elastic behaviour of orthotropic materials in a plane state of stress can be characterised by 4 independent engineering constants. The constants appear as complex values in the compliance matrix that describes the relation between the strain column and stress column:

In (3.1) the used symbols are: : normal strain component in the i-direction : shear strain component in the 12-plane : normal stress component in the i-direction : shear stress component in the 12-plane : Main Poisson's ratio in the 12-plane : Young's modulus in the i-direction : Shear modulus in the 12-plane The complex engineering constants can be written as:

The real part of the engineering constants describes the pure elastic behaviour of the material while the imaginary part describes the dissipative or damping behaviour. The dissipative part is often called 'the tangents delta'. The characterisation of orthotropic material hence requires the evaluation of 8 independent constants: 4 real values and 4 'tangents delta' values.


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The resonalyser procedure proceeds in two parts: the first part estimates the real part of the engineering constants using measured resonance frequencies and the second part estimates the imaginary part by measured modal damping ratios. The second part is optional and will be only executed by people who are also interested in the damping behaviour of the examined material.

3.2. ESTIMATION OF THE REAL PART OF THE ENGINEERING CONSTANTS The principle for the determination of the real part is to compare measured resonance frequencies of a rectangular test plate with frequencies computed with a numerical model of the test plate (Figure 2).

A thin plate model, based on the Love Kirchhoff theory, is used for the numerical model of the test plate [15], [16]. Extensions of the resonalyser procedure use thick plate models with linear and higher order shear corrections [11]. The resonance frequencies and mode shapes of the numerical model can be found by the solution of an eigenvalue problem. The parameter column contains the 4 real parts of the orthotropic engineering constants (hence NP = 4):

The 5 first resonance frequencies of the freely suspended test plate are stored in the output column {y} and the measurement column {m} (hence NM = 5). The relation between {p} and the computed resonance frequencies in the numerical

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model is non-linear. The estimation of {p} is thus obtained using the iterative estimator (2.17). The formula yields good results if the parameters can be observed through the measured resonance frequencies. This means that at least one of the five measured resonance frequencies must sufficiently change for variations of each of the 4 material parameters. It can be shown [16] that this demand requires a length/width ratio a/b of the test plate close to the value given by:

The preparation of such a test plate however requires the knowledge of the ratio which is basically unknown. The value of this ratio can be found by cutting two test beams in perpendicular directions from the test plate (Figure 3).

The measurement of the fundamental (first) resonance frequencies and of this two beams allows the computation of fair values for and with (3.5) and hence of the ratio a/b using (3.4):

(EI is the bending stiffness, the weight per unit length, L the beam length). The obtained values for and provide at the same time excellent starting values for the inverse identification procedure. A freely suspended test plate according to (3.4) has a torsional mode shape associated to the first resonance an anticlastic bending mode shape associated to the second resonance and a synclastic bending mode shape associated to the third resonance The stress pattern of the torsion mode shape is dominated by shear stresses and hence the resonance frequency is very sensitive for variations of The difference between the resonance frequencies and is mainly due to the value of Poisson's


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ratio. (for the imaginary case of zero Poisson's value, both mode shapes will have the same resonance frequency). These observations provide an easy way of finding good starting values for and

in which: The first eigenvalue of the test plate The second eigenvalue of the test plate The third eigenvalue of the test plate The sensitivity of the eigenvalue associated with the torsional mode shape for

variations

The sensitivity of the difference of the synclastic and anticlastic eigenvalues for variations of Poisson's ratio The empirical evaluation of

and

is explained in [16].

The identification of the elastic parts of the engineering constants is performed using the 5 plate frequencies:

The components of the (5 x 4) sensitivity matrix [S] are recomputed numerically in every iteration step (see e.g. [16], [9], [11]):

in which is the i-th computed mode shape and is the stiffness matrix of the numerical model of the test plate (both computed with the value of the parameters at a given j-th iteration step). The parameter value in the final iteration is the result of the first stage of the resonalyser procedure.

3.3. ESTIMATION OF THE COMPLEX PART OF THE ENGINEERING CONSTANTS The complete identification of the complex moduli requires also the knowledge of the imaginary (or dissipative) part of the engineering constants. In the orthotropic material case this includes the determination of 4 tangents delta. These tangents values can be identified by the measurement of the modal damping ratios associated with the 5 plate resonance frequencies. The (4x1)

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parameter column {p} and the (5x1) measurement column {m} for the second stage hence are:

The modal damping ratios can be evaluated by curve fitting the decaying sinusoidal signal in the time domain after an excitation of the test specimen with the envisaged resonance frequencies (see Figure 4). This signal can be curve fitted with the formula (4.11):

in which

is the modal damping ratio and is the circular resonance frequency The procedure must be repeated for each of the five resonance

frequencies.

The curve fitting in the time domain allows to obtain accurate results even for lightly damped test specimen. The main problems for the measurement of the modal damping ratios are situated on the experimental level [9]. Small variations in the boundary conditions (suspension threats, accelero meter wiring,..) induce external damping that can not be distinguished from the internal material damping. Therefore it is strongly advised to generate and measure the signals without physical contact. The excitation can be performed acoustically with a loudspeaker and the measurement of the decaying signal can be done with a laser vibrometer. The influence of the boundary conditions can be minimised by the suspension of the test specimen in the nodal points of the mode shapes. A simplifying element in the second stage is that the relation between the parameters and the measured modal damping ratios is linear:


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The coefficients in (3.12) can be computed with the numerical plate model from the modal strain distributions and modal damping energy expressions [9]. The sensitivity matrix [S] can be derived easily from (3.12):

The linear character of the equation (4.12) allows to start from arbitrary initial values for the parameters. They can thus be taken all as zero. The least squares estimator (3.10) can be used with and and will yield immediately the optimal parameter values.

3.4. PRACTICAL PROCEDURE AND AN EXAMPLE The above described resonalyser procedure is automated using a MS-windows PC software programme. The PC is equipped with a data acquisition card for the vibration measurements. The practical measurement procedure proceeds as follows: Machine two test beams from the original test plate as indicated in figure 3 Measure the first resonance frequency of both test beams and compute initial values for and Determine the dimensions of the test plate according to (3.4) and machine the test plate Measure the first 5 frequencies of the test plate Start the first stage of the resonalyser procedure for the identification of the real parts using (2.17) Measure the modal damping ratios associated with the measured frequencies of the test beams and plate Start the second stage of the resonalyser procedure for the identification of the imaginary parts using (2.10) At the end of the procedure the complete set of orthotropic complex engineering constants are identified. Table 1 gives an example of the results of the resonalyser procedure on a Carbon/Epoxy [0°, 90°]s test plate with dimensions (0.29 m, 0.26 m, 0.0018 m). Other examples from the resonalyser and validation of

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these results with results from traditional testing (ASTM tensile and DMA tests) is given in [9], [15], [16].

3.5. ERROR DISCUSSION Error discussion of the final results of material identification using inverse methods is a difficult matter because the physical definition of the desired material properties is given outside the framework of the identification procedure (e.g. a modulus is defined as a ratio of strain and stress while in the numerical model it acts as a model parameter). Inside the IMMI framework and with the assumption that the numerical model reflects perfectly the behaviour of the test specimens, the standard variation on the final results can be computed [5], [7] based on the standard variation of the measurements. In the resonalyser procedure the estimation of the real part is based on the measurement of resonance frequencies. The experimental error on measured frequencies is small, even if non-sophisticated measurement equipment is used. The results of the IMMI with high quality test plates (homogeneous elastic properties and constant thickness and thus an accurate numerical model) are accurate. A detailed error discussion on the real part errors can be found in [16]. The estimation of the complex part is based on the measurement of the modal damping ratios. This part is more sensitive to measurement noise and non-linear behaviour of the damping as a function of the excitation amplitude. The results are hence less accurate. A detailed discussion on the complex part errors can be found in [9]. More examples of IMMI procedures on different physical problems can be found in [17], [18].

5. Discussion: Resonalyser procedure versus standard testing It is useful to compare the resonalyser procedure with standard procedures to measure material stiffnesses like tensile and shear testing as described in ASTM procedures [3], [4]. A basic difference between standard testing and the resonalyser procedure is that standard testing aims to create a uniform field (uni-axial stress fields), while the resonalyser procedure analyses the complex multi-axial stress fields occuring in mode shapes. Indeed, standard testing is based on analytical processing of the measured results and thus requires data that can be described with a 'simple' formula. Uniform response fields are usually very difficult to obtain correctly from an experimental point of view and require often expensive equipment. IMMI in general can deal with complex heterogeneous fields containing


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a lot of information. Heterogeneous fields occur usually in a natural way and are thus easier to obtain experimentally than uniform fields. An advantage of standard testing is that the error can be observed straightforwardly thanks to the direct character of the procedure, while IMMI is indirect and thus makes the error more difficult to estimate. A important advantage of IMMI is that it is possible to identify many material properties simultaneously from one test specimen, while standard testing is limited to the determination of one material property from one test specimen. This can lead to considerable time savings with IMMI. Probably the biggest advantage of IMMI is that it can be tailored to any test specimen geometry and boundary conditions. Only the imagination of the experimentalist is the limit for the conception of a good experiment with IMMI.

6.

Conclusion

IMMI is an identification technique that offers a uniform approach for different physical systems. The most important advantages of the method are the freedom in experiment conception and the possibility of simultaneous identification of set of parameters from the same test-system A disadvantage is the indirect character of the method and thus the increased sensitivity to errors. IMMI also requires a sound understanding of the relation between the desired parameter values and the measured response and a good insight in the nature of experimental errors and the numerical model limitations. It can be foreseen that in the future the sources of errors of IMMI will become smaller and smaller with the rapid evolution of computer power (more accurate numerical models) and the technical evolution in the measurement techniques of field information (data acquisition and scanning techniques). This evolution, in combination with a better understanding of the principles of mixed methods, will without any doubt lead to improved results and new applications of IMMI in the future.

7. References 1. 2. 3.

4. 5. 6. 7. 8. 9.

Atrek, A. 1984. New directions in optimum structural design, John Wiley and Sons, ISBN 0471-90291-8. ASTM, American Society for testing Materials, http://www.astm.org. ASTM, Designation D 3039, Standard Test Method for Tensile properties of Fiber resin composites, revised 2000. ASTM, Designation E 756-83, Standard Method for Measuring Vibration Damping properties of Materials, 1983. Catin, D.E. 1989. Estimation Control and Discrete Kalman Filter, Springerverlag. Clough, R.W. 1960, The finite element in plane stress analysis, Proc. 2nd.ASCE conf. on electronic computation. Collins, J.D. & Hart, G.C. & Hasselman, T.K. & Kennedy B. 1974. Statistical Identification of Structures, AIAA Journal, 12(2), pp.185-190. Cottin, N & Natke, H.G. 1986. On the Parameter Identification of Elasto-mechanical Systems using Weighted Input and Model Residuals, Ingenier. Archiv.56, pp.106-113. De Visscher, J., 1995, Identification of the complex stiffness matrix of orthotropic materials by a mixed numerical experimental method, Ph.D. Thesis presented at the Vrije Universiteit Brussel, Belgium.

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10. De Wilde, W.P. 1992, Numerical Modelling of linear elastic and visco-elastic response of 11. 12.

13. 14. 15. 16. 17. 18. 19.

composite structures, Computational Methods in Applied science, Elsevier Science publishers, pp. 233-245. Hua, H. 1993. Identification of Plate Rigidities of Anisotropic Rectangular Plates, Sandwich Panels and Orthotropic Circular Disks using Vibration Data. Ph.D. Thesis presented at the Vrije Universiteit Brussel, Belgium. Mottershead, J.E., Friswell, M.I. 1993, Model updating in structural dynamics: a survey, Journal of sound and vibration 167 (2), pp 347-375. Natke, H.G. 1995, What is a True mathematical model?, Inverse problems i n engineering, Vol.1, pp.26 7-272. Reklaitis, G.V., Ravindran, A., Ragsdell, K.M. 1983, Engineering Optimisation, John Wiley and Sons ISBN 0-471-055794. Sol, H. et all., 1996, La procedure resonalyser, La revue des laboratoires d'Essais, n° 46, pp. 10-12. Sol,H., 1986, Identification of anisotropic plate rigidities using free vibration data, Ph.D.Thesis presented at the vrije Universiteit Brussel, Belgium. Sol, H. & Oomens, C. Editors 1997. Material Identification Using Mixed Numerical Experimental Methods, Kluwer Academic Publishers. Vautrin, A. & Sol, H Editors 1991 Mechanical Identification of Composites., Proc of Euromech 269, St.Etienne, France. Zienkiewicz, O.C. 1977, The finite element method, McGraw-Hill, ISBN 0-07-084072-5.

CONSIDERATIONS OF A FLUTTER PREDICTION METHODOLOGY USING A COMBINED ANALYTICAL-EXPERIMENTAL PROCEDURE

PIERGIOVANNI MARZOCCA Virginia Polytechnic Institute and State University Blacksburg, VA 24061 LIVIU LIBRESCU Virginia Polytechnic Institute and State University Blacksburg, VA 24061 WALTER A. SILVA NASA Langley Research Center 100 NASA Road Hampton, VA 23681-2199

Abstract A non-destructive procedure enabling one to predict the flutter instability boundary from the data acquired in the subcritical flight speed regime is presented. The proposed technique combines an analytical approach with the experimental tests carried out in flight or in a wind tunnel. The expected outcomes of this study are: a) to reduce the risks of flying in the proximity of the flutter critical boundary, and b) to reduce significantly the amount of flight tests required in any flight clearance test program, that are both time consuming and costly. 1. Flutter Prediction Using Combined Analytical-Experimental Procedure

1.1 INTRODUCTIVE CONSIDERATIONS One of the most important phases involving the aeroelastic design is the evaluation of the flutter instability boundary by means of flight and tunnel tests at subcritical speed [1,2]. The objective of the flutter analysis is to quantify the critical boundaries of flutter [3-5]. Even at the present time, when analytical predictive methodologies have become increasingly useful toward this goal, flight and wind tunnel testing in conjunction with data analysis are always required for the verification of the theoretical predictions. Attempts to formulate adequate methods that permit a reliable prediction of the flutter boundary at speeds well below the flutter speed were made in the past by several investigators [3,6-8] and recently by Lind [5]. As it was pointed out, in order to reduce the number of flight or wind tunnel tests, a reliable technique for real-time monitoring of the aeroelastic stability of the aircraft is required [9]. A systematic approach should 609 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 609–618. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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be adopted toward the accurate estimation of aeroelastic structural parameters (mass, damping, stiffness, etc.) and to use these findings to effectively predict flutter boundary. In [1] an analytical approach was developed to address this problem. At the same time, the possibility of obtaining the flutter boundary from the aeroelastic response was indicated in [10] for 2-D lifting surfaces, and in [11,12] for aircraft swept wings in a compressible flow. One of the key elements of the methodology presented in this paper consists of the use, in combination with a real-time parametric identification procedure, of the coefficients of the aeroelastic governing equation. These parameters will be used for an exact evaluation of the aeroelastic response from the combined analytical procedure with the real time experimental results. Only when the certitude of the correctness of the coefficient of the equation of motion is obtained, we can extrapolate the effective value of the flutter speed. In addition, the analytical procedure of determining the subcritical aeroelastic response can be applied at any flight speed regime, i.e. subsonic, transonic and supersonic, via the inclusion in the model of the pertinent aerodynamics based on indicial functions [11-13]. 1.2 FLIGHT AND WIND TUNNEL TESTS A particularly dangerous aeroelastic instability that one must avoid is the flutter, which results from the interaction of structural, inertial and aerodynamic forces. Flutter is a special case of self-excited oscillation in which the wing exposed to the airflow, absorbs energy from the airstream in such a way that the added energy overpowers the natural damping of the structure, and causes the amplitude of the oscillations to increase exponentially at time unfolds. Flight tests are performed in order to evaluate the characteristics of an aircraft and determine a range of flight conditions in which an aircraft can operate without the occurrence of any aeroelastic instability. The totality of flight conditions fulfilling this requirement constitute the flutter flight envelope, see Fig. 1.

Within the traditional methods of flight testing, the flight envelope is expanded successively via a series of test points, using measurements generated in response to some excitation. This procedure involving flight flutter testing is based on the

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experimental determination of the modal damping coefficients and on their variation with the flight speed [14]. Such dynamical properties as damping are estimated from the measurement data, and trends are used to determine whether the envelope may be further expanded. As is well known [3], the procedures based on the determination of the damping are efficient in the case of mild or moderate flutter. On the other hand, the flutter phenomena can be violent, i.e. explosive, and therefore it may exhibit, with the increase of the flight speed, a rapid deterioration of the modal damping. In addition, in order to be able to use the flight test data, one must ensure that the data points are closely spaced, although the trends do not guarantee what increases in flight conditions may be safely considered without the sudden occurrence of the flutter instability. In this sense, flight flutter testing is potentially dangerous because an unexpected flutter instability, namely one not predicted from the trends can occur during a transition to a new test point. On this basis, a prediction of the flutter speed can be hazardous with the risk of a loss of the vehicle and even of human lives. It clearly appears that these methods are not reliable enough. Several approaches have been applied to improve the deficiencies in the flutter testing. Roy and Walker [9], have formulated an improved method for the evaluation of damping, Zimmerman and Weissenburger [3], and Houbolt [7], have tried, with certain success, of predicting the flutter speed at a speed well below the flutter dynamic pressure. However these methods are working well for system up to two degrees of freedom (2-DOF). Nissim and Gilyard [15], have developed an identification procedure as to identify the coefficients of the equations of motion of M-DOF dynamic systems. Such a procedure can be used in conjunction with the present one as to arrive to a better prediction of the flutter dynamic pressure. The procedure presented here provides the possibility to establish the stability boundary in a direct way from both the flight tests and the theoretical prediction of the flutter speed, see Fig. 2. In addition, having in view the fact that the proposed procedure is based on data that are elaborated in real-time, an expanded flutter flight envelope, is likely to be obtained. The prediction of the flutter boundary is accomplished via its identification from the subcritical aeroelastic response of the wing structure to pulse loads applied in a fixed position, such as the wing tip, and for selected flight speeds. On the other hand, transfer functions between accelerometers and the excitation can be determined and theoretical models predicting the flutter speed can be improved based on the deviations between the aircraft vibration test and model transfer function results [16]. Flutter predictions via subcritical responses and eigenvalues analyses can be performed with the updated and adjusted aeroelastic model parameters. Wind tunnel and flight tests are efficient ways enabling one to quantify in a rather exact way the aeroelastic instability parameters of lifting surfaces [17,18]. In the general case, using the wind tunnel tests it is possible to arrive close to the critical value of the flutter speed without the loss of the model [19]. The advantage of this new proposed method and its potential application carried out in flight or in a wind tunnel toward exactly determining the flutter boundary is presented next.

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1.3. COMBINED ANALYTICAL –EXPERIMENTAL PROCEDURE 1.3.1. PRELIMINARIES

In this section, the problem of the determination and study of the aeroelastic response behavior and the use of these predictions towards determination in flight of the flutter critical speed is addressed. The procedure presented in this section is applicable to the full aircraft, in general, and to advanced aircraft lifting surfaces and empennage incorporating composite materials [20] toward determination of the flutter dynamic pressure, in particular. In the most general form, the aeroelasticity of dynamic systems can be written symbolically as [21]:

Herein S , A and I denote the structural, aerodynamical and inertial operator, respectively, whereas Q denotes the external disturbance forces, assumed to be known explicitly, while t is the time variable. In our procedure it is represented by a pulse load. Its expression can be found in [10]. Equation (1) represent the functional relation between the generalized displacement u, at a specified point and in a specified direction, and the external disturbance generalized forces.


1.3.2.

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SUGGESTED PROCEDURE FOR THE DETERMINATION OF THE FLUTTER SPEED

The proposed testing procedure avoids the risk of flying in the vicinity of the flutter speed and assists the tests toward determination of the flutter speed. A chart depicting the procedure enabling one to predict the flutter speed from this combined analytical – experimental procedure is included (Fig. 3). The theoretical analysis of the aeroelastic response to a delta pulse acting on the wing will provide the possibility to determine the critical flutter speed. At the same time, from the analysis it is possible, for selected values of the flight speed, to obtain information on the response behavior of the aeroelastic system. In order to supply the necessary elements enabling one to substantiate the method that will be described in the sequel, a simple application of the flutter instability and aeroelastic response of a swept wing operating in an incompressible flow and exposed to blast pulses is presented in Fig. 4. To do this, in the linear formulation, the aeroelastic governing equation (1) specialized for a swept aircraft wing [11,12], can be converted into the Laplace transformed space and solved for the unknown generalized displacements in bending and twist. Taking the inverse Laplace transform of these quantities, one obtain the bending and twist time-histories due to the external pressure pulse applied at the wing tip. Notice that, well established procedures for the identification of aerodynamic indicial Functions using flight test data [13] and of aerodynamic impulse response using digital filter technique [22], can be adopted for the evaluation of the aerodynamic load acting on the aircraft or on the lifting surface. This procedure can provide a model superior to the aerodynamic derivative model [11,12]. An accurate approach enabling one to evaluate the flutter speed is based on the recursive model parameter evaluation that can be extracted directly from system identification or modal test data [16,17,23], in conjunction with the analytical model. All the information extracted from the subcritical response can be used in the process depicted in Fig. 4 that highlights the aeroelastic response of the swept wing (the sweep angle being ), when the flight speed increases, in the subcritical speed range. The phase portraits, at the top of the graph and in the chart of Fig. 4, show how the plunging response time-history to a unitary pulse (at the zero time and located at the typical section of the wing) evolves with the increase of the flight speed V . Corresponding to the flutter speed, that coincides with that obtained from the eigenvalue analysis, the trajectory of motion describes an orbit with constant amplitude, the so called center. For as time unfolds, a decay of the amplitude is experienced, which reflects the fact that in this case a subcritical response is involved (stable focal point), while for the response becomes unbounded, implying that the occurrence of the flutter instability is impending (unstable focal point).

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Figures 5a,b and c illustrate the effect that the airflow might have on the vibrational behavior of the wing structure. Figure 5a shows how, when the aircraft is flying at a subcritical speed, the vibrations usually damp out. But, when the flow speed is increasing, the vibration amplitudes can remain constant, Fig. 5b, or increase without limit, Fig. 5c. This phenomenon is extremely dangerous and can result in the catastrophic failure of the structure.

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The critical value of the flutter speed in Fig. 4 is obtained here from both the aeroelastic response and the eigenvalue analysis of the homogeneous system of equations. As a preliminary requirement, it is necessary to choose Mach number that is kept constant throughout the procedure. During the flight tests, exciting the wing by a unitary pulse at selected values of the flight altitude [24], keeping the Mach number constant, is possible to measure in each specific condition of flight, the maximum acceleration or deflection at the wing tip. This procedure can be implemented via the use of the structural acceleration measures that can easily be evaluated with real time measurements via accelerometers placed on the aircraft wing. The data, corresponding to the acceleration/deflection at the wing tip, obtained from the test can be included in the analytical loop, Fig. 4, where the theoretical maximum acceleration ratio (normalized by the acceleration of gravity g) or deflection at the wing tip for selected sweep angle are indicated. When the values of the deflection provided via flight tests coincide with those in the plot, Fig. 4, it becomes clear that the real flutter speed is that obtained from the analytical prediction. Supposing that the predictions via experimental procedures provide different values of the maximum acceleration/amplitude for selected values of the flight speed, the input data have to be updated as to yield, for two flight speeds and the same pressure signature, correct values of the aeroelastic response. In this way, we can avoid the problem of the uncertainties that are involved in any aeroelastic analysis, theoretical or experimental. In such conditions, using the methodology presented in this section, a new line connecting the maximum amplitudes of the aeroelastic response, that is determined uniquely by two values of the flight speed can be traced. This line is labeled as TL (Theoretical Line). It is clearly seen that corresponding to EL1 (Experimental Line) and EL2, the wing will feature lower and larger stiffness characteristics, respectively. At the same time, the change in the slope of EL1 or EL2 as compared to that of TL, is due to a different damping or mass parameter as compared to that of the theoretical model. We assume that the coefficient of equations of motion can be reasonably identified, using for example the Nissim’s method [15], so that these parameters should be updated as to arrive to the convergence of the theoretical prediction with the experimental ones. The slope of the experimental line, the amplitude envelope of the response that bounds the oscillation in time, and a parametric study of the response in terms of the structural parameter will provide the possibility to update these coefficients. With an automatic iterative procedure, updating the parameter at each loop, the new theoretical line of the maximum amplitudes that coincide with the experimental ones can be determined. Consequently, the exact value of the flutter speed of the aeroelastic structure will be supplied from extrapolation and intersection with the updated maximum acceleration or deflection line. As clearly appears from this paper, the issue related with the test technique and the excitation system as to obtain a pulse was not addressed. This excitation can be produced via a control surface pulse, that can approximate a delta function at high frequency content and due to its short test duration can be repeated many times at each test point, or by thrusters, ballistic exciters or impulse generator, simple devices that can produce short transient response [24].


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2. Concluding Remarks

This paper suggests an alternative approach for the prediction of the flutter speed via a combined experimental-theoretical analysis. The approach presumes that modal parameters, such as frequencies, damping ratios, and aeroelastic responses in terms of deflections or accelerations (at sensor locations), can be identified during the subcritical flight speed range. The proposed approach is based on the possibility of updating a theoretical model in terms of its mass, damping, and stiffness, that enables an extrapolation of the flutter speed. Among the advantages of the proposed procedure there are: i) possibility of a systematic and accurate flutter prediction on the basis of the on-line real modal parameters; ii) can contribute to the expansion of the flight envelope without weight penalties; and, iii) possibility of reducing significantly the amount of flights required in the flight test program. In addition, this methodology may significantly improve the procedures for the flight flutter testing and reduce its cost. 3. Acknowledgment

The partial support of this research by the NASA Langley Research Center through Grant NAG-1-01007 is acknowledged. 4. 1. 2.

References

Houbolt, J.C. (1974) Subcritical flutter testing and system identification, NASA-CR-132480. Rosenbaum, R. (1974) Survey of aircraft subcritical flight testing methods, NASA-CR-132479, ARAP-218. 3. Zimmerman, N.H. and Weissenburger, J.T. (1964) Prediction of flutter onset speed based on flight testing at subcritical speeds, J. Aircraft 1 (1), 190-202. 4. Lind, R. and Brenner, M. (1999) Robust aeroservoelastic stability analysis, Springer-Verlag, London. Lind, R. (2001) Flight testing with the flutterometer, presented as AIAA Paper 2001-1654 at the 5. 42nd AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, Seattle, WA, 16-19 April. 6. Richardson, J.R. (1965) A more realistic method for routine flutter calculations, AIAA Symposium on Structural Dynamics and Aeroelasticity, Boston, Massachusetts, August 30 - September 1, 1017. 7. Houbolt, J.C. (1975) On identification frequencies and damping in subcritical flutter testing, NASA SP-415, Flutter testing techniques, A conference held at Dryden Flight Research Center, Edwards, CA, 9-10 October. 8. Bucharles, A., Cassan, H., and Roubertier, J. (1990) Advanced parameter identification techniques for near real time flight flutter test analysis, AIAA/SFTE/DGLR/SETP 5th Biannual Flight Test Conference, Ontario, CA, May 22-24. 9. Roy, R., and Walker, R. (1985) Real-time flutter identification, NASA-CR-3933. 10. Marzocca, P., Librescu, L., and Silva, W.A. (2000) Aerodynamic indicial functions and their use in aeroelastic formulation of lifting surfaces, presented as AIAA paper 2000-WIP at the 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Atlanta, GA, 3-6 April.

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11. Marzocca, P., Librescu, L., and Silva, W.A. (2001) Aeroelastic response of swept aircraft wings in a compressible flow field, presented as AIAA Paper 2001-0714 at the 39th AIA A Aerospace Sciences Meeting, Reno, NV, 8-11 January. 12. Marzocca, P., Librescu, L., and Silva, W.A., (2002) Unified approach of aeroelastic response and flutter of swept aircraft wings in an incompressible flow, AIAA Journal, in press. 13. Gupta, N.K. and Iliff, K.W. (1982) Identification of aerodynamic indicial function using flight data, presented as AIAA-82-1375 at the 9th AIAA Atmospheric Flight Mechanics Conference, San Diego, California, 9-11 August. 14. Russo, M.L., Richardson, P.T., and Perangelo, H.J. (1983) Identification of linear flutter model, presented as AIAA-83-2696 at the AIAA/AHS/IES/SETP/SFTE/DGLR 2nd Flight Testing Conference, Las Vegas, Nevada, 16-18 November. 15. Nissim, E. and Gilyard, G.B. (1989) Method for experimental determination of flutter speed by parameter identification, NASA-TP-2923. 16. Lacabanne, M. and Esquerre, J.P. (1995) Correlation between theoretical flutter models and tests for civil aircraft, Aerospatiale – Aircraft Division TR. 17. Dat, R., Dunoyer P. (1981) Identification des modes propres d’une structure a partir des reponses a une excitation non appropriee, ONERA TP-1981-25, 52ème réunion de la Comission Structures et Matériaux de l'AGARD, Cesme-Izmir, 6-11 April, in French. 18. Smith, M.S. (1999) Analysis of wind tunnel oscillatory data of the X-31A aircraft, NASA-CR1999-208725. 19. Klein, V. and Noderer, K.D. (1994) Modeling of aircraft unsteady aerodynamic characteristics, Part 1 - Postulated models, NASA-TM-109120. 20. Karpouzian, G. and Librescu, L. (1994) Comprehensive model of anisotropic composite aircraft wings suitable for aeroelastic analyses,” J. of Aircraft, 31 (3), 703–712. 21. Bisplinghoff, R.L. and Ashley, H. (1996) Principles of Aeroelasticity, Dover Publications, Inc. 22. Silva, W.A. (1997) Identification of linear and nonlinear aerodynamic impulse response using digital filter techniques, NASA-TM-112872, presented as AIAA Paper 97-3712 at the AIAA Atmospheric Flight Mechanics Conference, New Orleans, LA, 11-13 August. 23. Gaukroger, D.R., Skingle, C.W., and Heron, K.H. (1980) An application of system identification to flutter testing, J. Sound and Vibration 72 (2), 141-150. 24. Kehoe, M.W. (1995) A historical overview of flight flutter testing, NASA-TM-4720.

DISPLACEMENT-BASED SMOOTHING HYBRID FINITE-ELEMENT REPRESENTATION FOR STRESS ANALYZING PERFORATED COMPOSITES

K. Y. HE and R. E. ROWLANDS Department of Mechanical Engineering University of Wisconsin Madison, WI 53706

Abstract A reliable and effective displacement-based smoothing finite-element representation is demonstrated for determining stresses on, and near, the edge of a geometric discontinuity in finite, orthotropic composites. Relatively little measured displacement data are necessary and they originate away from the cutout. Data acquisition positions can be selected at the user’s discretion and scatter in the measured input information is automatically filtered.

1. Introduction Obtaining accurate stresses at holes or notches in finite orthotropic materials by purely theoretical or numerical techniques can be difficult and/or undependable, particularly when far-field loading and geometry are complicated or unknown. Although numerous experimental techniques are available for obtaining stress (strain, displacement) information, such measurements are often unreliable on the edge of a discontinuity. One superior approach is to evaluate stresses on the edge of a hole or notch from measured information away from the cutout. Rhee [1, 2] determined the stresses associated with cutouts in composites utilizing concepts from Gerhardt’s hybrid element [3]. However, Rhee’s method necessitates interpolating displacements at the external nodes of the hybrid element from adjacent moiré fringes. One has only finite flexibility as to where to locate these nodes and reliability of that approach is further eroded if fringe quality or density near a node is inferior. Such shortcomings are overcome here by actually embedding, numerically and experimentally, the external portion of the hybrid element amongst the measured displacements. Data acquisition positions now become quite arbitrary and can be selected at the user’s discretion (e.g., use input data having high resolution and reliability). The present technique enjoys the additional advantage of automatically smoothing experimental scatter from the measured input data. 619 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 619–628. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Analytical Background

2.1 GENERAL COMMENTS

Consider a loaded finite composite component containing an arbitrarily-shaped hole, Fig. 1. The edge of the hole is denoted by Moreover, (line DEF) is the external boundary of a hybrid element (region S) which surrounds the cutout, and the area between contours ABC and GHI forms a data acquisition region, R, throughout which one collects measured displacements. Stresses are be evaluated here throughout S, including along the edge of the hole, from the measured displacements in region R.

2.2 HYBRID ELEMENT

The stresses and displacements in a loaded homogeneous orthotropic material under plane stress are given by [4]

STRESSES AROUND CUTOUTS IN COMPOSITES

621

Re denotes the real part of a complex number, complex material properties are the roots of the characteristic equation associated with the compatibility relationship, and and are

Satisfying the boundary conditions on the edge of the cutout of Fig. 1 is aided by mapping its edge, in the physical plane into the unit circle, in the where (j = 1 and 2) maps the unit circle and its exterior in the into the edge of geometric discontinuity and its exterior in the physical plane, and and are related to material properties and location within S [1-3]. Traction-free conditions on the edge, of the cutout can be implemented using the principle of analytic continuation, such that [3]

where and are real numbers, n and m are positive integers, and B and C are material properties [3]. Combining Eqs. 1 through 5 gives the stresses and displacements within the hybrid element, i. e.,

where is the coefficient vector of Eqs. 4 and 5, and [U] and [S] depend on the cutout shape, material properties and k [5]. Terms with k = 0 contribute to rigid body motion and can consequently be omitted. As occurs here, for geometrical, mechanical and material symmetry about the x- and y-axes, Eqs. 4 and 5 involve only real and odd coefficients Taking m = n such that results in a total of (m+1) real coefficients. The elastic strain energy of a region, S, (hybrid element) surrounding a traction-free boundary, is related to {c} and the nodal displacement vector, on the external boundary, of the element [3,5]:

with

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and ds is an arc length of integral path along (DEF), Fig. 1. Matrix [L] of Eqs. 8 and 9 relates displacements, on the external boundary of the hybrid element to the nodal displacements, on and and are the direction cosines of the normal to A quadratic interpolation is utilized here for [L] [5]. Minimizing the potential energy in the hybrid element of Eq. 11 gives the following relationship between and

Matrices [H] and [G] depend on material and geometry of a given hybrid element. Combining Eqs. 6 through 10 gives

Quantity of Eq. 11 can be treated as a shape function similar to those in structural FEA. Once is known, Eq. 10 provides and the stress functions, stresses and displacements (hence strains) are available throughout S by Eqs. 1 through 11, including on Determining smoothed experimentally-based data, will now be discussed. 2.3 SMOOTHING HYBRID FINITE-ELEMENT REPRESENTATION

References 1 and 2 used the foregoing concept to determine stresses associated with cutouts from nodal values of on by interpolating from adjacent moiré fringes. That concept is enhanced here by embedding, numerically and physically, the outer boundary of S, within the data-acquisition region, R, of Fig. 1. The smoothing hybrid finite-element representation (SHFER) is well suited for smoothing measured displacements and providing accurate at the nodes along [5]. In addition to filtering noisy experimental input, SHFER enables the user to employ measured data from arbitrary locations throughout R and to accommodate a region S of Fig. 1 of widely-varying shape and/or size. The relative impact of individual measured input values can also be controlled numerically through a weighting factor. Denote the in-plane displacement vector of the two components u(x, y) and v(x, y) in region R of Fig. 1 as Measured displacements at locations in R are used to quantify the displacement representations throughout R and particularly to evaluate reliable on of region S of Fig. 1. It is advantageous to employ a predicting representation in R that does not necessarily agree exactly with the measured displacements at each input location but smoothes the measured information. Such a smoothing function can be obtained by minimizing a functional. Using concepts from Refs. 6 and 7, and discretizing region R, one obtains [5]


623

where are the local coordinates of the locations of measured input displacements in the surrounding isoparametric elements (between and ABC of Fig. 1), are the global coordinates of the input locations in that portion, of region S between curves GHI and of Fig. 1, J is the number of input locations in a surrounding element and H is the number of input locations in Vectors and represent measured input displacements in the surrounding elements and in respectively. Integer M is the total number of surrounding elements, and and are the nodal displacements of a surrounding element and on the external boundary of the hybrid element, respectively. Shape functions and (the latter given by Eq. 11) are evaluated at input locations in the surrounding elements and in respectively. Equation 12 can be rewritten as

or

where is the assembly of and and is the assembly of and in region R. Equation 14 resembles that for structural FEA, but the individual terms are physically different. Pseudo stiffness of Eq. 14 involves geometry, material properties, number of terms and coordinates of measured input values in region R. In addition to material properties, pseudo load vector, involves the locations and magnitudes of the input displacements in R. Solving Eq. 14 for gives the nodal displacements throughout R, including those on the external boundary of S, The total amount of measured input data utilized throughout region R should be no less than the total d.o.f. of the elements which make up region R.

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The method was used to determine the tangential stress on the edge of the central circular hole in the graphite/epoxy composite tensile member of Fig. 2. Elastic properties are and The strong, stiff orientation of the orthotropic composite of Fig. 2 is parallel to the applied load, P. The hole has diameter plate width W = 38.1 mm (1.5 in.), length of the plate between loading grips is 38.1 cm (15 in.) and composite thickness t = 2.2 mm (0.085 in.).

Four-beam interferometric moiré was used to record the u- and v-displacements. The specimen contained a 1200 lines/mm (30,480 lines/inch) crossed diffraction grating and the virtual analyzer had a frequency of 2400 lines/mm (60,960 lines/inch), for a displacement resolution of (16.4 per fringe. The moiré system, composite plate and loading frame were all supported on an isolated table. Moiré fringes associated with each of the x- and y-displacements were recorded throughout the shaded area of Fig. 2, and those for the v-field are shown in Fig. 3 at P = 1334.4 N (300 Ib) Figure 4 shows the configuration of the hybrid element (region S) and the dataacquisition region R utilized, with c = 2r = 11.25 mm. (0.443 in). Geometry and material symmetry enable one to work with only one quarter of the composite member. Both u- and v-displacements were employed at 69 locations in region R (of Fig. 4) of a quarter-member in Fig. 2 to provide 138 input values with which to evaluate the 76 unknown nodal displacements at the relevant 38 nodes of region R. However, measured displacements u and v were recorded at all 276 associated locations within the four quadrants around the hole and their values at corresponding locations were averaged to eliminate any non-symmetry. Fringe orders were obtained at the respective positions


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and at several load levels. Symmetry was imposed and 10 real and odd coefficients were used, i.e., m = n = 9. The 69 input locations were somewhat uniformly distributed throughout region R associated with Figs. 1 through 4. If uniformly distributed, and based on a total area of region R of Fig. 4 of this would represent an average input spacing of 1.4 mm (0.05 in.), corresponding to ~7 input readings per cm on the actual composite.

The 38 nodes of region R consist of 13 nodes (N1 through N13) on the outer boundary, of the hybrid element, S, (region denoted by DEFIQPGD of Fig 4) plus 25 nodes associated with the seven surrounding isoparametric elements contained within lines DEF and ABC of Fig. 4.

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Table 1 contains the stress concentration factors in the graphite/epoxy as obtained from the measured displacements and the smoothing hybrid finite-element representation, SHFER. These results are based on and various values of The locations of measured input displacements used closest to the hole are approximately 0.5r = 2.8 mm (0.11 in.) from its boundary, whereas the most distant measured input displacements originate at approximately 2.5r = 1.4 cm (0.55 in.) from the edge of the hole. Based on the geometry of Fig. 2 (r = 0.563cm), the external horizontal boundary, FE, of region S (the hybrid element) of Fig. 4 coincides approximately with the top edge of the fringe pattern of Fig. 3, whereas its vertical boundary, DE, is somewhat inside the right edge of this fringe photograph.

Current results are compared in Table 1 with those by Tan [8] and ANSYS. One quarter of the composite plate of Fig. 2 was modeled for the latter. The normalized tangential stresses, are plotted around the boundary of the hole in Fig. 5.


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Complete details, including those for the area integrals of the pseudo stiffness of that portion of S which helps form region R, are contained in Reference 5. One quarter of the composite plate of Fig. 2 was modeled (ANSYS) using from 3766 eight-node isoparametric elements (90 evenly spaced elements around the quarter-hole boundary and a total of 11,583 nodes) to 7374 elements (180 evenly spaced elements around the quarter-hole and a total of 22,567 nodes). These FEA models give a tensile stress concentration factor of to 7.741, respectively. 5.

Summary, Discussion and Conclusions

A simple, reliable and effective displacement-based technique is demonstrated for determining stresses on the boundary, and in the neighborhood, of geometric discontinuities in orthotropic composites from distant measured displacements. The method synergizes analytical, numerical and experimental techniques, and incorporates some previous concepts by Rhee et al. [1,2] and Feng [7]. Relatively few measured input data are needed and they originate away from the cutout. Whereas Rhee’s technique involved interpolating displacements at the nodes on the external boundary of a hybrid element from neighboring fringes, this shortcoming is overcome here by embedding the external boundary of hybrid element amongst the measured displacements. Moreover, the present technique filters experimental scatter and data acquisition positions can be selected largely at the user’s discretion. The latter permits one to omit questionable or unreliable data. Although Baek and Rowlands [9] used technology associated with Eqs. 1 through 5 to evaluate edge stresses at a hole from distance strain-gage readings, the present approach does a better job of filtering the input scatter. Unlike Refs 1, 2 and 9, the relative contribution of individual input values can also be controlled here through a weighting factor. These features enhance reliability. Only the tangential stress on the hole boundary is shown here. Once one knows the displacements of the external nodes of the hybrid element, individual components of stress, strain and displacement are available throughout the hybrid element. Input displacements range here between 3mm to 14 mm from the hole boundary. Although not assessed quantitatively, one could probably omit some of the current input values, particularly those closest to the hole, without degrading the results. In addition to employing fewer input values (locations) per node, one might consider reducing the number (increasing individual size) of the surrounding elements, thereby decreasing the number of nodes on of Fig. 1 and 4. Present analyses were conducted for various values of smoothing parameter, but no attempt was made here to evaluate an optimum value of As with most experimental methods, one neither needs to know the far-field physical boundary conditions nor analyze the entire structure. Moiré is used here, but displacements could also be recorded using methods such as speckle, holography or image correlation. The approach is applicable to multiple cutouts and/or other geometries.

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Acknowledgement

Ms. Lynda Litzkow proficiently prepared this manuscript. 7. References 1. 2.

3. 4. 5. 6. 7. 8. 9.

Rhee, J., (1995), Geometric Discontinuities in Orthotropic Composites, Ph.D. Thesis, University of Wisconsin, Madison. Rhee, J., He, S. and Rowlands, R. E.: Hybrid moiré-numerical stress analysis around cutouts in loaded composites, Experimental Mechanics 36(4) (1996), 379-387. Gerhardt, T. D.: A hybrid finite element approach for stress analysis of notched anisotropic materials, Jour, of Applied Mechanics 51 (1984), 804-810. Lekhnitskii, S. G., (1968), Anisotropic Plates, Gordon and Breach New York. He, K. Y., (2000), Hybrid Stress and Fracture Analysis of Orthotropic Media, Ph.D. Thesis, University of Wisconsin, Madison. Segalman, D. J., Woyak, D. B. and Rowlands, R .E.: Smooth spline-like finite element differentiation of experimental vector data over arbitrary geometry, Experimental Mechanics 19 (1979), 429-439. Feng, Z. and Rowlands, R. E.: Continuous full-field representation and differentiation of threedimensional vector data, Computers and Structures 26 (1987), 979-990. Tan, S. C., (1994), Stress Concentrations in Laminated Composites, Technomic Publishing, Co, Inc., Lancaster, PA. Baek, T. H. and Rowlands, R. E.: Hybrid stress analysis of perforated composites using strain gages, Experimental Mechanics 41(2) (2001), 195-203.

8. Composite Structures


FUTURE EXPERIMENTAL METHODS NEEDED TO VERIFY COMPOSITE LIFE-CYCLE SIMULATIONS

CHRISTOS C. CHAMIS NASA Glenn Research Center Cleveland, OH 44135, USA LEVON MINNETYAN Clarkson University Potsdam, NY 13699, USA

Abstract

The future experimental methods needed for composite life-cycle are identified by computationally simulating the fracture of an integrally stiffened composite structure. The simulation describes events occurring during the fracture progression at all composite structure scales, the fracture modes that contribute to those events and the respective local failure mechanisms. The fracture modes in their respective scales provide opportunities to suggest future testing techniques to measure them. For example, energies emitted can be calibrated to identify non-destructive techniques to measure corresponding energies such as acoustic, thermal and even optical. Successful testing methods can then be used to implement monitoring systems for in-service structural life-cycles. 1.

Introduction

The life-cycle of composite structures can be determined by evaluating its damage tolerance. One way to a-priori computationally simulate damage tolerance is by progressive structural fracture, which includes all failure mechanisms that contribute to progressive structural fracture. Since composite structures have a multitude of failure mechanisms, the simulation must combine composite structural analysis, composite laminate theory, composite macromechanics and composite micromechanics. This combination will allow the synthesis of local behavior and respective fabrication process through the various scales up to structural behavior (scale telescoping) and the decomposition of structural response through the various scales down to individual constituents (scale tunneling through point and deplying through layers). The quantification of composite scale telescoping and composite scale tunneling provides description of the various events that occur at each scale during the progressive fracture of the composite structure. Since each scale represents a respective physical entity, it is rather reasonable to design experiments to measure the events that occur at that scale 631 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 631–644. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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during progressive structural fracture. The objective of this paper is to computationally simulate the events that occur in each scale during the evaluation of the life-cycle of a select composite structural component. Other objectives are to identify the fracture modes that contribute to those events and to suggest possible non-destructive experiments to measure them. 2.

Select Composite Structural Component

Laminated composite structures are used in many aerospace applications such as rotorcraft components, advanced aircraft fuselage, rocket motor cases, pressure vessels, containment structures, and other components with various shapes and sizes. In these applications composite structures are required to withstand significant in-plane loads. Additionally, composite structures are required to possess sufficient bending stiffness to resist buckling. The component selected for this evaluation is an integrally stiffened composite structures subjected to in-plane loads and internal/external pressures. Damage initiation, growth, accumulation, and propagation to fracture are simulated for composite panels and cylindrical shells with and without integrated stiffening layups. The influence of integrated stiffeners is examined with regard to out of plane stiffness contribution as well as damage progression and structural durability assessment under applied loading. Changes in the damage initiation load and the structural fracture load are quantified due to the presence of integrated stiffeners in order to identify viable nondestructive test methods. Recent developments in computational simulation technology have made it possible to evaluate the details of progressive damage and fracture in composite structures. Computational simulation enables assessment of the damage initiation and propagation loads. A damage energy release rate is evaluated globally during simulation by computing the work done per unit damage created. The damage energy release rate is used to quantify the structural damage tolerance at different stages of degradation. The influence of local defects or flaws and effects of the fabrication process in terms of residual stresses are taken into account. An important feature of computational simulation is the assessment of damage stability or damage tolerance of a structure under loading. At any stage of damage progression, if there is a high level of structural resistance to damage progression under the service loading, the structure is stable with regard to fracture. The corresponding state of structural damage is referred to as stable damage. On the other hand, if damage progression does not encounter significant structural resistance, it corresponds to an unstable damage state. Unstable damage progression is characterized by very large increases in the amount of damage due to small increases in loading, enabling measuring of the events that cause those large damage volumes. Whereas, during stable damage progression, the amount of increase in damage is consistent with the increase in loading, providing challenging opportunities for measurement. 3.

Methodology

Computational simulation is implemented via the integration of three modules: (1) composite mechanics, (2) finite element analysis, and (3) damage progression tracking.

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The overall evaluation of composite structural durability is carried out in the damage progression module (Refs. 1 and 2) that keeps track of composite degradation for the entire structure. The damage progression module relies on the composite mechanics code (Ref. 3) for composite micromechanics, macromechanics and laminate analysis, and calls a finite element analysis module that uses anisotropic thick shell elements to model laminated composites (Ref. 4). CODSTRAN Computational Cycle is shown in Figure 1.

Details of the methodology are described in the references cited. Herein its application to special types of composite structures and relevance to nature experimental methods needed are described. 4.

Integrally Stiffened Panel

A graphite/epoxy laminated composite plate with integrated ±45° intermittent lattice stiffeners was investigated with damage and fracture propagation due to tension, compression, and in-place shear loads. The response of the integrally stiffened composite panel was compared with that of an unstiffened skin plate. The unstiffened plate was given additional skin thickness such that the material volume was the same as the material volume of the integrally stiffened plate. The additional plies of the unstiffened plate were given fiber orientations of ±45°, same as the fiber orientations of the integrated stiffeners. Both the unstiffened and integrally stiffened panels were made of AS-4 graphite fibers in a high modulus high strength epoxy matrix (AS-4HMHS). Ply layup of the unstiffened panel was The stiffened panel had a skin layup of The stiffener plies were added to the top of the skin as or as Ply thickness was 0.127 mm (0.005 in.). The fiber volume ratio was and the void volume ratio was The cure temperature was (350° F).

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The fiber and matrix properties were obtained from a databank of composite constituent material properties resident in the composite mechanics module (Ref. 3). The fiber and matrix properties corresponding to this case are given in Tables I and II, respectively.

The HMHS matrix properties are representative of the 3501-6 resin. These in-situ properties are similar to those identified by (Ref. 5) with experimental correlation. The panels had identical planar geometry with a width of 305 mm (12.0 in.) and length of 457 mm (18 in.). Each finite element model contained 117 nodes and 96 uniformly sized square elements. Figure 2 shows the finite element model with diagonal lines along the ±45° stiffener bands.


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Numbers at nodes indicate the laminate type number at that node for the integrally stiffened panel. Laminate type 1 had a layup of representing the skin. Laminate type 2 had the skin layup plus representing the +45 stiffened nodes. Laminate type 3 had the skin layup plus representing the –45 stiffened nodes. Laminate type 4 had the skin layup plus representing intersection nodes for +45 and –45 stiffeners. Panels were assumed to be simply supported for structural response; i.e. the left end nodes of the panel were restrained against all displacement components and the right end nodes were restrained against displacement normal to the plane of the panel via duplicate node specifications.

Structural response characteristics of the integrally stiffened panel were evaluated in terms of the buckling load and the bending stiffness. Analysis of the integrally stiffened panel under uniaxial compression indicated a buckling load of 1,114 N (250 lbs). The buckling load of the unstiffened panel was only 213 N (48 lbs). Therefore the buckling resistance of the integrally stiffened panel was 5.2 times that of the unstiffened panel with the same amount of composite material. Similarly, the bending rigidity was significantly improved due to integrated stiffeners. Figure 3 shows a comparison of the midspan deflections of unstiffened and integrally stiffened panels due to uniform bending moment applied to the simply supported right end of the panel. Figure 3 shows that the integrally stiffened panel was 7.2 times stiffer than the unstiffened panel in

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bending. The future experiments needed are those that can measure incipient buckling and the respective resistance of stiffeners and skin at that instant.

Next, the in-plane progressive damage and fracture responses were compared for the unstiffened and stiffened panels. Figure 4 shows damage progressions of unstiffened and integrally stiffened panels subjected to tension.


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Damage initiation for the integrally stiffened panel occurred in the 90° skin plies due to transverse tensile fractures. Damage initiation for the unstiffened panels was also in the 90° plies due to transverse tensile fractures. In both cases the combined stress failure criterion was activated. However, the damage initiation load for the integrally stiffened panel was approximately one third of the damage initiation load for the unstiffened panel with the same material volume. The transverse tensile fracture in the 90° plies is mainly controlled by the matrix and the interface. Microstresses are all predicted from ply stresses as well as energies emitted during fracture. Local experimental methods are needed to identify which failure mechanism was activated first. Figure 5 shows the displacements in tension.

The uniaxial stiffnesses of the unstiffened and integrally stiffened panels are approximately the same prior to damage initiation. However, stiffness of the integrally stiffened panel degrades quickly due to damage initiation and progression. The respective stresses at damage initiation are predicted and the corresponding fracture modes as well as energies emitted and temperature rises. Unique experiments are needed to measure them directly. Figure 6 shows damage progressions of the unstiffened and integrally stiffened panels under uniaxial compression. Damage initiation for the integrally stiffened panel occurred at the edge of the panel at midspan where +45 and -45 stiffeners converged (laminate type 4). The damage initiation modes included the transverse tensile and longitudinal compressive failures of the 90° skin plies, in-plane shear failures of the ±45° skin plies, as well as the in-plane shear failures of the +45 stiffener plies near the skin. Damage initiation for the unstiffened panels under compression was in the 0° plies due to longitudinal compressive fractures. The damage propagation load of the unstiffened panel was more than five times that of the integrally stiffened panel. The compression fracture modes predicted provide ample opportunities for innovative non-destructive experimental

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techniques to measure their magnitudes in terms of energies emitted in the form of temperature changes, or acoustic signatures or even optical changes.

The initial stiffnesses of the unstiffened and integrally stiffened panels in compression were the same as observable from Figure 7. However, the stiffness of the integrally stiffened panel degraded at a much lower loading compared to degradation load of the unstiffened panel. Figure 8 shows the damage progressions of unstiffened and integrally stiffened panels under in-plane simple shear loading.


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Damage initiation for the integrally stiffened panel was due to longitudinal compressive failure of a +45 skin ply at a type 3 stiffened node. On the other hand, damage initiation for the unstiffened panel under shear was due to transverse tensile failures of the 90° plies. As it was in tension and compression, also in shear the integrally stiffened panel degraded at a lower load compared to the unstiffened panel. Figure 9 shows the load-displacement relationships in shear. Computational simulation results depicted in Figure 9 indicate that in shear, the initial stiffness of the unstiffened panel was slightly higher than that of the integrally stiffened panel.

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Shear loadings offer challenging opportunities to measure fracture modes and respective failure mechanisms non-destructively. The predicted magnitudes and locations definitely provide guidance. 6. Integrally Stiffened Composite Shell A composite cylindrical shell with and without integrated ±45° intermittent lattice stiffeners was investigated with damage and fracture propagation due to axial tension, as well as internal and external pressure loads. The response of the integrally stiffened shell was compared with that of an unstiffened shell. The unstiffened shell was given additional skin thickness such that the material volume was the same as the material volume of the integrally stiffened shell. The additional plies of the unstiffened shell were given fiber orientations of ±45°, same as the fiber orientations of the integrated stiffeners. Both the unstiffened and integrally stiffened shells were made of AS4/HMHS composite. Ply layup of the unstiffened shell was The stiffened shell had a skin layup of The stiffener plies were added to the top of the skin as or as Ply thickness was 0.127 mm (0.005 in). The fiber volume ratio was and the void volume ratio was The cure temperature was (350° F). Their respective properties are listed in Tables I and II, as mentioned previously. Each cylindrical shell finite element model contained 400 nodes and 384 uniformly sized square elements. Figure 10 shows the finite element model for a cylindrical shell.

The shell was simulated subject to increasing levels of internal and external pressurizations as well as axial loading. To represent the axial stresses produced in the closed end pressure vessel, boundary nodes at one end of the cylinder were subjected to


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force resultants in the axial direction, whereas the boundary FEM nodes were restrained in the axial direction at the opposite end. The uniformly distributed axial tension was such that the generalized axial stresses in the shell wall were half those developed in the hoop direction. Figure 11 shows a comparison of damage progressions for the integrally stiffened and unstiffened composite shells under axial tension. Figure 11 indicates that ultimate strength of the integrally stiffened cylindrical shell is approximately 60 percent of the ultimate strength of shell without integral stiffeners.

Additionally, damage initiation for the integrally stiffened shell begins at only 10 percent of the ultimate load. On the other hand, damage initiation for the unstiffened shell corresponds to a loading level that is approximately 97 percent of its ultimate load. The integrally stiffened shell provides ample opportunity for innovative non-destructive methods that are energy related. While the unstiffened panel provides a very challenging task to devise a test that will capture the insipient fracture which is that close to structural fracture. Figure 12 shows the exhausted cumulative damage energy based on depleted energies of the local failure mechanisms. The exhausted damage energy represents a similar pattern of damage progression as the percent damage volume depicted in Figure 12. Figure 13 shows the damage energy release rates (DERR) based on the incremental work done by applied loading during damage progression of the integrally stiffened cylindrical shell.

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Peak values in the DERR levels indicate significant damage events. The first peak in DERR corresponding to damage initiation for the integrally stiffened shell occurred due to transverse tensile fractures in the 90° skin plies. The steep increase in DERR at approximately 11 kN (2.5 k) axial tension corresponds to a partial separation between skin and stiffener plies. The results in Figures 12 and 13 show the amount of detailed information that is obtained from the computational simulation. This type of


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information is valuable to guide and even identify non-destructive test methods to measure it. Figure 14 shows comparison of damage progressions for integrally stiffened and unstiffened cylindrical shells subjected to external pressures.

The ultimate pressure for the integrally stiffened shell is approximately 75 percent of the ultimate pressure for the unstiffened shell. Damage initiation for the integrally stiffened shell occurs at 23 percent of its ultimate pressure. On the other hand damage initiation for the unstiffened shell occurs at approximately 94 percent of its ultimate pressure. Also, the volume of damage in the integrally stiffened shell at ultimate pressure is much greater than the volume of damage in the unstiffened shell at ultimate pressure. The damaged volume difference provides opportunities for non-destructive test methods to track damage tolerance in the presence of considerable damage and very little if at all.

6. Conclusions On the basis of the results obtained from the investigated flat composite plate, integrally stiffened panel, integrally stiffened and unstiffened composite cylindrical shell structure examples, experimental methods, and from the general perspective of the available computational simulation method and suggested future test method needs, the following conclusions are drawn:

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1.

Computational simulation can be used to track the details of damage initiation, growth, and subsequent propagation to fracture for unstiffened and integrally stiffened composite structures, as well as provide quantifiable information for implementing non-destructive test methods to measure it.

2.

For the considered integral stiffening structures, out-of-plane structural response characteristics such as the buckling load and the bending stiffness are significantly improved. Respective test methods needed to evaluate the load sharing between panel and stiffeners.

3.

In-plane load carrying capability of a composite panel is reduced due to damage initiation and progression processes caused by the presence of integrated stiffeners. Unique non-destructive test methods are needed to isolate the fracture modes that are active during the fracture progression.

4.

Computational simulation, with the use of established composite mechanics and finite element modules, can be used to predict the influence of composite geometry as well as loading and material properties on the durability of composite structures. This type of simulation also provides energy emitted during the fracture progression which can be converted to acoustic, thermal or optical in order to measure it by using non-destructive testing techniques.

5.

The demonstrated procedure is flexible and applicable to all types of constituent materials, structural geometry, and loading. Hybrid composites and homogeneous materials, as well as laminated, stitched, woven, and braided composites can be simulated. Appropriate innovative and inventive nondestructive test methods need to be developed to measure the damage accumulated for life-cycle verification of predictions.

6.

Computational simulation by CODSTRAN represents a new global approach to progressive damage and fracture assessment for any structure. Complementing this with suitable non-destructive testing will complete quantifiable in-service monitoring systems for structural life-cycles.

7. 1. 2. 3. 4. 5.

References L. Minnetyan, P. L. N. Murthy and C. C. Chamis, "Progression of Damage and Fracture in Composites under Dynamic Loading," NASA TM–103118, April 1990, 16pp. L. Minnetyan, P. L. N. Murthy and C. C. Chamis, "Composite Structure Global Fracture Toughness via Computational Simulation," Computers & Structures, Vol. 37, No. 2, pp. 175180, 1990. P. L. N. Murthy and C. C. Chamis, Integrated Composite Analyzer (ICAN): Users and Programmers Manual, NASA Technical Paper 2515, March 1986. S. Nakazawa, J. B. Dias, and M. S. Spiegel, MHOST Users' Manual, prepared for NASA Glenn Research Center by MARC Analysis Research Corp., April 1987. L. Sobel, C. Buttitta, and J. Suarez, "Probabilistic & Structural Reliability Analysis of Laminated Composite Structures Based on IPACS Code," Proceedings of the 34th SDM Conference, LaJolla, California, April 19-22, 1993, Vol. 2, pp. 1207-1217.

EXPERIMENTAL OBSERVATIONS ON THE DELAMINATION BEHAVIOR IN COMPOSITE STRUCTURES

G. A. KARDOMATEAS, V. LA SAPONARA* and G. J. SIMITSES School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA 30332-0150, USA *Currently at the School of Civil and Environmental Engineering Georgia Institute of Technology Atlanta GA 30332-0355

Abstract

The objective of the current paper is to present experimental data of interface cracks (delaminations) in layered Glass/Epoxy and Graphite/Epoxy composites under cyclic compressive (fatigue) and static (Mixed Mode Bending and Double Cantilever Beam) loads. The growth behavior ranges from self-similar growth along the interface in the unidirectional configuration to branching out of the interface in 90 deg. and cross-ply configurations. In the case of cyclic compressive loading, the specimens undergo repeated buckling/unloading of the delaminated layer with a resulting reduction of the interlayer resistance. Also, for this case, equations describing the growth of the delaminations under cyclic loads are obtained on the basis of a combined delamination buckling/post-buckling and fracture mechanics model. The latter is based on a modedependent critical fracture energy concept and is expressed in terms of the spread in the energy release rate in the pre- and post- buckling state. The growth laws developed in this manner are integrated numerically, in order to produce the delamination growth vs. number of cycles curves. An important characteristic in all the configurations tested is that the state of stress near the delamination tip is of mixed mode, I and II. In the cases where branching (intra-layer cracking) occurs, there is some self-similar crack growth from the initial delamination (simulated as a Teflon sheet inserted in the plate) followed by branching of the crack through the thickness of the plate. Observed behavior ranges from sudden and catastrophic failure induced by the intra-layer crack in the 90 deg. specimen to the alternate successive initiation of intra-layer cracks and secondary interlayer cracks (delaminations) in the cross-ply specimens. Experiments and statistics show that there is a critical branching angle above which crack growth is greatly accelerated. The particular details of the experimental study, including the difficulties and challenges, are discussed. 645 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 645–660. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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G. A. KARDOMATEAS, et al.

1. Introduction One of the most serious problems associated with laminated composites is the formation of an internal delaminated zone, i.e. two adjacent layers are partially debonded at their interface. This occurs most commonly as a consequence of low-velocity impact, but it may also be due to pre-existing manufacturing imperfections. There may be other reasons that cause delaminations, such as vibrations excited by the propulsion systems. In particular, under compression loading, the delaminated layer may buckle. This local instability does not necessarily lower the ultimate load, and usually the laminate is capable of carrying on in the post-buckling phase under higher loading. However, the buckling-induced stresses at the tip of the delamination may cause growth of the interlayer crack and eventual complete separation of the layer. Besides strength and stiffness, delaminations can influence other performance characteristics, such as the energy absorption capacity of composite beam systems [1]. The prospect of local delamination buckling, and especially the possibility of growth, limits the high potential of composites and needs to be well-understood so that composites can be safely used in compressive load bearing applications. This is one of the composite failure modes under compression loading, other frequently observed modes include micro-cracking in the matrix and kinking zones. Self-similar growth was observed in the compression fatigue unidirectional experiments, [2] and [3]. Another type of growth behavior observed was crack branching in static compressive, static Double Cantilever Beam (DCB) or compressive fatigue tests of cross-ply configurations, [4]-[7]. A crack growing along an interface is said to branch when it deviates from the interface into the contiguous ply. Typically it returns to grow along the original interface. In the case of compression loading and the ensuing buckling, a geometrically nonlinear model should be adopted to determine the post-buckling deflections of a thin debonded layer when a plate element is subjected to cyclic compression. The explicit presence of additional parameters with the dimension of the length (e.g. the thickness of the separated layer) makes this a nontraditional problem of fracture mechanics. Additional complications are introduced by the circumstance that the process of separated-layer growth may be accompanied by the phenomenon of elastic instability of the entire structural element. The initial post-buckling behavior of general delaminated composites (i.e. with no restrictive assumptions on the delamination dimensions) was studied in [8], by using a perturbation procedure based on an asymptotic expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. The analysis leads to closed form solutions for the load versus applied compressive displacement and the near tip resultant moments and forces. In these studies, the bimaterial interface crack solutions for the mode mixity and the energy release rate in terms of the resultant moments and forces, as derived in [9], were subsequently employed. The growth law proposed is a function of the spread in the energy release rate in the pre- and postbuckled states and the maximum value of the energy release rate in the cycle, both normalized with the mode-dependent interface fracture toughness. The exponent and the constant of the growth law are also taken to be mode-dependent. The delamination

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growth vs. number of cycles curves are produced by integrating numerically the growth laws developed in this manner. Regarding crack branching, data for cross-ply specimens of Glass/Epoxy and Graphite/Epoxy under static and fatigue loads will be presented in this paper. Cracks branched with angles that could be as big as 90 deg., and they crossed a distance that was of the order of 1-2 ply thickness, [6] and [7]. The mechanism of this failure is far from being well understood and predicted, but these experiments provide much needed information on the crack branching behavior. Statistical analysis based on the experiments will also be discussed. 2.

Self-Similar Growth

The experimental study was conducted first on unidirectional graphite/epoxy, procured from Amoco in the form of prepreg tape (commercial specification T50 6K ERL 19393) with about 33 per cent resin content and ply thickness of about 0.10 mm. Test panels were laid up by hand and cured in the autoclave according to the manufacturer's recommended cure cycle, i.e. at a temperature of 177C (350F) and a pressure of 414 KPa (60 psi). The material was put in a vacuum bag prior to the curing cycle. The delaminations were introduced by placing at the desired location, through the width, a Dupont Teflon film of 0.025 mm (0.001 in) thickness. Since the curing process affects the final dimensions of the specimens due to resin bleeding, the thicknesses of the specimens were measured after curing with a micrometer. The thickness measurements were taken at different points through the width and length to insure overall uniformity of the thickness and they were found to be satisfactory. The cured panels were carefully inspected for possible abnormalities, and were subsequently cut to specimen size using a silicon carbide blade. The equipment used consists of an Instron 8501 dynamic testing machine and a Questar remote video measurement system. The system allows tri-axial motion of the microscope and a simultaneous digital measurement of distances. The cyclic compression tests were conducted on these unidirectional graph ite/epoxy specimens at a frequency of 1 Hz, and the delamination growth was monitored using the Questar. The experiments were at a constant maximum strain and the characteristic properties of the specimens were found to be as follows: longitudinal modulus of elasticity GPa; critical energy release rates, exponent ratio (see growth law below) constant ratio, 10.01. These constants, which are typical of graphite/epoxy, were found from independent Mode I and Mode II tests. The specimens had a width w = 12.7 mm and a half-length between the grips of L = 50.8 mm. A mode-dependent cyclic growth law is suggested in [2] as follows:

This growth law is expressed in terms of the spread release rate g in the pre- and post-buckling state:

of the normalized energy

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is the mode mixity and


is the mode-dependent fracture toughness. Now the

exponent and the constant are also mode-dependent. The exponent's dependence is according to:

and a similar formula can be written for the constant Five delamination configurations were tested. In the following list, h/T is the ratio between h, the number of plies in the delaminated plate, and the total number of plies T. a) 30 plies, specimen thickness T = 2.87 mm, delamination of half-length 21.25 mm, between fourth and fifth ply, hence h/T = 4/30, and at a maximum compressive strain This specimen configuration is denoted as 4/30(A). b) 15 plies, specimen thickness T = 1.55 mm, delamination of half-length 25.86 mm, between fourth and fifth ply, hence h/T = 4/15, and at a maximum compressive strain (denoted as 4/15). 30 plies, specimen thickness T = 2.75 mm,delamination of half-length c) 26.15 mm, between sixth and seventh ply, hence h/T = 6/30 and at a maximum compressive strain (denoted as 6/30). 30 plies, specimen thickness T = 2.87 mm, delamination of half-length d) 18.75 mm, between fourth and fifth ply, hence h/T = 4/30, and at a maximum compressive strain This specimen configuration is denoted as 4/30(B). e) 36 plies, specimen thickness T = 3.00 mm, delamination ofhalf-length 33.02 mm, between tenth and eleventh ply, hence h/T = 10/36 and at a maximum compressive strain (denoted as 10/36). These five test configurations exhibited different mode mixities and energy release rate spreads; the 10/36 and 6/30 specimens exhibited the highest Mode I components. All specimens were subjected to less than 20 per cent of the critical energy release rate, with the 10/36 and 6/30 specimens under a larger spread Two data points, (l; N), from the 4/30(A) specimen were used to obtain the constants mI and CI. These data points are: (30.48 mm; 231,354 cycles) and (30.734 mm; 253,155 cycles). The values obtained are: and Based on these values, Fig. 1 shows the actual experimental data and the predicted cycles for all five specimens configurations on a semi-logarithmic plot. In all the experiments the delamination grew straight along the interface and the delamination length was measured with the help of the Questar. It should be first emphasized that the same exponent and constant in the growth law are used for all delaminations, although each delamination configuration is characterized by a different location through the thickness, different initial length, different applied peak strain and different mode mixity. Since measurements were performed at discrete points, the measured data are given by discrete data points, whereas the predictions are represented by the lines. It is


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seen that the data form five distinct groups, each for each specimen type. First, an immediate observation can be made that the closer the delamination to the surface (i.e. the lower h/T), the slower the growth. Second, the experimental data seem to correlate adequately with the predicted values. The proposed fatigue growth law is shown to correlate well with the experiments, in spite of the different geometry (location of the delamination through the thickness, initial delamination length, plate thickness) and applied loading in each of the five delamination configurations for each material tested (Fig.1).

Of the five specimen configurations, the mode mixity is fairly similar between the 4/30(A) and 4/15 specimens, but distinctly different between the 4/30(A) and the 6/30 or 10/36 or 4/30(B) specimens. In this context, in order to examine the importance of assuming a mode-dependent growth law, the predicted number of cycles for the corresponding mode-independent version of the growth law was calculated and compared to the experimental data. The following conclusions were made: if modeindependence was to be assumed in the growth law, which means that and then fitting the two constants, m and C, from the 4/30(A) data would fail to adequately predict the 6/30 data. Specifically, this would predict a number of cycles less than half those predicted by the present mode-dependent approach and the ones obtained from the experiments. Subsequently, the same study was conducted on unidirectional Glass/Epoxy specimens. The Glass/Epoxy used was the S2/SP250 made by 3M Co., and was supplied in the form of prepreg tape by NASA Langley. The average ply thickness of the S2/SP250 was 0.2413 mm and the mechanical properties of this material are as

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follows: moduli (in GPa) in-plane Poisson’s ratio resin content by weight 33±3 percent. The specimens were made by hand lay-up and curing in the autoclave according to the cure cycle provided by the manufacturer of the prepreg tape: a cure temperature of 120°C (250°F) and a pressure of 345 KPa (50 psi). The cured laminates which were 304.8 by 50.8 mm (12 by 2 inches), were carefully inspected for possible abnormalities and were subsequently cut into 152.4 by 12.7 mm (6 by 0.5 inch) pieces using a tungsten-carbide tipped tool. The delaminations were introduced by placing at the desired location, through the width, a Dupont Teflon film of 0.0254 mm (0.001 in) thickness. Since the curing process affects the final dimensions of the specimens due to resin bleeding, the thicknesses of the specimens were measured after curing with a micrometer. At this point it should be mentioned that the preparation of the Glass/Epoxy specimens required more effort and care than the Graphite/Epoxy ones because of the larger ply thickness and the different resin flow properties. The testing of the specimens was in compression at moderate load and displacement levels. Compressive testing is often done by using tabs or an interleaf between the specimen and the jaws of the testing machine in order to better transmit the applied load to the specimen through friction. However, neither tabs nor interleafs were found to be needed in these experiments. The tab-less specimens used in this experimental study proved to be very effective, besides being less expensive and requiring less time. Testing in compression fatigue was carried out in the Instron 8501 mentioned above. The experiments were done in displacement control. A sine wave of 5 Hz was applied. Regarding the toughness properties and fatigue growth parameters, these were as follows: critical energy release rates, exponent ratio, constant ratio, The specimens had a width w = 12.7 mm and a slightly varying half-length between the grips, L, between 50 and 60 mm, as reported below. The specimens consisted of 24 plies with the delamination implanted either between the fourth and fifth ply (4/24) or between the fifth and sixth ply (5/24). Different applied maximum compressive strains and different initial delamination lengths were used, which resulted in the following test configurations: a) 5/24 A: 24 plies, specimen thickness T = 4.98 mm, specimen half-length between grips L = 57.6 mm, initial delamination of half-length mm, between fifth and sixth ply, hence h/T = 5/24, and at a maximum compressive strain The delamination extended in fatigue up to half-length of l = 37.173 mm requiring 281,371 cycles. b) 5/24 B: 24 plies, specimen thickness T = 4.98 mm, specimen half-length between grips L = 54.6 mm, initial delamination of half-length mm, between fifth and sixth ply (h/T = 5/24), and at a maximum compressive strain Delamination extension up to l = 37.871 mm in 54,675 cycles. c) 5/24 C: 24 plies, specimen thickness T = 4.98 mm, specimen half-length between grips L = 58.4 mm, initial delamination of half-length mm, between fifth and sixth ply, hence h/T = 5/24, and at a maximum compressive strain Delamination extension up to l = 31.255 mm in 71,504 cycles.


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d) 4/24 A: 24 plies, specimen thickness T = 4.98 mm, specimen half-length

between grips L = 55.6 mm, initial delamination of half-length mm, between fourth and fifth sixth ply (h/T = 4/24), and at a maximum

compressive strain

In this configuration the delamination

extended in fatigue up to l = 27.775 mm in 252,260 cycles. 4/24 B: 24 plies, specimen thickness T = 5.10 mm, specimen half-length e) between grips L = 56.0 mm, initial delamination of half-length mm, between fourth and fifth sixth ply (h/T = 4/24), and at a maximum compressive strain In this configuration the delamination extended in fatigue up to l = 30.099 mm in 406,724 cycles. Two data points (l ; N) from the 5/24 A glass/epoxy specimen were used to obtain the constants mI and These data points are: (29.604 mm; 51,601 cycles) and (37.173 mm; 281,371 cycles). The values obtained are: and m/cycle. Results are shown in the following Fig. 2.

3. Non Self-Similar Growth Patterns When the composite construction is other than unidirectional, non self-similar growth patterns involving crack branching are observed. Specifically, experiments with growth of delaminations under fatigue or monotonic loading in cross-ply composite specimens revealed that intra-layer angle cracks often emanate from the delamination tip and propagate into the layer until they meet the next interface and then they may turn again

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into delaminations. The development and growth of the delamination/intra-layer crack system was examined under (i) axial compressive cyclic (fatigue) loading, (ii) off-axis, i.e. out-of-plane or transverse, monotonic, tensile loading of Glass/Epoxy and Graphite/Epoxy cross-ply laminates. Fig. 3 shows a sketch with the nomenclature that will be used to describe crack branching. Intra-layer and branching crack are two equivalent terms; kink angle is equivalent to branching angle.

In [5], symmetry was observed in crack branching in Glass/Epoxy specimens subject to fatigue loading, as shown in Fig. 4-5. This symmetry was not observed in [6] and [7], in Glass/Epoxy and Graphite/Epoxy specimens, subject to both static and fatigue loading. It should be pointed out that it is very difficult to calculate fracture toughness from the experiments, due to the presence of non self-similar crack growth. As it will be seen later on, several branchings occur, typically as soon as the crack grows from the delamination starter. For this reason, it is not possible, at this state, to present a mode mixity analysis nor to give an estimate of fracture toughness. However, experimental data and statistical analysis based on them will be presented and discussed. The tests performed in [6] and [7] were of different types: a) static Mixed Mode Bending (MMB) on: unidirectional 90 deg. S2/ SP250 Glass/Epoxy (24 plies, h/T=12/24); cross-ply S2/ SP250 Glass/Epoxy (lay-up h/T=15/30; 0], h/T=15/29). The average dimensions were 25.5 x 254x4.67 mm, with ply thickness = 0.205 mm. The average delamination starter length was mm. b) compressive fatigue tests on T7G145/F1914 Graphite/Epoxy specimens. The material was donated by Hexcel. The dimensions were 12.6 x 152.4x1.93 mm, the specimens had lay-up and h/T=4/15. The average delamination starter length, ao, was 50 mm and was located in the middle of the specimen. Crack growth could occur on either side of ao. The average ply thickness was 0.129 mm.


653

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c)


static DCB test on cross-ply TOHO UT500 Graphite/Epoxy specimens. The specimens had dimensions 25.4 x 137x2.55 mm and the delamination starter lengths were ao= 46 mm and 90 mm, and were located either in the mid-plane (h/T=10/20) or two plies away from it (h/T=12/20). The material has properties

The tests were performed using the Instron 8501 machine and the Questar microscope. Moreover, the Mixed Mode Bending (MMB) and the fatigue tests were video-recorded, and the images had been analyzed on a Silicon Graphics O2 computer. All the specimens were coated with white primer for visualization of the crack. This did not allow the identification of the plies during the test. The DCB specimens were polished after the tests and observed with an Olympus BH2-UMA microscope connected with a CCD camera, a monitor and a Polaroid-type printer. A fixture has been used for the MMB tests, as described in [10] and [11]. In the static DCB tests and in the fatigue tests, the piano hinges (for the DCB tests) and the specimens (for the fatigue tests) were directly clamped in the grips of the testing machine. The delamination starter consisted of a Teflon film inserted during the hand lay-up preparation of the specimens, as in the previous cases. In the MMB tests, a rate of 0.00254 mm/s (0.0001 in./s) was used. The Mode I and Mode II components of the applied load P, called and depend on two specific lengths in the MMB fixture. An expression for the ratio of the strain energies of Mode I and Mode II, i.e. under the simplification of negligible weight of the fixture, is also given in the previously mentioned references. By changing the fixture’s lengths, it is possible to have predominance of Mode I or Mode II. The work described in [10] and [11] is applied to unidirectional specimens as self-similarity of the crack growth is sought. An estimate of the mode mixity for multi-directional specimens requires a numerical solution and is not available at this point. In the MMB tests, crack branching was observed in all the specimens. In the unidirectional 90 deg. specimens, the crack branched away from the interface between two contiguous 90 deg. plies, and broke the specimen (Fig. 6). The branching angles, measured with respect to the original delamination direction, varied between 70 deg. and 110 deg. In the cross-ply specimens, the crack branched away from the interface between a 0 and 90 deg. ply, went into the 90 deg. and then turned parallel to the original delamination direction. The phenomenon was confined into an area of the order of 1-2 ply thickness, as the crack never crossed a 0 deg. ply. Several branching were observed, and the angles varied between 29 and 55 deg. (Fig. 7). The fatigue tests were done in displacement control, with a 0.0381 mm (0.0015 in.) sine wave and a frequency of 5 Hz. The twelve specimens tested were gripped and brought to buckling in static condition. Then, cycling was started. The branching angles varied between 12 deg. and 90 deg. (Fig. 8).


655

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Statistical analysis was needed to understand the effect of branching on life and crack propagation, as the scatter did not allow to draw conclusions. The statistical technique used belongs to the field of Exploratory Data Analysis, and consists of the socalled smoothing by running median of 3 repeated, introduced in [12], and applied in [6]. Medians were chosen for two reasons: they are more robust than mere averages, and the Exploratory Data Analysis technique is based on medians. Each specimen was associated to: a) a median branching angle (median of the branching angles observed in that specimen); b) a median crack growth, measured as projection of the crack along the direction of the starter. As mentioned earlier, the delamination starter was located in the middle of the specimen, hence crack growth could occur on either side of the starter, on both sides, or on neither. Consequently, a median of these growths was calculated. Zero crack growths and zero branching angles were considered in the calculations when present. The crack growth data at 100,000 cycles (or the available closest number from the experiments) were chosen. The number of cycles for the twelve specimens ranged from 25,267 (corresponding to catastrophic crack growth) to 250,000. 100,000 cycles represented a reasonable number to compare different specimens, as branching occurred before 100,000 cycles, and there was enough crack growth data for the statistical analysis. Table 1 reports the original fatigue data for the twelve specimens. “Total cycles” is the total number of cycles for the specimen, “Kink angle (side 1)” and “Kink angle (side 2)” are the branching (or kink) angles observed on either side of the delamination. “Cycles kink” is the number of cycles at which a specific kink occurred. Table 2 shows the data used for the statistical analysis, ordered in ascending crack growths.


657

The results of the smoothing by running median of 3 repeated are the following: there is a critical branching angle above which crack growth is greatly accelerated [6]. If the median of all the branching angles in a specimen is above such critical branching angle, then branching will considerably affect the life of the specimen and accelerate its

658


failure. If the median of the branching angles in a specimen is below such value, branching is not an important failure mode. In the twelve specimens analyzed, this critical angle is 33 deg. This behavior may be explained with the presence of contact inside the branching crack: it is hypothesized that for angles below the critical branching angle, contact in the branching crack slows down crack propagation; for angles above the critical angle, the branching crack opens and therefore crack propagation is accelerated. Verification of this hypothesis on the amount of contact zone would require detailed investigation of the branching crack, which was not possible with the current capabilities and resolutions. Moreover, a similar study was performed at the onset of crack branching, in the same set of twelve specimens. Crack growth was analyzed around 50,000 cycles, half of the life considered earlier. It is found that the critical branching angle is the same, 33 deg. Hence, this phenomenon seems not to be affected by the specimen’s life [13]. Finally, eight DCB static specimens were tested in displacement control with a rate of mm/s (0.02 in./min). Fig. 9 shows one of the specimens, polished and observed with an Olympus BH2-UMA microscope after the test. Several branchings occurred per specimen, with a considerable scatter of data: from 11 deg. to 78 deg. The DCB specimens were designed according to a 2-level 2-parameters, factorial design of experiments based on a 8x8 Hadamard matrix. The method is described in [14], and was utilized in [7]. It was chosen in order to minimize the number of specimens to prepare and test, while obtaining statistically valid results. The effects of two delamination lengths (46 and 90 mm) and two positions of delamination in the specimen (h/T=10/20 and h/T= 12/20) were studied, and crack growth and branching angles were monitored in the experiments.

The results were obtained with 90% confidence and an estimate of the variance with four degrees of freedom. They can be applied to a population (theoretically, an infinite number) of specimens with the same characteristics. Moreover, the same smoothing technique utilized for the fatigue tests was applied to the DCB data: the goal was to identify the trend of median crack speeds (total crack growth divided by total


time of the experiment in a specimen, calculated at the rate of

659

and

median branching angle. The outcome of both design of experiments analysis and smoothing is the following: there is a critical branching angle that affects the behavior of the population of DCB specimens. Such angle is between 39 and 40 deg. In addition, when the median branching angle (median of all the branching angles in a specimen) is above such critical angle, crack growth is greatly accelerated. This behavior is similar to what was found in the analysis of the fatigue data, with the following differences: in the T7G145/F1914 Graphite/Epoxy specimens subject to compressive fatigue tests, the critical angle was 33 deg., while in the TOHO UT500 Graphite/Epoxy subject to static DCB tests, the critical angle was 39-40 deg. Geometry and loading conditions seems not to significantly affect the critical branching angle, if one considers the amount of scatter present in such experiments. A future goal is to establish whether this behavior could be repeated in other specimens’ configurations and other materials. Scatter of data seems to show that the crack growth is greatly influenced by the local conditions around the crack tip and heterogeneities like fiber clusters, inclusions and so on. Also, a study is required to observe the amount of roughness-induced closure in the branching crack, and correlate it to the branching angle. This can be done numerically with an accurate micro-mechanical model and a statistical model to describe the heterogeneities in the composite. 4.

Conclusions

This paper presented a review of experimental data and modeling in delaminated unidirectional and cross-ply composites. In unidirectional Glass/Epoxy and Graphite/Epoxy composites, a growth law was proposed, which is based on the spread in the energy release rate in the pre- and post-buckled states and the maximum value of the energy release rate in the cycle. The exponent and the constant of the growth law are mode-dependent. The crack growth vs. number of cycles curves given in the paper show a good correlation between analytical prediction and actual crack growth. Delaminated cross-ply specimens exhibited non self-similar crack growth, which did not allow a measurement of fracture toughness and mode mixity. However, experimental data were presented in the paper for Glass/Epoxy and Graphite/Epoxy specimens under static and fatigue loading. Scatter greatly affected the results. Two statistical techniques were utilized: an Exploratory Data Analysis technique called smoothing by running median of 3 repeated; a 2-level 2-parameters, factorial design of experiment based on a 8x8 Hadamard matrix. A critical branching angle is obtained as a result of the analyses, above which crack growth is greatly accelerated. Hence, branching could considerably affect the life of a multi-directional composite structure.

5.

Acknowledgements

The financial support of the Office of Naval Research, Ship Structures S&T Division, Grants N00014-90-J-1995 and N00014-00-10323, and the interest and encouragement of the Grant Monitor, Dr. Y.D.S. Rajapakse, is gratefully acknowledged.

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6. 1. 2.

3. 4. 5.

6.

7. 8.


References Kardomateas, G.A. and Schmueser, D.W. (1988) Buckling and Post-buckling of Delaminated Composites Under Compressive Loads Including Transverse Shear Effects, AIAA Journal 26, 337-343. Kardomateas G.A., Pelegri A.A. and Malik B. (1995) Growth of Internal Delaminations Under Cyclic Compression in Composite Plates, Journal of the Mechanics and Physics of Solids 43, 847-868. Kardomateas G.A. and B. Malik B. (1997) Fatigue Delamination Growth Under Cyclic Compression in Glass/Epoxy Composite Beam/Plates, Polymer Composites 18, 169-178. Chai H. (1984) The characterization of Mode I delamination failure in non-woven, multidirectional composites, Composites 15, 277-290. Pelegri A.A., Kardomateas G.A. and Malik B.U. (1997) The Fatigue Growth of Internal Delaminations Under Compression in Cross Ply Composite Plates, Composite Materials: Fatigue and Fracture (Sixth Volume), ASTM STP 1285, E.A. Armanios Ed., American Society for Testing and Materials, 143-161. La Saponara, V. and Kardomateas, G. A. (2000) Statistical considerations in the analysis of data from fatigue tests on delaminated cross-ply graphite/epoxy composites, ASME Journal of Engineering Materials and Technology, 122, 409-414. La Saponara, V. and Kardomateas, G.A. (2001) Crack Branching in Layered Composites: An Experimental Study, Composite Structures 53, 333-344. Kardomateas G.A. (1993) The Initial Post-buckling and Growth Behavior of Internal

Delaminations in Composite Plates, Journal of Applied Mechanics (ASME) 60, 903-910. 9. 10. 11. 12. 13.

14.

Suo, Z. and Hutchinson, J.W. (1990) Interface crack between two elastic layers, International Journal of Fracture 43, 1-18. Reeder, J. R. and Crews Jr., J. H. (1990) Mixed Mode Bending Method for Delamination Testing, AIAA Journal 28, 1270-1276. Reeder, J. R. (1992) An evaluation of mixed-mode delamination failure criteria, NASA Technical Memorandum 104210. Tukey, J. W. (1977) Exploratory Data Analysis, Addison-Wesley. La Saponara, V. (2001) Crack Branching in Cross-Ply Composites, Ph.D. Thesis in Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150. Diamond, W. J. (1989) Practical Experiment Designs for Engineers and Scientists, John Wiley & Sons.

DIRECT IDENTIFICATION OF ELASTIC PROPERTIES OF COMPOSITE STRUCTURES – A WAVE-CONTROLLED IMPACT APPROACH

SERGE ABRATE Department of Technology Southern Illinois University Carbondale, IL62901-6603

Abstract

The determination of the four elastic constants for an orthotropic layer of composite materials usually requires that specimens be prepared and several tests be performed. This can be costly and time consuming and a number of investigators have sought to infer those properties from the free vibration characteristics of a single specimen. While this approach is successful tests are still performed on a specimen that must be assumed to be representative of the properties of the actual structure. In this paper, we investigate the possibility of generating impact that produce very localized bending deformations in the actual structure and monitoring the sound generated during the contact phase of the event. This approach could be used for in-situ determination of material properties, which is important with composite materials since the properties of a specimen can differ from those of an actual structure. 1. Introduction

Fiber reinforced composite materials are characterized by four independent elastic constants. These constants can be determined from static tests performed on specially prepared specimens. This approach has two drawbacks: (1) is requires that several tests be performed; (2) the specimens are usually made separately and have different layups than the actual structure. There is interest in developing a procedure to determine the material properties from tests on a single specimen that has the same layup as the actual structure, or, if possible, directly on the actual structure. In recent years several investigators used experimentally determined natural frequencies of composite plates to estimate the elastic properties of the material. In many cases the specimen is excited through an impact generated by an instrumented hammer or by a spherical projectile. The response of the plate can be sensed by a surface mounted accelerometer or by non-contact means: fiber optic sensor, holographic interferometry, optical interferometry [1], microphone [2]. Impacts can be classified into several categories [3]. With high-velocity impacts only a very small region near the projectile is under stress while the projectile is in contact with the target and bending motion of the target has not been established. With wave-controlled impacts, bending deformation takes place in a zone surrounding the 661 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 661–670. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

S. ABRATE

662

point of impact but does not reach the boundaries during the contact phase. For boundary-controlled impacts, waves have reached the boundary and have been reflected back and forth several times so that the entire plate participates in the response. Both wave-controlled and boundary-controlled impacts are called low velocity impacts. The response of a structure to an impact by a projectile generally consists of two phases: (1) the contact phase during which the projectile is applying a load on the structure; and (2) the subsequent free vibration phase. Previous investigators measure the response of a specimen during the free vibration phase in order to determine its natural frequencies and mode shapes. In this paper we investigate the possibility of using information from the contact phase of the impact in order to determine the elastic properties of the composite. Tests are designed to induce only local deformations to that they can be performed on actual structures regardless of the complexity of their geometry or their support conditions. 2.

Impact Dynamics

Several approaches are available for studying the dynamics of foreign object impacts with varying degrees of approximation to the dynamics of the target [3,4]. Three such approaches will be considered here. In the first one the deformation of the target is neglected, the second assumes the plate to be of infinite extend, and the third model accounts for the full dynamics of finite-sized plates.

2.1. IMPACT ON HALF-SPACE If the overall deformations of the plate are negligible, the impact causes only local deformations in the contact zone. The relationship between the contact force F and the indentation is governed by Hertz's contact law [3, 4]

where the contact stiffness k is given by

where R is the radius of the impactor, E is the modulus of elasticity and is Poisson's ratio. The subscripts p and s refer to the plate and the spherical indentor respectively. The equation of motion and the initial condition of the projectile are

The maximum contact force

and the contact duration

are given by

Eqs. (5,6) provide a quick means for estimating the magnitude of the maximum contact force and the contact duration.

ELASTIC PROPERTIES OF COMPOSITE STRUCTURES

663

2.2. IMPACT ON INFINITE PLATE

Zener [5] showed that the dynamics for foreign object impacts on infinite isotropic plates is governed by the single differential equation

which is written in terms of the non-dimensional indentation time that are defined as

and the non-dimensional

with the time constant

Eq. (7) depends on the single non-dimensional parameter

where D is the bending rigidity and m is the mass per unit area of the plate. The initial conditions are

Note that Eq. (3) is a special case of Eq. (9) for Solving Eqs. (7, 12) for several values of shows the effect of the different factors on the qualitative response of the plate. As shown in Fig. 1, when the maximum non-dimensional force is the largest and the loading and unloading phases are symmetrical (Fig. 1). When the maximum contact force decreases, the contact duration becomes longer, and the unloading phase is longer than the loading phase. Olsson [6] extended this model to orthotropic plates and showed that in that case the equivalent rigidity of the plate should be

where with isotropic or quasi-isotropic laminates and can vary between 0 and 3 for general layups [7]. The contact force and the deflection of the structure is obtained from the solution of Eq. (7) using

664

S. ABRATE

During impact, the wave front is assumed to be elliptical and the analysis is based on expanding the transverse displacements in terms eigenfunctions of a rectangular simply supported plate extending from –a/2 to a/2 and –b/2 to b/2 impacted in the center

Both i and j must be odd numbers and i=j. The axes of the ellipse are given by

ELASTIC PROPERTIES OF COMPOSITE STRUCTURES

665

More recently, Olsson [8] proposed a mass criterion to discriminate between wavecontrolled and boundary-controlled impacts. Elliptical bending waves propagate away from the impact point and eventually reach the boundary of the plate as illustrated in Fig.2. The shaded area in that figure represents the portion of the plate in which bending waves can propagate freely response before reaching the boundary. If is the mass of that portion of the plate, wave-controlled impacts occur when

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